# Analysis of the Decoder Width for Parametric Partial Differential Equations

Zhanhong Ye  Beijing International Center for Mathematical Research, Peking
University, Beijing, China<EMAIL_ADDRESS>,  Hongsheng Liu  Central
Software Institute, Huawei Technologies Co. Ltd, Hangzhou, China
<EMAIL_ADDRESS>,  Zidong Wang  Central Software Institute, Huawei
Technologies Co. Ltd, Hangzhou, China<EMAIL_ADDRESS>and  Bin Dong  Beijing
International Center for Mathematical Research, Peking University, Beijing,
China  Center for Machine Learning Research, Peking University, Beijing, China
<EMAIL_ADDRESS>
###### Abstract.

Recently, Meta-Auto-Decoder (MAD) was proposed as a novel reduced order model
(ROM) for solving parametric partial differential equations (PDEs), and the
best possible performance of this method can be quantified by the decoder
width. This paper aims to provide a theoretical analysis related to the
decoder width. The solution sets of several parametric PDEs are examined, and
the upper bounds of the corresponding decoder widths are estimated. In
addition to the elliptic and the parabolic equations on a fixed domain, we
investigate the advection equations that present challenges for classical
linear ROMs, as well as the elliptic equations with the computational domain
shape as a variable PDE parameter. The resulting fast decay rates of the
decoder widths indicate the promising potential of MAD in addressing these
problems.

###### Key words and phrases:

Parametric partial differential equations, reduced order modeling, $n$-widths,
nonlinear approximation, solution set, variable-domain PDEs

###### 2010 Mathematics Subject Classification:

41A46, 35A17

Bin Dong is the corresponding author.

##  1\. Introduction

Many important phenomena in science and engineering can be modeled by the so-
called parametric partial differential equations (PDEs), which can be
formulated as

(1.1) $\mathcal{L}_{\widetilde{\boldsymbol{x}}}^{\gamma_{1}}u=0,\
{\widetilde{\boldsymbol{x}}}\in\Omega\subset\mathbb{R}^{d},\qquad\mathcal{B}_{\widetilde{\boldsymbol{x}}}^{\gamma_{2}}u=0,\
{\widetilde{\boldsymbol{x}}}\in\partial\Omega.$

Here, $\mathcal{L}^{\gamma_{1}}$ and $\mathcal{B}^{\gamma_{2}}$ are
differential operators parametrized by $\gamma_{1}$ and $\gamma_{2}$,
respectively, and ${\widetilde{\boldsymbol{x}}}$ denotes the independent
variable in spatiotemporal-dependent PDEs. Let $\mathcal{A}$ be the space of
parameters. For a given set of parameters of the PDEs
$\eta=(\gamma_{1},\gamma_{2},\Omega)\in\mathcal{A}$, we assume (1.1) to admit
a unique solution $u^{\eta}\in\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})$, where
$\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})$ is a Banach space of functions
defined on $\Omega$. In the simpler case when the computational domain
$\Omega$ is fixed, we shall denote for simplicity
$\mathcal{U}=\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})$. Taking $S:\eta\mapsto
u^{\eta}$ to be the solution mapping, the set of solutions is then given by
$S(\mathcal{A})=\\{u^{\eta}\mid\eta\in\mathcal{A}\\}$.

Solving the parametric PDEs (1.1) quickly for a series of $\eta$’s is crucial
in many application domains, such as inverse problems [15], inverse design
[27], optimal control [6] and uncertainty quantification [3, 23]. In order to
obtain an efficient solver for the parametric PDEs, the reduced order modeling
technique aims to construct a suitable low-dimensional trial manifold
containing (at least approximately) the solution set $S(\mathcal{A})$. By
restricting the original system (1.1) or its discretized version to this trial
manifold, a reduced order model (ROM) is produced, whose low-dimensional
nature would enable faster solving. However, taking into account the intrinsic
complexity of the solution set, the solution provided by the ROM may not
achieve the desired accuracy, especially when the dimensionality of the trial
manifold is insufficient. The concept of _widths_ is then introduced to
quantify the best possible accuracy using an $n$-dimensional trial manifold.
Different types of trial manifolds are considered in different ROMs, and the
definitions of widths would vary accordingly.

A widespread family of ROMs, such as reduced basis methods (RBMs) [32],
utilize only those trial manifolds that are linear subspaces, and represent
the approximate solution as a linear combination of basis functions
$\hat{u}=\sum_{i=1}^{n}\lambda_{i}u_{i}.$ The Kolmogorov width (2.1) serves as
a theoretical tool for analyzing such ROMs. As the solution set of advection-
dominated parametric PDEs typically exhibits a slow decaying Kolmogorov width
[30, 19], these linear ROMs would struggle to balance between efficiency and
accuracy when solving these equations, and this has become a long-standing
challenge in the field of parametric PDEs.

Making use of the nonlinear representation of the neural networks (NNs), many
novel ROMs are proposed [26, 17, 18, 21, 5, 35]. Specifically, the recently
proposed ROM named Meta-Auto-Decoder (MAD) [22, 35] seeks to construct a
Lipschitz continuous mapping $D:Z\to\mathcal{U}$, where $Z=\mathbb{R}^{n}$
represents the latent space. Denoting
$Z_{B}=\\{\boldsymbol{z}\mid\|\boldsymbol{z}\|_{2}\leq 1\\}$ to be the closed
unit ball of $Z$, MAD takes $D(Z_{B})$ as the trial manifold, and solves (1.1)
by searching on this trial manifold (MAD-L) or possibly in a neighborhood of
it (MAD-LM). The corresponding width associated with this novel ROM is termed
the _decoder width_ , which will be further introduced in Section 3. In
contrast to the Kolmogorov width, the decoder width exhibits a much faster
decay for certain advection equations (Section 4.4), indicating the
superiority of MAD over classical linear ROMs.

Moreover, MAD is mesh-free since it represents the decoder mapping
$D:Z\to\mathcal{U}$ as $z\mapsto u_{\theta}(\cdot,z)$, where $u_{\theta}$ can
be modeled (in the simplest case) as a neural network that takes the
concatenation of $\widetilde{\boldsymbol{x}}$ and $\boldsymbol{z}$ as input.
On the contrary, typical ROMs are based on classical numerical methods such as
finite difference method (FDM) and finite element method (FEM), in which the
computational domain $\Omega$ is often discretized, and we cannot evaluate the
solution $u^{\eta}$ to (1.1) at arbitrary spatiotemporal points
$\widetilde{\boldsymbol{x}}$ without interpolation. In the case of parametric
PDEs on a variable domain, i.e. those including $\Omega$ as part of the
variable parameter $\eta$, further convenience would be brought by the mesh-
free nature. Typical discretization-based ROMs need to assume the existence of
a _reference domain_ $\Omega^{\text{ref}}$, and associate to each possible
$\Omega$ with an explicit domain transformation
$T_{\Omega}:\Omega^{\text{ref}}\to\Omega$, which could be hard to design by
hand for complex-shaped domains, and has the additional prerequisite that no
topology changes in $\Omega$ should occur. These are no longer necessary for
the MAD method, as any $\widetilde{\boldsymbol{x}}\in\mathbb{R}^{d}$ can be
taken as the input of a neural network. The corresponding notion of decoder
width can be introduced by using a _master domain_ $\Omega^{\text{m}}$, as
will be further explained in Section 3.2. To the best of our knowledge, this
is the first notion of width that can deal with domain deformation without
using a reference domain.

The current manuscript primarily focuses on analyzing the decoder width, which
quantifies the best possible performance of the MAD method. In comparison to
the stable manifold width (2.3), the decoder width does not exceed its value
(Proposition 3.2), and the difference is sometimes real (Example 3.4). We also
estimate the decoder widths of the solution sets of several common parametric
PDEs. For the elliptic and the parabolic equations, including those on
variable domains, the decoder widths would become zero for finite values of
$n$ if $\mathcal{A}$ is finite-dimensional (Example 4.3, 4.6, 4.12 and 4.14).
Even when $\mathcal{A}$ is infinite-dimensional, the decoder width can still
exhibit a fast decay rate, depending on the internal complexity of
$\mathcal{A}$ (Example 4.7 and 4.15). In the case of the advection equations,
an exponential decay rate is guaranteed when using the $L^{2}$-norm (Example
4.23), and the decoder width may become zero if the $L^{1}$-norm is employed
instead (Example 4.21). We also compare the decay rates of the decoder width
with that of the Kolmogorov width and the manifold width in Table 1. These
results demonstrate the promising potential of the MAD method in solving these
parametric PDEs, and motivate the choice of the latent dimension $n$ in its
implementation.

The rest of the paper is orgnized as follows. In Section 2 we provide a brief
review of the related work, including the definitions of existing widths. In
Section 3, we present the definition of the decoder width along with its
fundamental properties, covering both the fixed domain case and the variable
domain case. In Section 4, we examine the solution sets of several parametric
PDEs, and estimate the upper bounds of the corresponding decoder widths. We
shall only state the conclusions in Section 3 and 4, and defer all the proofs
till Section 5. Our conclusions as well as further discussions will be given
in Section 6.

Table 1. A comparison of the decay rate of different widths, taking
$\mathcal{K}$ as the solution sets of several parametric PDEs (on a fixed
domain $\Omega$). We assume $\mathcal{A}$ has finite-dimensionality
$\mathsf{d}^{\mathcal{A}}=\dim\mathcal{A}<\infty$. A “–” means the result is
currently unknown to the authors.

PDE type | Kolmogorov width $d_{n}$ | manifold width $d_{n}^{\text{Mani}}$ | decoder width $d_{n,l}^{\text{Deco}}$
---|---|---|---
elliptic |  $O(n\exp(-cn^{1/\mathsf{d}^{\mathcal{A}}}))$ [34] | 0 as $n\geq\mathsf{d}^{\mathcal{A}}$ [16] | 0 as $n\geq\mathsf{d}^{\mathcal{A}}$ (Eg.4.3)
parabolic | – | – | 0 as $n\geq\mathsf{d}^{\mathcal{A}}$ (Eg.4.6)
advection ($L^{1}$) |  $O(1/n)$ [31] | – | 0 as $n\geq\mathsf{d}^{\mathcal{A}}$ (Eg.4.21)
advection ($L^{2}$) |  $O(1/\sqrt{n})$ [30] |  $O(e^{-cn})$* |  $O(e^{-cn})$ (Eg.4.23)

  * *

The proof is the same as Example 4.23 if we apply 2.5 instead of Corollary
3.3.

##  2\. Related Work

###  2.1. Widths and Parametric PDEs

Let $\mathcal{U}$ be a Banach space, and $\mathcal{K}\subset\mathcal{U}$ be a
compact subset. If we aim at approximating $\mathcal{K}$ using a linear trial
manifold as in classical ROMs, then the best possible accuracy can be measured
by the well-known Kolmogorov width [25]. To be more specific, given a linear
subspace $U_{n}\subset\mathcal{U}$ of dimension $n$, its performance in
approximating the elements of $\mathcal{K}$ is evaluated by the worst case
error

$\sup_{u\in\mathcal{K}}d_{\mathcal{U}}(u,U_{n})=\sup_{u\in\mathcal{K}}\inf_{v\in
U_{n}}\|u-v\|_{\mathcal{U}}.$

The _Kolmogorov width_ is then defined by taking infimum over all such linear
subspaces

(2.1)
$d_{n}(\mathcal{K})_{\mathcal{U}}=\inf_{U_{n}}\sup_{u\in\mathcal{K}}\inf_{v\in
U_{n}}\|u-v\|_{\mathcal{U}}.$

We may write $d_{n}(\mathcal{K})=d_{n}(\mathcal{K})_{\mathcal{U}}$ for
simplicity when $\mathcal{U}$ is clear from the context, and this applies to
all notions of widths to be introduced below. In the context of parametric
PDEs, the case of particular interest is taking $\mathcal{K}=S(\mathcal{A})$
to be the solution set. For the elliptic equations, the corresponding
Kolmogorov width of the solution set exhibits rapid decay [8, 34], indicating
that conventional linear ROMs can achieve a satisfactory accuracy with a
reasonable latent dimension $n$. However, the case is quite different for
advection-dominant problems [30, 19], posing significant performance
challenges for linear ROMs.

In light of the limitations of the linear ansatz, nonlinear alternatives to
the Kolmogorov width are considered, and the manifold width proposed in [12]
is a noticeable candidate. Let $Z=\mathbb{R}^{n}$ be a fixed $n$-dimensional
latent space, and $E:\mathcal{K}\to Z$, $D:Z\to\mathcal{U}$ be two continuous
mappings, we consider the worst case reconstruction error given by

$\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{\mathcal{U}}=\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{\mathcal{U}},$

and the _manifold width_ is defined by taking infimum over all possible
mapping pairs $(E,D)$:

(2.2)
$d_{n}^{\mathrm{Mani}}(\mathcal{K})=\inf_{E,D}\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{\mathcal{U}}.$

For the solution set of certain elliptic equations, this width is known to
become zero even for finite $n$ [16]. If we further require these two mappings
to be $l$-Lipschitz continuous, a variant named the _stable manifold width_
[7] is then obtained as111The original definition given in [7] actually takes
infimum over all possible norms $\|\cdot\|_{Z}$ on $Z$ along with $E,D$. Here
we choose a fixed norm $\|\cdot\|_{Z}=\|\cdot\|_{2}$ as in the decoder width
(3.2), which won’t make a difference for our purpose, and (2.5) is still
valid.

(2.3) $d_{n,l}^{\mathrm{SMani}}(\mathcal{K})=\inf_{E,D\
l\text{-Lip}}\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{\mathcal{U}}.$

We note that some recently proposed ROMs [26, 17, 18, 21, 5] find the trial
manifold by training a convolutional autoencoder, and these notions of widths
can serve as a quantified criterion for their best possible performance.

Another notion termed the entropy number, though not named as a “width”, is
also noteworthy due to its close relationship to these concepts. Given an
integer $n>0$, the _entropy number_ is defined to be the infimum of all
$\epsilon>0$ such that $2^{n}$ balls of radius $\epsilon$ cover $\mathcal{K}$
[4], or equivalently

(2.4) $\epsilon_{n}(\mathcal{K})=\inf_{V_{n}}\sup_{u\in\mathcal{K}}\inf_{v\in
V_{n}}\|u-v\|_{\mathcal{U}},$

where the first infimum is taken over all subsets $V_{n}\subset\mathcal{U}$
containing no more than $2^{n}$ points. If the additional restriction
$V_{n}\subset\mathcal{K}$ is imposed, we obtain the so-called _inner entropy
number_ $\tilde{\epsilon}_{n}(\mathcal{K})$, and
$\epsilon_{n}(\mathcal{K})\leq\tilde{\epsilon}_{n}(\mathcal{K})\leq
2\epsilon_{n}(\mathcal{K})$ holds. When $\mathcal{U}$ is a Hilbert space, [7,
Theorem 4.1] asserts the inequality

(2.5) $d^{\text{SMani}}_{26n,2}(\mathcal{K})\leq 3\epsilon_{n}(\mathcal{K}).$

More variants of widths exist in the realm of metric geometry, including the
Alexandroff width [1] and the Urysohn width [20]. The interested readers may
refer to [11] for a further discussion on this topic.

###  2.2. PDEs on a Variable Domain

Some parametric PDEs may take the shape of the computational domain as the
variable parameter, i.e. $\eta=\Omega$, which have attracted attention due to
both theoretical and engineering interests. Solving shape optimization
problems typically requires to apply sensitivity analysis to the solutions to
(1.1). To be more specific, we consider a class of domains $\Omega_{\epsilon}$
parametrized by a small parameter $|\epsilon|\ll 1$, and the derivative of
$u^{\Omega_{\epsilon}}$ (after transformed to a reference domain) with respect
to $\epsilon$ is analyzed [24, 33, 2]. Another bunch of work investigates
scenarios where the domain is perturbed singularly. Given a series of domains
$\Omega_{n}$ approaching $\Omega$, the convergence of the corresponding
solutions $u^{\Omega_{n}}$ is studied [9, 10, 13], and the solutions on
different domains are compared via direct restrictions to subdomains. However,
these works primarily focus on the limiting behavior of a varying domain. In
contrast, the concept of widths is related to the whole solution set
$S(\mathcal{A})=S(\\{\Omega\\})$, which takes into account the entire
distribution of all possible domains. The main challenge in analyzing widths
in this context arises from the fact that the solutions involved are defined
on different domains and, consequently, belong to different function spaces.
Transforming all these solutions to a fixed reference domain appears to be one
resolution to this issue [32], but lacks flexibility as we have discussed in
Section 1. To the best of our knowledge, we are the first to introduce a
version of width for variable-domain PDEs without involving a reference domain
(Section 3.2), and provide a theoretical estimation of it for several special
cases (Section 4.3).

##  3\. Definition and Basic Properties of the Decoder Width

For the rest of this paper, we denote $\|\cdot\|_{L^{p}}$ to be the
$L^{p}$-norm of a function, and $\|\cdot\|_{H_{0}^{1}}$ similarly. The
$\ell^{p}$-norm of a vector or matrix is denoted as $\|\cdot\|_{p}$, making
$\ell^{p}(\mathbb{R}^{d})=(\mathbb{R}^{d},\|\cdot\|_{p})$ to be a finite-
dimensional Banach space, as well as $\ell^{p}(\mathbb{R}^{d\times
d})=(\mathbb{R}^{d\times d},\|\cdot\|_{p})$ endowed with the matrix norm. If
$\mathcal{V}$ is a Banach space,

(3.1) $\bar{B}_{R}(\mathcal{V})=\\{v\in\mathcal{V}\mid\|v\|_{\mathcal{V}}\leq
R\\}$

represents the closed ball of radius $R>0$ in $\mathcal{V}$.

###  3.1. The Decoder Width on a Fixed Domain

We recall the notion of the decoder width initially introduced in the context
of the MAD method [35]. Given a fixed domain $\Omega$,
$\mathcal{U}=\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})$ is a fixed Banach space,
and we are interested in the approximation of a compact subset
$\mathcal{K}\subset\mathcal{U}$. MAD constructs the trial manifold by using a
decoder mapping $D:Z\to\mathcal{U}$ similar to the autoencoders, but does not
involve the encoder mapping $E:\mathcal{K}\to Z$. The corresponding _decoder
width_ would thereby be different from the stable manifold width (2.3), and is
defined as

(3.2) $\begin{split}d_{n,l}^{\mathrm{Deco}}(\mathcal{K})&=\inf_{D\
l\text{-Lip}}\sup_{u\in\mathcal{K}}d_{\mathcal{U}}(u,\\{D(\boldsymbol{z})\mid\|\boldsymbol{z}\|_{2}\leq
1\\})\\\ &=\inf_{D\ l\text{-Lip}}\sup_{u\in\mathcal{K}}\inf_{\boldsymbol{z}\in
Z_{B}}\|u-D(\boldsymbol{z})\|_{\mathcal{U}},\end{split}$

where the first infimum is taken over all mappings $D:Z\to\mathcal{U}$ that
are $l$-Lipschitz continuous. The constraint $\|\boldsymbol{z}\|_{2}\leq 1$
and the Lipschitz continuity condition are required to avoid highly irregular
mappings resembling the space-filling curves, which will be further discussed
in Section 6.

For the case of a fixed domain, the decoder width is very similar to the
Lipschitz width proposed in [31]. In the definition of the Lipschitz width,
the latent space $Z=\mathbb{R}^{n}$ is assumed to be a general finite-
dimensional Banach space, and an additional infimum over all possible norms
$\|\cdot\|_{Z}$ on $Z$ is taken before the first infimum in (3.2), that is,

$d_{n,l}^{\mathrm{Lip}}(\mathcal{K})=\inf_{\|\cdot\|_{Z}}\inf_{D\
l\text{-Lip}}\sup_{u\in\mathcal{K}}\inf_{\boldsymbol{z}\in
Z_{B}}\|u-D(\boldsymbol{z})\|_{\mathcal{U}}.$

However, the decoder width sticks to the $\ell^{2}$-norm on $Z$. As the
$\ell^{2}$-norm is typically more convenient for numerical optimization, we
take the decoder width as the more suitable notion of width related to the MAD
method. Indeed, it is worth noting that the Lipschitz width is not greater
than the decoder width, and any upper bound derived for the decoder width
automatically serves as an upper bound for the Lipschitz width as well.

In order to give an analysis of the decoder width, we find it useful to
introduce the concept of the _restricted decoder width_
$d_{n,l}^{\text{rDeco}}(\mathcal{K})$, which shares the same expression as
(3.2) but with the first infimum taken over all $l$-Lipschitz mappings
$D:Z\to\mathcal{K}$. In other words, it imposes the additional requirement
that the image of $D$ should lie inside $\mathcal{K}$, and we have the obvious
inequality $d_{n,l}^{\text{Deco}}(\mathcal{K})\leq
d_{n,l}^{\text{rDeco}}(\mathcal{K})$.

The following properties will be useful for the analysis of the decoder width:

######  Proposition 3.1.

Assume $\mathcal{K}=\mathcal{F(K_{V})}$, where
$\mathcal{K}_{\mathcal{V}}\subset\mathcal{V}$ is a compact subset,
$\mathcal{V}$ is another Banach space, and $\mathcal{F:K_{V}\to U}$ is an
$L_{\mathcal{F}}$-Lipschitz mapping. Then we have
$d_{n,L_{\mathcal{F}}l}^{\text{rDeco}}(\mathcal{K})\leq
L_{\mathcal{F}}d_{n,l}^{\text{rDeco}}(\mathcal{K}_{\mathcal{V}}).$ The obvious
corollary is $d_{n,L_{\mathcal{F}}l}^{\text{Deco}}(\mathcal{K})\leq
L_{\mathcal{F}}d_{n,l}^{\text{rDeco}}(\mathcal{K}_{\mathcal{V}}).$

######  Proposition 3.2.

Assume the compact subset $\mathcal{K}$ has diameter
$\delta_{\mathcal{K}}=\mathrm{diam}(\mathcal{K})=\sup_{u,v\in\mathcal{K}}\|u-v\|_{\mathcal{U}}<+\infty$.
Then $d_{n,\delta_{\mathcal{K}}l^{2}}^{\mathrm{Deco}}(\mathcal{K})\leq
d_{n,l}^{\mathrm{SMani}}(\mathcal{K}).$

######  Corollary 3.3.

If $\mathcal{U}$ is a Hilbert space, and $\mathcal{K}$ has diameter
$\delta_{\mathcal{K}}=\mathrm{diam}(\mathcal{K})<+\infty$, then we shall have
$d^{\text{Deco}}_{26n,4\delta_{\mathcal{K}}}(\mathcal{K})\leq
3\epsilon_{n}(\mathcal{K}).$

Compared with the stable manifold width (2.3), the decoder width would not
take a greater value according to Proposition 3.2. Here a strict inequality
can appear, as is shown in the following simple example:

######  Example 3.4.

Let $\mathcal{U}=\ell^{2}(\mathbb{R}^{2})$ be the Euclidean plane, and
$\mathcal{K}=\\{u\in\mathcal{U}\mid\|u\|_{2}=1\\}=S^{1}$ be the unit circle.
Then we have $d_{1,l}^{\text{SMani}}(\mathcal{K})\geq 1$ for any $l>0$, but
$d_{1,\pi}^{\text{Deco}}(\mathcal{K})=0$. See Figure 1 for an illustration.

Figure 1. An illustration of Example 3.4. With an appropriately chosen decoder
mapping $D_{1}:Z\to\mathcal{U}$, $\mathcal{K}$ can be fitted perfectly by the
trial manifold $D_{1}(Z_{B})$, and the decoder width is therefore zero.

######  Remark 3.5.

The Lipschitz width exhibits similar properties to the decoder width. The
inequality presented in Proposition 3.2 also appears in [31, Theorem 5.1], and
[31, Theorem 5.3] generalizes Example 3.4 to the case of all $n\geq 1$.
However, for the Lipschitz width, an inequality stronger than Corollary 3.3
can be given (see [31, Theorem 3.3]), and the proof relies on choosing a norm
different than $\ell^{2}$ (more specifically, $\ell^{\infty}$) on $Z$.

Corollary 3.3 can be derived directly by combining Proposition 3.2 with (2.5).
The proof of the other assertions above will be given in Section 5.

###  3.2. The Decoder Width on a Variable Domain

Another more complicated case of interest would involve a variable domain,
rather than a fixed one. Now let $\mathcal{K}$ be a set of functions, and each
element $u\in\mathcal{K}$ is associated with a domain
$\Omega_{u}\subset\mathbb{R}^{d}$ such that
$u\in\mathcal{U}(\Omega_{u};\mathbb{R}^{d_{u}})$222Here $d_{u}>0$ is a fixed
integer independent of $u$. . This scenario arises when the shape of the
computational domain $\Omega$ is part of the variable parameter in the
parametric PDEs. In this case, the unique solution to (1.1) for any PDE
parameter $\eta=(\gamma_{1},\gamma_{2},\Omega)\in\mathcal{A}$ is denoted as
$u^{\eta}$, and we have $\Omega_{u^{\eta}}=\Omega$. This case poses a
challenge, as different elements of $\mathcal{K}$ may lie in different
function spaces, and the distance between them is not readily defined. The
natural resolution to this issue is to modify all these functions to fit into
the same function space, which can be achived by using either a reference
domain or a master domain.

In the first case, we assume there is a _reference domain_
$\Omega^{\text{ref}}$, and every function $u\in\mathcal{K}$ is associated with
a diffeomorphism $T_{u}:\Omega^{\text{ref}}\to\Omega_{u}$, which satisfies
certain regularity conditions (depending on $\mathcal{U}$). Making use of this
diffeomorphism, we may obtain another function on the reference domain

$\bar{u}(\widetilde{\boldsymbol{x}})=u(T_{u}(\widetilde{\boldsymbol{x}}))\in\mathcal{U}(\Omega^{\text{ref}};\mathbb{R}^{d_{u}}).$

The modified set of functions
$\bar{\mathcal{K}}=\\{\bar{u}(\widetilde{\boldsymbol{x}})\mid
u\in\mathcal{K}\\}$ is now a subset of the function space
$\mathcal{U}(\Omega^{\text{ref}};\mathbb{R}^{d_{u}})$, and all previous
definitions of widths (including the decoder width) can be applied. This
reformulation is convenient for classical mesh-based ROMs [32], but may lack
flexibility as discussed in Section 1.

In the second case, we assume instead there is a _master domain_
$\Omega^{\text{m}}$ containing all possible $\Omega$’s as its subdomain, i.e.,
$\Omega_{u}\subseteq\Omega^{\text{m}}$ for any $u\in\mathcal{K}$. Using
restrictions instead of domain transformations, the decoder width is now
defined as

(3.3) $d_{n,l}^{\mathrm{Deco}}(\mathcal{K})=\inf_{D\
l\text{-Lip}}\sup_{u\in\mathcal{K}}\inf_{\boldsymbol{z}\in
Z_{B}}\Bigl{\|}u-D(\boldsymbol{z})|_{\Omega_{u}}\Bigr{\|}_{\mathcal{U}(\Omega_{u};\mathbb{R}^{d_{u}})},$

where the first infimum is taken over all mappings
$D:Z\to\mathcal{U}(\Omega^{\text{m}};\mathbb{R}^{d_{u}})$ that are
$l$-Lipschitz continuous. Conceptually, it aims to approximate the target
function $u$ by restricting the decoded function $D(\boldsymbol{z})$ to the
subdomain $\Omega_{u}\subseteq\Omega^{\text{m}}$ on which $u$ is defined. This
alternative reformulation is prefered in the context of the MAD method, since
we take neural networks as the mesh-free ansatz of PDE solutions, which can be
evaluated for all $\widetilde{\boldsymbol{x}}\in\mathbb{R}^{d}$.

In order to estimate this type of decoder width of the solution set
$S(\mathcal{A})$, we may seek to construct an extended solution mapping
$\bar{S}:\mathcal{A\to
U}(\Omega^{\text{m}};\mathbb{R}^{d_{u}}),\eta\mapsto\bar{u}^{\eta}$ which
satisfies $\bar{u}^{\eta}|_{\Omega}=u^{\eta}$ for any
$\eta=(\gamma_{1},\gamma_{2},\Omega)\in\mathcal{A}$. Let
$\bar{\mathcal{K}}=\\{\bar{u}^{\eta}\mid\eta\in\mathcal{A}\\}=\bar{S}(\mathcal{A})$,
it is then easy to see that $d_{n,l}^{\mathrm{Deco}}(\mathcal{K})\leq
d_{n,l}^{\mathrm{Deco}}(\bar{\mathcal{K}})$ holds, since we have

$\Bigl{\|}u^{\eta}-D(\boldsymbol{z})|_{\Omega}\Bigr{\|}_{\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})}=\Bigl{\|}(\bar{u}^{\eta}-D(\boldsymbol{z}))|_{\Omega}\Bigr{\|}_{\mathcal{U}(\Omega;\mathbb{R}^{d_{u}})}\leq\|\bar{u}^{\eta}-D(\boldsymbol{z})\|_{\mathcal{U}(\Omega^{\text{m}};\mathbb{R}^{d_{u}})}.$

In this way, we have essentially converted the variable domain case to the
simpler fixed domain case.

##  4\. Decoder Width Estimation for Several Parametric PDEs

In this section, we shall examine several common parametric PDEs, and estimate
the decoder widths of the corresponding solution sets. The proofs of these
assertions can be found in Section 5.

Before diving into the rigorous statements and proofs, we shall first give a
brief informal description of the intuition behind. For the elliptic and the
parabolic equations on a fixed domain, the solution mappings are Lipschitz
continuous, i.e. $\|u-\bar{u}\|\leq C\|\eta-\bar{\eta}\|$ holds. By
Proposition 3.1, this would imply $d_{n,Cl}^{\text{Deco}}(\\{u^{\eta}\\})\leq
Cd_{n,l}^{\text{rDeco}}(\\{\eta\\})$. When the specific form of
$\mathcal{A}=\\{\eta\\}$ is given (in the examples), the specific upper bounds
for $d_{n,Cl}^{\text{Deco}}(\\{u^{\eta}\\})$ can be derived accordingly.

In terms of the scenario involving a variable domain, we only present the
results for the elliptic equations, and the case of the parabolic equations
could be tackled in a similar manner. The overall strategy for deriving upper
bounds remains consistent with the fixed-domain case, but additional efforts
are required to establish the Lipschitz continuity of the solution mapping,
since the variable PDE parameter $\eta$ now incorporates the shape of the
computational domain $\Omega$. To address this, we first examine how the
solution $u$ is affected by a small perturbation of $\Omega$ (Section 5.4.1).
A master domain $\Omega^{\text{m}}$ is predefined, and the perturbed domain
$\Omega_{2}=\beta(\Omega)$ is given by an automorphism
$\beta:\Omega^{\text{m}}\to\Omega^{\text{m}}$ that is close to the identity.
Subsequently, we compare solutions on two arbitrary domains by introducing a
sequence of intermediate domains, and conclude that the solution mapping is
indeed Lipschitz continuous (Proposition 4.10).

The case appears to be slightly different for the advection equations on a
fixed domain, as the solution mapping is Lipschitz only when the function
space $\mathcal{U}$ is endowed with the $L^{1}$-norm. If we switch to the
$L^{2}$-norm that is more convenient for numerical computation, the solution
mapping becomes only $\frac{1}{2}$-Hölder continuous, i.e. $\|u-\bar{u}\|\leq
C\sqrt{\|\eta-\bar{\eta}\|}$. We would then recourse to the entropy number as
the alternative tool for analysis, and show $\epsilon_{n}(\\{u^{\eta}\\})\leq
C\sqrt{\tilde{\epsilon}_{n}(\\{\eta\\})}$ (Lemma 5.16). Since
$d^{\text{Deco}}_{26n,4\delta}(\\{u^{\eta}\\})\leq
3\epsilon_{n}(\\{u^{\eta}\\})$ holds according to Corollary 3.3, the specific
form of the upper bounds can be established.

###  4.1. Elliptic Equations on a Fixed Domain

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with Lipschitz
continuous boundary, we consider the elliptic equation

(4.1) $\displaystyle Lu$ $\displaystyle=f$ $\displaystyle\text{in }\Omega,$
$\displaystyle u$ $\displaystyle=0$ $\displaystyle\text{on }\partial\Omega,$

where the differential operator $L$ is defined as

$Lu=-\nabla\cdot(\boldsymbol{a}(\boldsymbol{x})\nabla
u(\boldsymbol{x}))+c(\boldsymbol{x})u(\boldsymbol{x})$

with variable coefficients $\boldsymbol{a}\in\Sigma^{d}_{r,R}(\Omega)$,
$c\in\mathsf{D}^{c}$, $f\in\mathsf{D}_{R}^{f}$. Here

(4.2)
$\begin{split}\Sigma^{d}_{r,R}(\Omega)=\\{\boldsymbol{a}(\boldsymbol{x})\mid&r\xi^{\mathrm{T}}\xi\leq\xi^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\xi\leq
R\xi^{\mathrm{T}}\xi,\\\
&\boldsymbol{a}(\boldsymbol{x})^{\mathrm{T}}=\boldsymbol{a}(\boldsymbol{x})\in\mathbb{R}^{d\times
d},\forall\xi\in\mathbb{R}^{d},\boldsymbol{x}\in\Omega\\}\end{split}$

is a subset of $L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times d}))$ with
$0<r<R<\infty$,

$\mathsf{D}^{c}=\\{c(\boldsymbol{x})\in L^{\infty}(\Omega)\mid
c(\boldsymbol{x})\geq 0,\forall\boldsymbol{x}\in\Omega\\},$

and $\mathsf{D}_{R}^{f}=\bar{B}_{R}(L^{2}(\Omega))$ using the notation (3.1).
The weak solution $u\in H^{1}_{0}(\Omega)$ to (4.1) should satisfy

(4.3) $\int_{\Omega}\left((\nabla u)^{\mathrm{T}}\boldsymbol{a}\nabla
v+cuv\right)\,\mathrm{d}\boldsymbol{x}=\int_{\Omega}fv\,\mathrm{d}\boldsymbol{x},\quad\forall
v\in H^{1}_{0}(\Omega).$

We have the following proposition:

######  Proposition 4.1.

Let $\bar{u}\in H_{0}^{1}(\Omega)$ be the weak solution to (4.1) with
alternative coefficients $\boldsymbol{\bar{a}}\in\Sigma^{d}_{r,R}(\Omega)$,
$\bar{c}\in\mathsf{D}^{c}$, $\bar{f}\in\mathsf{D}_{R}^{f}$. Then we shall have

$\begin{split}\|u-\bar{u}\|_{H_{0}^{1}}\leq\frac{C_{\text{P}}}{r^{2}}\Bigl{(}R\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}+RC_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}}+r\|\bar{f}-f\|_{L^{2}}\Bigr{)},\end{split}$

where $C_{\text{P}}$ is the Poincaré constant of $\Omega$.

If we denote
$\mathcal{A}_{0}=\Sigma^{d}_{r,R}(\Omega)\times\mathsf{D}^{c}\times\mathsf{D}_{R}^{f}$
with metric333This metric can be viewed as inherited from the norm of a Banach
space. Therefore, Proposition 3.1 is applicable.  given as

$d_{\mathcal{A}_{0}}((\boldsymbol{a},c,f),(\boldsymbol{\bar{a}},\bar{c},\bar{f}))=R\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}+RC_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}}+r\|\bar{f}-f\|_{L^{2}},$

then Proposition 4.1 indicates that the (extended) solution mapping
$S_{0}:\mathcal{A}_{0}\to H_{0}^{1}(\Omega),\quad(\boldsymbol{a},c,f)\mapsto
u$ is $\frac{C_{\text{P}}}{r^{2}}$-Lipschitz continuous. The following theorem
is then direct from Proposition 3.1:

######  Theorem 4.2.

Let $\mathcal{A}\subset\mathcal{A}_{0}$ be the set of parameters taken into
consideration. Then we shall have

$d_{n,C_{\text{P}}l/r^{2}}^{\text{Deco}}(S_{0}(\mathcal{A}))\leq\frac{C_{\text{P}}}{r^{2}}d_{n,l}^{\text{rDeco}}(\mathcal{A}).$

Intuitively speaking, the decoder width of the solution set
$S_{0}(\mathcal{A})$ is controlled by the internal complexity (measured by the
restricted decoder width) of the parameter set $\mathcal{A}$.

One simple special case is given as follows:

######  Example 4.3.

Let $\Omega=(0,1)^{2}$ be a square domain, $r=1,R=2$, and fix
$c\in\mathsf{D}^{c}$, $f\in\mathsf{D}_{2}^{f}$. Given a matrix
$\lambda\in\Lambda=[1,2]^{K\times K}$, we take
$\boldsymbol{a}^{\lambda}=(\sum_{i,j=1}^{K}\lambda_{ij}\chi_{D_{ij}})I_{2}$ to
be a piecewise constant field, where
$D_{ij}=[\frac{i-1}{K},\frac{i}{K})\times[\frac{j-1}{K},\frac{j}{K})$ (see
Figure 2). Let $u^{\lambda}\in\mathcal{U}=H_{0}^{1}(\Omega)$ be the weak
solution to (4.1) with coefficients $(\boldsymbol{a}^{\lambda},c,f)$. Then we
shall have
$d_{K^{2},KC_{\text{P}}}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})=0$,
which indicates that the solution set can be perfectly fitted using a
$K^{2}$-dimensional latent space.

Figure 2. The piecewise constant field $\boldsymbol{a}^{\lambda}$ used in
Example 4.3, showing the case $K=3$. Note that $\boldsymbol{a}^{\lambda}$ is
in fact a matrix-valued field.

###  4.2. Parabolic Equations on a Fixed Domain

Letting $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with Lipschitz
continuous boundary, $0<r<R<\infty$, $T>0$, we consider the following
parabolic equation on $\Omega_{T}=[0,T]\times\Omega$:

(4.4) $\displaystyle\partial_{t}u+Lu$ $\displaystyle=f\quad$
$\displaystyle\text{in }\Omega_{T}$ $\displaystyle u$ $\displaystyle=0$
$\displaystyle\text{on }[0,T]\times\partial\Omega$ $\displaystyle u$
$\displaystyle=g$ $\displaystyle\text{on }\\{t=0\\}\times\Omega.$

Here, we define the differential operator $L$ in the divergence form to be

$Lu=-\nabla\cdot(\boldsymbol{a}\nabla u)+\boldsymbol{b}\cdot\nabla u+cu$

with variable coefficients $\boldsymbol{a}\in\Sigma^{d}_{r,R}(\Omega_{T})$,
$\boldsymbol{b}\in\bar{B}_{R}^{d}(\Omega_{T})$,
$c\in\bar{B}_{R}^{1}(\Omega_{T})$, where

$\begin{split}\bar{B}_{R}^{d}(\Omega_{T})&=\bar{B}_{R}(L^{\infty}(\Omega_{T};\ell^{2}(\mathbb{R}^{d})))\\\
&=\left\\{\boldsymbol{b}\in L^{\infty}(\Omega_{T};\mathbb{R}^{d})\ \middle|\
\operatorname*{ess\,sup}_{t\in[0,T],\boldsymbol{x}\in\Omega}\|\boldsymbol{b}(t,\boldsymbol{x})\|_{2}\leq
R\right\\}\end{split}$

using the notation given in (3.1), and similarly

$\begin{split}\bar{B}_{R}^{1}(\Omega_{T})&=\bar{B}_{R}(L^{\infty}(\Omega_{T}))\\\
&=\left\\{c\in L^{\infty}(\Omega_{T})\ \middle|\ \|c\|_{L^{\infty}}\leq
R\right\\}.\end{split}$

Furthermore, we assume $g\in L^{2}(\Omega)$, and
$f(t,\boldsymbol{x})=\mathbf{f}(t)(\boldsymbol{x})$ for some $\mathbf{f}\in
L^{2}([0,T];H^{-1}(\Omega))$. The solution $u$ can also be associated with a
mapping $\mathbf{u}$ by $u(t,\boldsymbol{x})=\mathbf{u}(t)(\boldsymbol{x})$.

We say a function $\mathbf{u}\in L^{2}([0,T];H_{0}^{1}(\Omega))$ with
$\mathbf{u}^{\prime}=\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}\in
L^{2}([0,T];H^{-1}(\Omega))$ is a weak solution to (4.4) if

(4.5)
$\displaystyle\langle\mathbf{u}^{\prime}(t),v\rangle+A[\mathbf{u}(t),v;t]$
$\displaystyle=\langle\mathbf{f}(t),v\rangle,$ $\displaystyle\mathbf{u}(0)$
$\displaystyle=g$

holds444According to Theorem 3 in Section 5.9.2 of [14], we shall have
$\mathbf{u}\in C([0,T];L^{2}(\Omega))$, and therefore the equality
$\mathbf{u}(0)=g$ makes sense.  for all $v\in H_{0}^{1}(\Omega),t\in[0,T]$,
where the time-dependent bilinear form $A[u,v;t]$ is given by

$\begin{split}A[u,v;t]&=\int_{\Omega}(\nabla
v(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(t,\boldsymbol{x})(\nabla
u(\boldsymbol{x}))+v(\boldsymbol{x})\boldsymbol{b}(t,\boldsymbol{x})^{\mathrm{T}}\nabla
u(\boldsymbol{x})+c(t,\boldsymbol{x})u(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\end{split}$

for $u,v\in H_{0}^{1}(\Omega)$ and $t\in[0,T]$.

######  Proposition 4.4.

Denote $\mathsf{D}_{R}^{\mathbf{f}}=\bar{B}_{R}(L^{2}([0,T];H^{-1}(\Omega)))$,
$\mathsf{D}_{R}^{g}=\bar{B}_{R}(L^{2}(\Omega))$ using the notation (3.1), and

$\mathcal{A}_{0}=\Sigma^{d}_{r,R}(\Omega_{T})\times\bar{B}_{R}^{d}(\Omega_{T})\times\bar{B}_{R}^{1}(\Omega_{T})\times\mathsf{D}_{R}^{\mathbf{f}}\times\mathsf{D}_{R}^{g}.$

Then the (extended) solution mapping

$\begin{split}S_{0}:\mathcal{A}_{0}\to
L^{2}([0,T];H_{0}^{1}(\Omega)),\quad(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)\mapsto\mathbf{u}\end{split}$

is Lipschitz continuous. Specifically, let
$(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)$ and
$(\boldsymbol{\bar{a}},\boldsymbol{\bar{b}},\bar{c},\bar{\mathbf{f}},\bar{g})$
be two sets of coefficient fields, with the corresponding weak solutions
$\mathbf{u}$ and $\bar{\mathbf{u}}$, respectively. Then we should have

(4.6)
$\begin{split}\|\mathbf{u}-\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}&\leq
C(\|\boldsymbol{a}-\boldsymbol{\bar{a}}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d\times
d}))}+\|\boldsymbol{b}-\boldsymbol{\bar{b}}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d}))}\\\
&\quad+\|c-\bar{c}\|_{L^{\infty}(\Omega_{T})}+\|\mathbf{f}-\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}\\\
&\quad+\|g-\bar{g}\|_{L^{2}}),\end{split}$

where the constant $C>0$ depends only on $r,R,T$ and $\Omega$.

The metric on $\mathcal{A}_{0}$ is chosen according to the right-hand-side of
(4.6) (inside the parenthesis). Similar to the elliptic equations, the decoder
width of the solution set admits an upper bound given by the complexity of the
parameter set $\mathcal{A}$, as the following theorem would imply:

######  Theorem 4.5.

Let $\mathcal{A}\subset\mathcal{A}_{0}$ be the set of parameters taken into
consideration. Then $d_{n,Cl}^{\text{Deco}}(S_{0}(\mathcal{A}))\leq
Cd_{n,l}^{\text{rDeco}}(\mathcal{A}),$ where $C$ is the same as in Proposition
4.4.

This theorem is easily obtained by combining Proposition 3.1 with the
$C$-Lipschitz continuity of $S_{0}$ (Proposition 4.4), and we shall omit the
proof.

####  4.2.1. Special Cases

Assume we are given a sequence of functions
$\\{(\boldsymbol{\phi}_{k}^{a},\boldsymbol{\phi}_{k}^{b},\phi_{k}^{c},\phi_{k}^{\mathbf{f}},\phi_{k}^{g})\\}_{k=1}^{\infty},$
satisfying:

  1. (a)

(boundedness) $\boldsymbol{\phi}_{k}^{a}\in\bar{B}_{(R-r)/2}^{d}(\Omega_{T})$,
$\boldsymbol{\phi}_{k}^{b}\in\bar{B}_{R}^{d}(\Omega_{T})$,
$\phi_{k}^{c}\in\bar{B}_{R}^{1}(\Omega_{T})$.

  2. (b)

(disjoint support) For $k\neq l$, we have
$\mathrm{supp}\boldsymbol{\phi}_{k}^{a}\cap\mathrm{supp}\boldsymbol{\phi}_{l}^{a}=\varnothing$,
where $\mathrm{supp}\boldsymbol{\phi}_{k}^{a}$ represents the essential
support of the function
$\boldsymbol{\phi}_{k}^{a}:\Omega_{T}\to\mathbb{R}^{d}$. Similar assumptions
are made for $\boldsymbol{\phi}_{k}^{b}$ and $\phi_{k}^{c}$.

  3. (c)

$\phi_{k}^{\mathbf{f}}\in\mathsf{D}_{R}^{\mathbf{f}}$,
$\sum_{k=1}^{\infty}\|\phi_{k}^{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}^{2}\leq
R^{2}$.

  4. (d)

$\phi_{k}^{g}\in\mathsf{D}_{R}^{g}$,
$\sum_{k=1}^{\infty}\|\phi_{k}^{g}\|_{L^{2}}^{2}\leq R^{2}$.

Let
$\lambda=\\{(\lambda_{k}^{a},\lambda_{k}^{b},\lambda_{k}^{c},\lambda_{k}^{\mathbf{f}},\lambda_{k}^{g})\\}_{k=1}^{\infty}$
be a sequence of scalars, and we denote

$\lambda_{k}^{2}=(\lambda_{k}^{a})^{2}+(\lambda_{k}^{b})^{2}+(\lambda_{k}^{c})^{2}+(\lambda_{k}^{\mathbf{f}})^{2}+(\lambda_{k}^{g})^{2}$

for simplicity. We further assume that the corresponding norm

$\|\lambda\|=\sqrt{\sum_{k=1}^{\infty}\lambda_{k}^{2}}$

is finite. The mapping
$S_{\Lambda}:\lambda\mapsto(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)$ is
defined as

$\begin{split}\boldsymbol{a}&=\frac{1}{2}(R+r)I_{d}+\mathrm{diag}\biggl{(}\sum_{k=1}^{\infty}\lambda_{k}^{a}\boldsymbol{\phi}_{k}^{a}\biggr{)},\quad\boldsymbol{b}=\sum_{k=1}^{\infty}\lambda_{k}^{b}\boldsymbol{\phi}_{k}^{b},\\\
c&=\sum_{k=1}^{\infty}\lambda_{k}^{c}\phi_{k}^{c},\quad\mathbf{f}=\sum_{k=1}^{\infty}\lambda_{k}^{\mathbf{f}}\phi_{k}^{\mathbf{f}},\quad
g=\sum_{k=1}^{\infty}\lambda_{k}^{g}\phi_{k}^{g},\end{split}$

and we denote $u^{\lambda}=S_{0}(S_{\Lambda}(\lambda))$ to be the weak
solution to (4.4) for specific $\lambda$. Two special cases are given as
follows:

######  Example 4.6.

Let $\Lambda_{K}=\\{\lambda\mid\lambda_{k}^{2}=0\text{ for
}k>K,\quad\sum_{k=1}^{K}\lambda_{k}^{2}\leq 1\\},$ which essentially consists
of sequences of finite length. Then (1)
$S_{\Lambda}(\lambda)\in\mathcal{A}_{0}$ holds for any
$\lambda\in\Lambda_{K}$, (2) the mapping $S_{0}\circ
S_{\Lambda}:\Lambda_{K}\to\mathcal{U}=L^{2}([0,T];H_{0}^{1}(\Omega))$ is
$5CR$-Lipschitz continuous, where the constant $C$ is the same as in
Proposition 4.4, and (3) $d_{5K,5CR}^{\text{Deco}}(\mathcal{K}_{K})=0$, where
$\mathcal{K}_{K}=\\{u^{\lambda}\mid\lambda\in\Lambda_{K}\\}=S_{0}\circ
S_{\Lambda}(\Lambda_{K})$ is the solution set.

######  Example 4.7.

Let $\\{w_{k}\\}$ be a decreasing sequence with $w_{1}=1$,
$\lim_{k\to\infty}w_{k}=0$, and let
$\Lambda_{w}=\\{\lambda\mid\sum_{k=1}^{\infty}\lambda_{k}^{2}/w_{k}\leq 1\\}.$
Then the solution set
$\mathcal{K}_{w}=\\{u^{\lambda}\mid\lambda\in\Lambda_{w}\\}=S_{0}\circ
S_{\Lambda}(\Lambda_{w})$ satisfies
$d_{5n,5CR}^{\text{Deco}}(\mathcal{K}_{w})\leq 5CR\sqrt{w_{n}}$, where the
constant $C$ is the same as in Proposition 4.4.

###  4.3. Elliptic Equations on a Variable Domain

Assume we are given a fixed _convex_ master domain $\Omega^{\text{m}}$, and
the subdomain of interest $\Omega\subset\Omega^{\text{m}}$ may vary. All
possible choices of $\Omega$ as well as $\Omega^{\text{m}}$ have a Lipschitz
continuous boundary. In this case, the PDE parameter
$\eta=(\boldsymbol{a},f,\Omega)$ would involve the shape of the computational
domain. We consider the elliptic equation

(4.7) $\displaystyle Lu$ $\displaystyle=f$ $\displaystyle\text{in }\Omega,$
$\displaystyle u$ $\displaystyle=0$ $\displaystyle\text{on }\partial\Omega,$

where

$Lu=-\nabla\cdot(\boldsymbol{a}(\boldsymbol{x})\nabla
u(\boldsymbol{x}))=-\sum_{i,j=1}^{d}\partial_{j}(\boldsymbol{a}^{ij}(\boldsymbol{x})\partial_{i}u(\boldsymbol{x})),$

and the following assumptions are satisfied:

######  Assumption 4.8.

(1) $\boldsymbol{a}:\Omega^{\text{m}}\to\ell^{2}(\mathbb{R}^{d\times d})$ is
Lipschitz continuous with Lipschitz constant $R$, (2)
$\boldsymbol{a}\in\Sigma_{r,R}^{d}(\Omega^{\text{m}})$, i.e. for all
$\boldsymbol{x}\in\Omega^{\text{m}}$, $\boldsymbol{a}(\boldsymbol{x})$ is a
symmetric matrix satisfying

$r\xi^{\mathrm{T}}\xi\leq\xi^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\xi\leq
R\xi^{\mathrm{T}}\xi,\quad\forall\xi\in\mathbb{R}^{d},$

and (3) $f\in\bar{B}_{R}(L^{2}(\Omega))$ (using the notation (3.1)), i.e.
$f\in L^{2}(\Omega^{\text{m}})$, $\|f\|_{L^{2}(\Omega^{\text{m}})}\leq R$.

In order to estimate the decoder width of the set of weak solutions
$\\{u^{\eta}\\}$ to 4.7, we consider the following PDEs on the extended domain

(4.8) $\displaystyle Lu$ $\displaystyle=f$ $\displaystyle\text{in
}\Omega^{\text{m}}\setminus\partial\Omega,$ $\displaystyle u$
$\displaystyle=0$ $\displaystyle\text{on
}\partial\Omega^{\text{m}}\cup\partial\Omega.$

Then, if $\bar{u}\in H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega)$ is
the weak solution to (4.8), i.e.,

$\int_{\Omega^{\text{m}}}(\nabla\bar{u}(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})(\nabla
v(\boldsymbol{x}))\,\mathrm{d}\boldsymbol{x}=\int_{\Omega^{\text{m}}}f(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},\quad\forall
v\in H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega),$

it is easy to see that $u=\bar{u}|_{\Omega}$ is the weak solution to (4.7),
and we only need to examine the set of these extended solutions
$\\{\bar{u}^{\eta}\\}$. For notational convenience, we do not explicitly
distinguish the solutions to (4.7) and (4.8) unless specified otherwise, and
denote $u$ to be the weak solution to (4.8).

Now we consider a class of domains $\Omega_{\lambda}\subset\Omega^{\text{m}}$
parametrized by $\lambda\in\Lambda$, where
$\Lambda\subset\mathbb{R}^{d_{\lambda}}$ is a bounded convex set that contains
the origin. The following assumptions are made on the parametrized domains:

######  Assumption 4.9.

There exist a parametrized class of $C^{1}$-diffeomorphisms555Explicit domain
transformations are required in the theoretical estimation of the decoder
width, but not in the numerical computation to solve the parametric PDEs. MAD
is still a flexible algorithm in this sense.
$T_{\lambda}:\bar{\Omega}^{\text{m}}\to\bar{\Omega}^{\text{m}}$ satisfying:

  1. (1)

$T_{0}=\mathrm{id}$, $T_{\lambda}(\Omega_{0})=\Omega_{\lambda}$.

  2. (2)

$\sigma_{\min}(\frac{\partial T_{\lambda}(x)}{\partial x})\geq 1/2$ holds for
any $x\in\bar{\Omega}^{\text{m}},\lambda\in\Lambda$, where $\sigma_{\min}$
denotes the smallest singular value of a matrix.

  3. (3)

Given $x\in\bar{\Omega}^{\text{m}}$, both $T_{\lambda}(x)$ and $\frac{\partial
T_{\lambda}(x)}{\partial x}$ are $1$-Lipschitz continuous with respect to
$\lambda$, i.e.,

$\|T_{\lambda}(x)-T_{\lambda^{\prime}}(x)\|_{2}\leq\|\lambda-\lambda^{\prime}\|_{2},\quad\left\lVert\frac{\partial
T_{\lambda}(x)}{\partial x}-\frac{\partial T_{\lambda^{\prime}}(x)}{\partial
x}\right\rVert_{2}\leq\|\lambda-\lambda^{\prime}\|_{2}$

for any $\lambda,\lambda^{\prime}\in\Lambda$.

####  4.3.1. Fixed Parameters

In the first case, the parameters $\boldsymbol{a},f$ are fixed for simplicity,
and the weak solution to (4.8) with domain $\Omega=\Omega_{\lambda}$ is
denoted as
$u^{\lambda}\in{H_{0}^{1}(\Omega^{\text{m}}\setminus\partial\Omega_{\lambda})}$.
We have the following proposition on the corresponding solution mapping:

######  Proposition 4.10.

Under Assumption 4.8 and 4.9, the solution mapping
$S:\Lambda\to\mathcal{U}=L^{2}(\Omega^{\text{m}})$, $\lambda\mapsto
u^{\lambda}$ is $C$-Lipschitz continuous, where the constant $C$ depends only
on $r,R$ and $\Omega^{\text{m}}$. Specifically, we can take

$C=\frac{4RC_{\text{P}}}{r^{2}}(3r+(\tfrac{3}{8}(9+\delta_{0}^{-1})+1)RC_{\text{P}}),$

where $C_{\text{P}}$ is the Poincaré constant of $\Omega^{\text{m}}$, and
$\delta_{0}=\min(1-(8/9)^{2/d},1/9)$.

The following theorem then follows from Proposition 3.1:

######  Theorem 4.11.

Under Assumption 4.8 and 4.9, we have $d_{n,Cl}^{\text{Deco}}(S(\Lambda))\leq
Cd_{n,l}^{\text{rDeco}}(\Lambda),$ where the constant $C$ is the same as in
Proposition 4.10.

######  Example 4.12 (moving disk).

We take the master domain to be the two dimensional unit disk
$\Omega^{\text{m}}=\\{\boldsymbol{x}\in\mathbb{R}^{2}\mid\|\boldsymbol{x}\|_{2}<1\\}$.
Given $\lambda\in\Lambda=[-\pi,\pi]$, the corresponding domain is

$\Omega_{\lambda}=\\{\boldsymbol{x}\in\mathbb{R}^{2}\mid(x_{1}-\tfrac{1}{3}\cos\lambda)^{2}+(x_{2}-\tfrac{1}{3}\sin\lambda)^{2}<\tfrac{1}{9}\\},$

which rotates around the origin as $\lambda$ varies. Then
$d_{1,C\pi}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})=0$ ($C$ being
the same as in Proposition 4.10), indicating that the solution set can be
perfectly fitted using a one-dimensional latent space.

######  Example 4.13 (unit disk with a moving hole).

We take
$\Omega^{\text{m}}=\\{\boldsymbol{x}\in\mathbb{R}^{2}\mid\|\boldsymbol{x}\|_{2}<1\\}$,
$\lambda\in\Lambda=[-\pi,\pi]$ as in the previous example, but replace
$\Omega_{\lambda}$ by the interior of its complement, namely
$\Omega_{\lambda}=\\{\boldsymbol{x}\in\Omega^{\text{m}}\mid(x_{1}-\frac{1}{3}\cos\lambda)^{2}+(x_{2}-\frac{1}{3}\sin\lambda)^{2}>\frac{1}{9}\\}$.
The resulting domain is the unit disk with a hole inside, and the hole changes
its position as $\lambda$ varies. Then we shall have
$d_{1,C\pi}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})=0$ as in the
previous example.

######  Example 4.14 (square with a deformable rectangular hole).

The master domain is taken to be $\Omega^{\text{m}}=(-\pi,\pi)^{2}$, and we
obtain $\Omega_{\lambda}$ by hollowing out a rectangular hole in the form

$\Omega^{\text{m}}\setminus\Omega_{\lambda}=\left[-\frac{\pi}{2}-\lambda_{1},\frac{\pi}{2}+\lambda_{1}\right]\times\left[-\frac{\pi}{2}-\lambda_{2},\frac{\pi}{2}+\lambda_{2}\right],$

where
$\lambda=(\lambda_{1},\lambda_{2})\in\Lambda=[-\frac{1}{2},\frac{1}{2}]^{2}$
is the parameter. Then we shall have
$d_{2,C/\sqrt{2}}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})=0$.

######  Example 4.15 (area under a variable curve).

Let $\lambda=(a_{1},b_{1},a_{2},\dots)$ be an infinite sequence of real
numbers with norm defined as
$\|\lambda\|^{2}=\sum_{k=1}^{\infty}(a_{k}^{2}+b_{k}^{2})$, and

$\Lambda=\left\\{\lambda\ \middle|\
\sum_{k=1}^{\infty}\frac{a_{k}^{2}+b_{k}^{2}}{w_{k}}\leq\frac{9}{64}\right\\},$

where $w_{k}$ is a decreasing sequence with $w_{1}=1$,
$\lim_{k\to\infty}w_{k}=0$. Letting

$\rho_{\lambda}(x_{1})=\sum_{k=1}^{\infty}\frac{\sqrt{6}}{k^{2}\pi}(a_{k}\cos
kx_{1}+b_{k}\sin kx_{1}),$

we take $\Omega^{\text{m}}=(-\pi,\pi)\times(0,\pi)$, and

$\Omega_{\lambda}=\left\\{\boldsymbol{x}\in\mathbb{R}^{2}\middle|-\pi<x_{1}<\pi,0<x_{2}<\frac{\pi}{2}+\rho_{\lambda}(x_{1})\right\\}.$

Then $d_{2n,3C/8}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})\leq
3C\sqrt{w_{n}}/8$.

Figure 3. Illustrative plot of the $\Omega_{\lambda}$’s in Example 4.12
(left), 4.14 (middle), and 4.15 (right).

####  4.3.2. Variable Parameters

Now both $\boldsymbol{a}$ and $f$ in the PDEs (4.8) can vary along with the
shape of the solution domain $\Omega$ (or its parameter $\lambda$). According
to Assumption 4.8, we denote $\mathsf{D}_{r,R}^{a}$ as the set of all possible
$\boldsymbol{a}$’s, and $\mathsf{D}_{R}^{f}=\bar{B}_{R}(L^{2}(\Omega))$ the
set of all possible $f$’s. Then we shall have the following proposition:

######  Proposition 4.16.

Let
$\mathcal{A}_{0}=\mathsf{D}_{r,R}^{a}\times\mathsf{D}_{R}^{f}\times\Lambda$
with the metric given as

(4.9)
$d_{\mathcal{A}_{0}}((\boldsymbol{a},f,\lambda),(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda}))=\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}+\|\bar{f}-f\|_{L^{2}}+\|\bar{\lambda}-\lambda\|_{2}.$

Then, under Assumption 4.9, the (extended) solution mapping
$S_{0}:\mathcal{A}_{0}\to\mathcal{U}=L^{2}(\Omega^{\text{m}})$,
$(\boldsymbol{a},f,\lambda)\mapsto u^{(\boldsymbol{a},f,\lambda)}$ is
$C_{1}$-Lipschitz continuous, where
$C_{1}=\max\left(\frac{C_{\text{P}}^{2}R}{r^{2}},C\right)$, and $C$ is the
same as in Proposition 4.10

Again, the following theorem is direct according to Proposition 3.1:

######  Theorem 4.17.

Let $\mathcal{A}\subset\mathcal{A}_{0}$ be the set of parameters taken into
consideration. Then, under Assumption 4.9, we have
$d_{n,C_{1}l}^{\text{Deco}}(S_{0}(\mathcal{A}))\leq
C_{1}d_{n,l}^{\text{rDeco}}(\mathcal{A}),$ where the constant $C_{1}$ is the
same as in Proposition 4.16.

######  Example 4.18.

We take the domain to be a square with a deformable rectangular hole, and
$\Omega^{\text{m}},\Omega_{\lambda}$, $\Lambda$ are the same as in Example
4.14. Taking $r=1,R=2$, we parametrize the PDE coefficients as
$\boldsymbol{a}=(\frac{3}{2}+\sum_{k=1}^{K}\mu_{k}^{a}\phi_{k}^{a})I_{2}$,
$f=\sum_{k=1}^{K}\mu_{k}^{f}\phi_{k}^{f}$, where

  1. (a)

$\phi_{k}^{a}:\Omega^{\text{m}}\to\mathbb{R}$ are 4-Lipschitz continuous,
$\|\phi_{k}^{a}\|_{L^{\infty}}\leq 1$, $\phi_{k}^{a}$ and $\phi_{l}^{a}$ has
disjoint supports for $1\leq k<l\leq K$;

  2. (b)

$\phi_{k}^{f}\in L^{2}(\Omega^{\text{m}})$,
$\sum_{k=1}^{K}\|\phi_{k}^{f}\|_{L^{2}}^{2}\leq 1$;

  3. (c)

$\mu=(\mu_{k}^{a},\mu_{k}^{f})_{k=1}^{K}\in\Lambda_{\mu}$,
$\Lambda_{\mu}=\\{\mu\mid\sum_{k=1}^{K}(\mu_{k}^{a})^{2}\leq\frac{1}{4},\sum_{k=1}^{K}(\mu_{k}^{a})^{2}\leq\frac{1}{4}\\}\subset\mathbb{R}^{2K}$.

Denote the corresponding solution as $u^{\mu,\lambda}$, we shall have
$d_{2K+2,3C_{1}}^{\text{Deco}}(\\{u^{\mu,\lambda}|\mu\in\Lambda_{\mu},\lambda\in\Lambda\\})=0,$
with $C_{1}$ given as in Proposition 4.16.

###  4.4. Advection Equations on a Fixed Domain

Consider the equation

(4.10) $u_{t}+c(x)u_{x}=0$

on the spatial-temporal domain $(t,x)\in[0,1]\times[-1,1]$, with initial
condition $u(0,x)=\chi_{[-1,0]}(x)$ and boundary condition $u(t,-1)=1$. Here,
the variable coefficient is $c\in\mathcal{A}_{0}$, where

$\mathcal{A}_{0}=\bigl{\\{}c\in C^{1}([-1,1])\mid r\leq c(x)\leq R,\forall
x\in[-1,1]\bigr{\\}},$

and $0<r<R<\infty$ holds. We shall view $\mathcal{A}_{0}$ as a subset of the
Banach space $L^{1}([-1,1])$ (inheriting its metric), and denote the
(extended) solution mapping to be $S_{0}:c\mapsto u$. By the method of
characteristics, it is easy to show that the weak solution to the PDEs is
given by

$u(t,x)=\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\}}=\begin{cases}1,&t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\\
0,&t>\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\end{cases}.$

######  Proposition 4.19.

Let $\bar{u}$ be the weak solution to (4.10) with alternative coefficient
$\bar{c}\in\mathcal{A}_{0}$, i.e. $\bar{u}=S_{0}(\bar{c})$. Viewing
$u,\bar{u}\in L^{p}([0,1]\times[-1,1])$ for $p\in[1,+\infty)$, we have

$\|u-\bar{u}\|_{L^{p}}\leq r^{-2/p}\|c-\bar{c}\|_{L^{1}([-1,1])}^{1/p}.$

Due to the particularity of the advection equations, the decoder widths using
$L^{1}$-norm ($p=1$) and $L^{2}$-norm ($p=2$) will be estimated separately.
The decoder widths can achieve zero for finite $n$ in the former case, while
in the latter case, only an exponential decay is guaranteed, as the Lipschitz
continuity of the solution mapping is no longer valid.

####  4.4.1. The Case of Using $L^{1}$-norm

Taking $p=1$ in Proposition 4.19, we apply Proposition 3.1 to get the
following theorem:

######  Theorem 4.20.

Let $\mathcal{A}\subset\mathcal{A}_{0}$ be the set of parameters taken into
consideration, and $\mathcal{U}=L^{1}([0,1]\times[-1,1])$. Then we shall have
$d_{n,l/r^{2}}^{\text{Deco}}(S_{0}(\mathcal{A}))_{L^{1}}\leq
r^{-2}d_{n,l}^{\text{rDeco}}(\mathcal{A}).$

######  Example 4.21.

Assume the velocity field $c(x)=c^{\lambda}(x)$ is parametrized as

$c^{\lambda}(x)=\phi_{0}(x)+\sum_{k=1}^{K}\lambda_{k}\phi_{k}(x),\quad\lambda\in\Lambda=[-1,1]^{K},$

where $\phi_{0},\dots,\phi_{K}$ are all continuous functions, and are chosen
appropriately such that $c^{\lambda}(x)\in C([-1,1];[r,R])$ holds for any
$\lambda\in\Lambda$. Let $u^{\lambda}=S_{0}(c^{\lambda})\in\mathcal{U}$ be the
weak solution corresponding to $\lambda$. Then, the set of weak solutions
$\mathcal{K}=\\{u^{\lambda}\mid\lambda\in\Lambda\\}$ satisfies
$d_{K,l/r^{2}}^{\mathrm{Deco}}(\mathcal{K})_{L^{1}}=0,$ where
$l=\sqrt{K\sum_{k=1}^{K}\|\phi_{k}\|_{L^{1}([-1,1])}^{2}}$ is a constant.

####  4.4.2. The Case of Using $L^{2}$-norm

When $p=2$, Proposition 4.19 only indicates the $\frac{1}{2}$-Hölder
continuity of the (extended) solution mapping $S_{0}:c\mapsto u$, and an
analogy of Theorem 4.20 is not available. In this case, the decoder width can
be estimated using the following theorem:

######  Theorem 4.22.

Let $\mathcal{A}\subset\mathcal{A}_{0}$ be the set of parameters taken into
consideration, and take $\mathcal{U}=L^{2}([0,1]\times[-1,1])$. Then we shall
have $d_{26n,8}^{\text{Deco}}(S_{0}(\mathcal{A}))_{L^{2}}\leq
3r^{-1}\sqrt{\tilde{\epsilon}_{n}(\mathcal{A})}.$

######  Example 4.23.

We consider the same solution set $\mathcal{K}$ as in Proposition 4.21, but
within a different function space $\mathcal{U}=L^{2}([0,1]\times[-1,1])$. Then
$d_{26n,8}^{\mathrm{Deco}}(\mathcal{K})_{L^{2}}\leq C\cdot 2^{-n/2K}$ for
sufficiently large $n$, where
$C=6r^{-1}\sqrt{K}\left(\prod_{k=1}^{K}\|\phi_{k}\|_{L^{1}([-1,1])}\right)^{1/2K}$
is a constant.

##  5\. Proofs

###  5.1. Basic Properties of the Decoder Width

###### Proof of Proposition 3.1.

Given an $l$-Lipschitz mapping $D:Z\to\mathcal{K_{V}}$, the mapping
$\mathcal{F}\circ D:Z\to\mathcal{K}$ is $L_{\mathcal{F}}l$-Lipschitz. The
definition of the restricted decoder width gives

$\begin{split}d_{n,L_{\mathcal{F}}l}^{\text{rDeco}}(\mathcal{F(K_{V})})&\leq\sup_{v\in\mathcal{K_{V}}}\inf_{\boldsymbol{z}\in
Z_{B}}\|\mathcal{F}(v)-\mathcal{F}\circ D(\boldsymbol{z})\|_{\mathcal{U}}\\\
&\leq L_{\mathcal{F}}\sup_{v\in\mathcal{K_{V}}}\inf_{\boldsymbol{z}\in
Z_{B}}\|v-D(\boldsymbol{z})\|_{\mathcal{V}}.\end{split}$

We conclude the proof by taking infimum over all such mappings $D$. ∎

###### Proof of Proposition 3.2.

Let $E:\mathcal{K}\to Z$ and $D:Z\to\mathcal{U}$ be two $l$-Lipschitz
continuous mappings, and fix an arbitrary element $v\in\mathcal{K}$. We define
a new mapping $\bar{D}:Z\to\mathcal{U}$ by
$\bar{D}(\boldsymbol{z})=D(l\delta_{\mathcal{K}}\boldsymbol{z}+E(v))$. Then
$\bar{D}$ is $l^{2}\delta_{\mathcal{K}}$-Lipschitz continuous. For given
$u\in\mathcal{K}$,
$\bar{\boldsymbol{z}}=\frac{1}{l\delta_{\mathcal{K}}}(E(u)-E(v))$ satisfies

$\|\bar{\boldsymbol{z}}\|_{2}=\frac{1}{l\delta_{\mathcal{K}}}\|E(u)-E(v)\|_{2}\leq\frac{1}{\delta_{\mathcal{K}}}\|u-v\|_{\mathcal{U}}\leq
1,$

and

$\bar{D}(\bar{\boldsymbol{z}})=D(l\delta_{\mathcal{K}}\bar{\boldsymbol{z}}+E(v))=D(E(u)-E(v)+E(v))=D(E(u)).$

This gives

$\inf_{\boldsymbol{z}\in
Z_{B}}\|u-\bar{D}(\boldsymbol{z})\|_{\mathcal{U}}\leq\|u-\bar{D}(\bar{\boldsymbol{z}})\|_{\mathcal{U}}=\|u-D(E(u))\|_{\mathcal{U}}.$

Taking supremum over all $u\in\mathcal{K}$, and then infimum over all possible
mappings $E,D,\bar{D}$ would then give

$\begin{split}d_{n,l^{2}\delta_{\mathcal{K}}}^{\mathrm{Deco}}(\mathcal{K})&=\inf_{\bar{D}\
l^{2}\delta_{\mathcal{K}}\text{-Lip}}\sup_{u\in\mathcal{K}}\inf_{\boldsymbol{z}\in
Z_{B}}\|u-\bar{D}(\boldsymbol{z})\|_{\mathcal{U}}\\\ &\leq\inf_{E,D\
l\text{-Lip}}\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{\mathcal{U}}\\\
&=d_{n,l}^{\mathrm{SMani}}(\mathcal{K}).\qed\end{split}$

###### Proof of the assertion in Example 3.4.

For any pair of $l$-Lipschitz continuous mappings $E:\mathcal{U}\to
Z=\mathbb{R}$, $D:\mathbb{R}\to\mathcal{U}$, the Borsuk-Ulam theorem asserts
that there exists a point $v\in\mathcal{K}=S^{1}$ such that $E(v)=E(-v)$, and

$\begin{split}\sup_{u\in\mathcal{K}}\|u-D(E(u))\|_{2}&\geq\max(\|v-D(E(v))\|_{2},\|-v-D(E(-v))\|_{2})\\\
&\geq\frac{1}{2}(\|v-D(E(v))\|_{2}+\|-v-D(E(-v))\|_{2})\\\
&=\frac{1}{2}(\|v-D(E(v))\|_{2}+\|v+D(E(v))\|_{2})\\\
&\geq\frac{1}{2}\|v-D(E(v))+v+D(E(v))\|_{2}\\\ &=\|v\|_{2}=1.\end{split}$

We conclude that $d_{n,l}^{\mathrm{SMani}}(\mathcal{K})\geq 1$ by taking
infimum over all possible pairs $(E,D)$. As for the decoder width, take
$\bar{D}:\mathbb{R}\to\mathcal{U}$ as $z\mapsto(\cos\pi z,\sin\pi z)$. Then
$\bar{D}$ is $\pi$-Lipschitz continuous, and
$\bar{D}(Z_{B})=\bar{D}([-1,1])=\mathcal{K}$ holds, which implies
$d_{1,\pi}^{\mathrm{Deco}}(\mathcal{K})=0$, and the first infimum in (3.2) is
achieved at $D=\bar{D}$. ∎

###  5.2. Elliptic Equations on a Fixed Domain

######  Lemma 5.1.

The weak solution to (4.1) satisfies $\|u\|_{H_{0}^{1}}\leq C_{\text{P}}R/r$,
where $C_{\text{P}}$ is the Poincaré constant of $\Omega$.

###### Proof.

Take the test function to be $v=u\in H^{1}_{0}(\Omega)$, then

$\begin{split}\int_{\Omega}(\nabla u)^{\mathrm{T}}\boldsymbol{a}\nabla
u+cu^{2}\,\mathrm{d}\boldsymbol{x}=\int_{\Omega}fu\,\mathrm{d}\boldsymbol{x}\leq\|f\|_{L^{2}}\|u\|_{L^{2}}\leq
RC_{\text{P}}\|u\|_{H_{0}^{1}},\end{split}$

where we used $\|f\|_{L^{2}}\leq R$ by the assumption
$f\in\mathsf{D}_{R}^{f}$. In the mean time,

$\begin{split}\int_{\Omega}(\nabla u)^{\mathrm{T}}\boldsymbol{a}\nabla
u+cu^{2}\,\mathrm{d}\boldsymbol{x}\geq\int_{\Omega}(\nabla
u)^{\mathrm{T}}\boldsymbol{a}\nabla
u\,\mathrm{d}\boldsymbol{x}\geq\int_{\Omega}r(\nabla u)^{\mathrm{T}}\nabla
u\,\mathrm{d}\boldsymbol{x}=r\|u\|_{H_{0}^{1}}^{2},\end{split}$

since $\boldsymbol{a}\in\Sigma^{d}_{r,R}(\Omega)$ and $c(\boldsymbol{x})\geq
0$. Combining these two inequalities would give $\|u\|_{H_{0}^{1}}\leq
C_{\text{P}}R/r$. ∎

###### Proof of Proposition 4.1.

Similar to (4.3), the alternative weak solution $\bar{u}\in H_{0}^{1}(\Omega)$
should satisfy

(5.1)
$\int_{\Omega}\left((\nabla\bar{u})^{\mathrm{T}}\boldsymbol{\bar{a}}\nabla
v+\bar{c}\bar{u}v\right)\,\mathrm{d}\boldsymbol{x}=\int_{\Omega}\bar{f}v\,\mathrm{d}\boldsymbol{x}.$

for any $v\in H_{0}^{1}(\Omega)$. Subtracting (5.1) from (4.3) would give

$\begin{split}&\quad\int_{\Omega}(\nabla
u-\nabla\bar{u})^{\mathrm{T}}\boldsymbol{a}\nabla
v+c(u-\bar{u})v\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega}(\nabla\bar{u})^{\mathrm{T}}(\boldsymbol{\bar{a}}-\boldsymbol{a})\nabla
v+(\bar{c}-c)\bar{u}v+(f-\bar{f})v\,\mathrm{d}\boldsymbol{x}.\end{split}$

Taking $v=w=u-\bar{u}$, we have

(5.2)
$\begin{split}&\quad\int_{\Omega}(\nabla\bar{u})^{\mathrm{T}}(\boldsymbol{\bar{a}}-\boldsymbol{a})\nabla
w\,\mathrm{d}\boldsymbol{x}\\\
&\leq\int_{\Omega}\|\nabla\bar{u}\|_{2}\cdot\|\boldsymbol{\bar{a}}(x)-\boldsymbol{a}(x)\|_{2}\cdot\|\nabla
w\|_{2}\,\mathrm{d}\boldsymbol{x}\\\
&\leq\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}\int_{\Omega}\|\nabla\bar{u}\|_{2}\cdot\|\nabla
w\|_{2}\,\mathrm{d}\boldsymbol{x}\\\
&\leq\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}\|\bar{u}\|_{H_{0}^{1}}\|w\|_{H_{0}^{1}}\\\
&\leq\frac{C_{\text{P}}R}{r}\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}\|w\|_{H_{0}^{1}}\quad\text{by Prop.\ref{lem:ellpH01u}},\end{split}$
$\begin{split}&\quad\int_{\Omega}(\bar{c}-c)\bar{u}w+(f-\bar{f})w\,\mathrm{d}\boldsymbol{x}\\\
&\leq\|\bar{c}-c\|_{L^{\infty}}\|\bar{u}\|_{L^{2}}\|w\|_{L^{2}}+\|\bar{f}-f\|_{L^{2}}\|w\|_{L^{2}}\\\
&\leq
C_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}}\|\bar{u}\|_{H_{0}^{1}}\|w\|_{H_{0}^{1}}+C_{\text{P}}\|\bar{f}-f\|_{L^{2}}\|w\|_{H_{0}^{1}}\\\
&\leq\frac{C_{\text{P}}}{r}\Bigl{(}RC_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}}+r\|\bar{f}-f\|_{L^{2}}\Bigr{)}\|w\|_{H_{0}^{1}},\end{split}$

and thus

$\begin{split}&\quad\int_{\Omega}(\nabla w)^{\mathrm{T}}\boldsymbol{a}\nabla
w+cw^{2}\,\mathrm{d}\boldsymbol{x}\\\
&\leq\frac{C_{\text{P}}}{r}\Bigl{(}R\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{d\times
d}))}+RC_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}}\\\ &\hskip
40.00006pt+r\|\bar{f}-f\|_{L^{2}}\Bigr{)}\|w\|_{H_{0}^{1}}.\end{split}$

The condition $\boldsymbol{a}\in\Sigma^{d}_{r,R}(\Omega)$ and
$c(\boldsymbol{x})\geq 0$ further guarantees

$\int_{\Omega}(\nabla w)^{\mathrm{T}}\boldsymbol{a}\nabla
w+cw^{2}\,\mathrm{d}\boldsymbol{x}\geq r\int_{\Omega}(\nabla
w)^{\mathrm{T}}\nabla w\,\mathrm{d}\boldsymbol{x}=r\|w\|_{H_{0}^{1}}^{2},$

and we conclude the proof by combining the two inequalities. ∎

If $Z_{\Lambda}$ is a convex compact subset of $Z=\mathbb{R}^{n}$ (with
$\ell^{2}$-norm by default), the metric projection $P_{Z_{\Lambda}}:Z\to
Z_{\Lambda}$ is given by

(5.3) $P_{Z_{\Lambda}}(z)=\operatorname*{arg\,min}_{\bar{z}\in
Z_{\Lambda}}\|\bar{z}-z\|_{2},$

which is 1-Lipschitz continuous.

###### Proof of the assertion in Example 4.3.

We first show that $d_{K^{2},K/2}^{\text{rDeco}}(\Lambda)=0$ holds. Let
$Z=\mathbb{R}^{K^{2}}$, $Z_{\Lambda}=[-1/K,1/K]^{K^{2}}$. The mapping

$D_{1}:Z_{\Lambda}\to(\mathbb{R}^{K\times K},\|\cdot\|_{F}),\quad
z\mapsto\lambda$

given as $\lambda_{ij}=\frac{3}{2}+\frac{K}{2}z_{Ki+j-K}$ is
$\frac{K}{2}$-Lipschitz continuous, and $D_{1}(Z_{\Lambda})=\Lambda$ holds.
Take $P_{Z_{\Lambda}}:Z\to Z_{\Lambda}$ to be the metric projection given by
5.3. Then $D=D_{1}\circ P_{Z_{\Lambda}}$ is also $\frac{K}{2}$-Lipschitz
continuous, and $D(Z_{B})=\Lambda$ since $Z_{\Lambda}\subset Z_{B}$. The
assertion $d_{K^{2},K/2}^{\text{rDeco}}(\Lambda)=0$ then follows from the
definition of the restricted decoder width.

Now we consider the variable coefficients of the PDEs. It is direct to check
that $\boldsymbol{a}^{\lambda}\in\Sigma_{1,2}^{2}(\Omega)$ holds for any
$\lambda\in\Lambda$. Given $\lambda,\bar{\lambda}\in\Lambda$, we have

$\begin{split}\|\boldsymbol{a}^{\lambda}-\boldsymbol{a}^{\bar{\lambda}}\|_{L^{\infty}(\Omega;\ell^{2}(\mathbb{R}^{2\times
2}))}&=\operatorname*{ess\,sup}_{\boldsymbol{x}\in\Omega}\left\lVert\biggl{(}\sum_{i,j=1}^{K}(\lambda_{ij}-\bar{\lambda}_{ij})\chi_{D_{ij}}(\boldsymbol{x})\biggr{)}I_{2}\right\rVert_{2}\\\
&=\operatorname*{ess\,sup}_{\boldsymbol{x}\in\Omega}\left\lvert\sum_{i,j=1}^{K}(\lambda_{ij}-\bar{\lambda}_{ij})\chi_{D_{ij}}(\boldsymbol{x})\right\rvert\\\
&=\max_{i,j}|\lambda_{ij}-\bar{\lambda}_{ij}|\quad\text{(as }D_{ij}\text{'s
are disjoint)}\\\ &\leq\|\lambda-\bar{\lambda}\|_{F},\end{split}$

and the mapping
$S_{\Lambda}:\Lambda\to\mathcal{A}_{0},\lambda\mapsto(\boldsymbol{a}^{\lambda},c,f)$
is therefore $R$-Lipschitz if we endow $\Lambda$ with the matrix Frobenius
norm $\|\cdot\|_{F}$. By Proposition 3.1, the set of possible parameters
$\mathcal{A}=S_{\Lambda}(\Lambda)$ satisfies
$d_{K^{2},KR/2}^{\text{rDeco}}(\mathcal{A})=0$. We then apply Theorem 4.2, and
insert $r=1,R=2$ to conclude the proof. ∎

###  5.3. Parabolic Equations on a Fixed Domain

######  Lemma 5.2.

For all $u\in H_{0}^{1}(\Omega)$, $t\in[0,T]$,

$\frac{r}{2}\|u\|_{H_{0}^{1}(\Omega)}^{2}\leq A[u,u;t]+C_{B}\|u\|_{L^{2}}^{2}$

holds, where $C_{B}=R+\frac{R^{2}}{2r}.$

###### Proof.

Omitting the time $t$ for simplicity, we have

$\begin{split}r\|u\|_{H_{0}^{1}(\Omega)}^{2}&=\int_{\Omega}r\|\nabla
u\|_{2}^{2}\,\mathrm{d}x\\\ &\leq\int_{\Omega}(\nabla
u)^{\mathrm{T}}\boldsymbol{a}(\nabla u)\,\mathrm{d}\boldsymbol{x}\\\
&=A[u,u]-\int_{\Omega}(u\boldsymbol{b}^{\mathrm{T}}(\nabla
u)+cu^{2})\,\mathrm{d}\boldsymbol{x}.\end{split}$

The second term in the last equation above can be estimated as

$\begin{split}-\int_{\Omega}u\boldsymbol{b}^{\mathrm{T}}(\nabla
u)\,\mathrm{d}\boldsymbol{x}&\leq\int_{\Omega}\|b\|_{2}\cdot\|\nabla
u\|_{2}\cdot|u|\,\mathrm{d}\boldsymbol{x}\\\ &\leq R\int_{\Omega}\|\nabla
u\|_{2}\cdot|u|\,\mathrm{d}\boldsymbol{x}\\\
&\leq\frac{r}{2}\int_{\Omega}\|\nabla
u\|_{2}^{2}\,\mathrm{d}\boldsymbol{x}+\frac{R^{2}}{2r}\int_{\Omega}u^{2}\,\mathrm{d}\boldsymbol{x}\\\
&=\frac{r}{2}\|u\|_{H_{0}^{1}(\Omega)}^{2}+\frac{R^{2}}{2r}\|u\|_{L^{2}}^{2},\end{split}$

and the third term as

$\begin{split}-\int_{\Omega}cu^{2}\,\mathrm{d}\boldsymbol{x}&\leq\|c\|_{L^{\infty}}\int_{\Omega}u^{2}\,\mathrm{d}\boldsymbol{x}\\\
&\leq R\|u\|_{L^{2}}^{2}.\end{split}$

Combining the inequalities above, the lemma is proved. ∎

######  Lemma 5.3.

The weak solution to (4.4) satisfies

$\|\mathbf{u}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}\leq
C_{1}(\|\mathbf{f}\|_{L^{2}([0,T];H^{-1}(\Omega))}+\|g\|_{L^{2}}),$

where $C_{1}>0$ is a constant depending only on $r,R$ and $T$.

###### Proof.

Taking $v=\mathbf{u}(t)$ in (4.5), we have

$\langle\mathbf{u}^{\prime}(t),\mathbf{u}(t)\rangle+A[\mathbf{u}(t),\mathbf{u}(t);t]=\langle\mathbf{f}(t),\mathbf{u}(t)\rangle.$

This gives

(5.4)
$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\bigl{(}\|\mathbf{u}(t)\|_{L^{2}}^{2}\bigr{)}&=2\langle\mathbf{u}^{\prime}(t),\mathbf{u}(t)\rangle\\\
&=2\langle\mathbf{f}(t),\mathbf{u}(t)\rangle-2A[\mathbf{u}(t),\mathbf{u}(t);t]\\\
&\leq
2\|\mathbf{f}(t)\|_{H^{-1}(\Omega)}\|\mathbf{u}(t)\|_{H_{0}^{1}(\Omega)}-2A[\mathbf{u}(t),\mathbf{u}(t);t]\\\
&\leq\frac{r}{2}\|\mathbf{u}(t)\|_{H_{0}^{1}(\Omega)}^{2}+\frac{2}{r}\|\mathbf{f}(t)\|_{H^{-1}(\Omega)}^{2}-2A[\mathbf{u}(t),\mathbf{u}(t);t]\\\
&\leq-\frac{r}{2}\|\mathbf{u}(t)\|_{H_{0}^{1}(\Omega)}^{2}+\frac{2}{r}\|\mathbf{f}(t)\|_{H^{-1}(\Omega)}^{2}+2C_{B}\|\mathbf{u}(t)\|_{L^{2}}^{2}.\end{split}$

The rest part of the proof has much in common with that of [14, Theorem 2,
Section 7.1.2], and we include the details for conveniece to the readers. We
denote $\alpha(t)=\|\mathbf{u}(t)\|_{L^{2}}^{2}$,
$\beta(t)=\frac{2}{r}\|\mathbf{f}(t)\|_{H^{-1}(\Omega)}^{2}$. Then (5.4)
implies $\alpha^{\prime}(t)\leq 2C_{B}\alpha(t)+\beta(t)$, and Gronwall’s
inequality yields the estimate

$\alpha(t)\leq\exp(2C_{B}t)\left(\alpha(0)+\int_{0}^{t}\beta(s)\,\mathrm{d}s\right).$

As $\alpha(0)=\|\mathbf{u}(0)\|_{L^{2}}^{2}=\|g\|_{L^{2}}^{2}$, we have

$\alpha(t)\leq\exp(2C_{B}T)\left(\|g\|_{L^{2}}^{2}+\frac{2}{r}\|\mathbf{f}\|_{L^{2}([0,T];H^{-1}(\Omega))}^{2}\right)$

for any $t\in[0,T]$. We integrate (5.4) from $0$ to $T$, and employ the
inequality derived above to get

$\|\mathbf{u}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}^{2}\leq
C_{1}^{2}(\|\mathbf{f}\|_{L^{2}([0,T];H^{-1}(\Omega))}^{2}+\|g\|_{L^{2}}^{2}),$

which directly implies the assertion of the lemma. ∎

###### Proof of Proposition 4.4.

We denote the bilinear forms corresponding to the two sets of coefficient
fields as $A[\cdot,\cdot;t]$ and $\bar{A}[\cdot,\cdot;t]$, respectively. Now
for any $v\in H_{0}^{1}(\Omega)$, we let
$\mathbf{w}=\mathbf{u}-\bar{\mathbf{u}}$, and substract the two equations

$\begin{split}\langle\mathbf{u}^{\prime}(t),v\rangle+A[\mathbf{u}(t),v;t]&=\langle\mathbf{f}(t),v\rangle,\\\
\langle\bar{\mathbf{u}}^{\prime}(t),v\rangle+\bar{A}[\bar{\mathbf{u}}(t),v;t]&=\langle\bar{\mathbf{f}}(t),v\rangle\end{split}$

to get

(5.5)
$\langle\mathbf{w}^{\prime}(t),v\rangle+A[\mathbf{w}(t),v;t]=\langle\mathbf{f}_{1}(t),v\rangle,$

where $\mathbf{f}_{1}:[0,T]\to H^{-1}(\Omega)$ is given by

(5.6)
$\langle\mathbf{f}_{1}(t),v\rangle=\bar{A}[\bar{\mathbf{u}}(t),v;t]-A[\bar{\mathbf{u}}(t),v;t]+\langle(\mathbf{f}-\bar{\mathbf{f}})(t),v\rangle$

for any $v\in H_{0}^{1}(\Omega)$. Now

$\begin{split}&\quad\bar{A}[u,v;t]-A[u,v;t]\\\ &=\int_{\Omega}((\nabla
v)^{\mathrm{T}}(\boldsymbol{a}-\boldsymbol{\bar{a}})(\nabla u)\\\
&\quad+v(\boldsymbol{\bar{b}}-\boldsymbol{b})^{\mathrm{T}}(\nabla
u)+(\bar{c}-c)uv)\,\mathrm{d}\boldsymbol{x}\\\
&\leq\int_{\Omega}\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{2}\|\nabla
u\|_{2}\|\nabla v\|_{2}\,\mathrm{d}\boldsymbol{x}\\\
&\quad+\int_{\Omega}\bigl{(}\|\boldsymbol{\bar{b}}-\boldsymbol{b}\|_{2}\|\nabla
u\|_{2}|v|+|\bar{c}-c|\cdot|u|\cdot|v|\bigr{)}\,\mathrm{d}\boldsymbol{x}\\\
&\leq\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d\times
d}))}\|u\|_{H_{0}^{1}(\Omega)}\|v\|_{H_{0}^{1}(\Omega)}\\\
&\quad+\|\boldsymbol{\bar{b}}-\boldsymbol{b}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d}))}\|u\|_{H_{0}^{1}(\Omega)}\|v\|_{L^{2}}\\\
&\quad+\|\bar{c}-c\|_{L^{\infty}(\Omega_{T})}\|u\|_{L^{2}}\|v\|_{L^{2}}\\\
&\leq\delta_{abc}\|u\|_{H_{0}^{1}(\Omega)}\|v\|_{H_{0}^{1}(\Omega)},\end{split}$

where

(5.7)
$\begin{split}\delta_{abc}&=\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d\times
d}))}+C_{\text{P}}\|\boldsymbol{\bar{b}}-\boldsymbol{b}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d}))}\\\
&\quad+C_{\text{P}}^{2}\|\bar{c}-c\|_{L^{\infty}(\Omega_{T})},\end{split}$

and $C_{\text{P}}>0$ is the Poincaré constant of the domain $\Omega$. By
(5.6), we have

$\begin{split}\|\mathbf{f}_{1}(t)\|_{H^{-1}(\Omega)}&=\sup_{\|v\|_{H_{0}^{1}(\Omega)}=1}\langle\mathbf{f}_{1}(t),v\rangle\\\
&\leq\delta_{abc}\|\bar{\mathbf{u}}(t)\|_{H_{0}^{1}(\Omega)}+\|\mathbf{f}(t)-\bar{\mathbf{f}}(t)\|_{H^{-1}(\Omega)},\end{split}$

which holds for all $t\in[0,T]$. These are non-negative functions with respect
to $t$, and taking $L^{2}([0,T])$-norm gives

(5.8)
$\|\mathbf{f}_{1}\|_{L^{2}([0,T];H^{-1}(\Omega))}\leq\delta_{abc}\|\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}+\|\mathbf{f}-\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}.$

By Lemma 5.3 and the condition
$\|\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}\leq R$,
$\|\bar{g}\|_{L^{2}}\leq R$, we know

(5.9) $\|\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}\leq 2C_{1}R.$

Applying Lemma 5.3 to (5.5) with initial condition
$\mathbf{w}(0)=\mathbf{u}(0)-\bar{\mathbf{u}}(0)=g-\bar{g}$ would give

$\begin{split}&\quad\|\mathbf{u}-\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}\\\
&=\|\mathbf{w}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}\\\ &\leq
C_{1}(\|\mathbf{f}_{1}\|_{L^{2}([0,T];H^{-1}(\Omega))}+\|g-\bar{g}\|_{L^{2}})\\\
&\leq
C_{1}\delta_{abc}\|\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}+C_{1}\|\mathbf{f}-\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}\\\
&\quad+C_{1}\|g-\bar{g}\|_{L^{2}}\hskip
60.00009pt\text{(by~{}\eqref{eq:parab_f1est})}\\\ &\leq
2RC_{1}^{2}\|\boldsymbol{a}-\boldsymbol{\bar{a}}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d\times
d}))}+2RC_{1}^{2}C_{\text{P}}\|\boldsymbol{b}-\boldsymbol{\bar{b}}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d}))}\\\
&\quad+2RC_{1}^{2}C_{\text{P}}^{2}\|c-\bar{c}\|_{L^{\infty}(\Omega_{T})}+C_{1}\|\mathbf{f}-\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}\\\
&\quad+C_{1}\|g-\bar{g}\|_{L^{2}}\hskip
50.00008pt\text{(by~{}\eqref{eq:parab_wu_est}
and~{}\eqref{eq:parab_dabc_def})}.\end{split}$

The proof is completed by taking
$C=\max(2RC_{1}^{2},2RC_{1}^{2}C_{\text{P}}^{2},C_{1})$. ∎

###### Proof of the assertion in Example 4.6.

(1) Given $\lambda\in\Lambda_{K}$, we denote
$S_{\Lambda}(\lambda)=(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)$. Since
$\boldsymbol{\phi}_{k}^{b}$, $k=1,\dots,K$ have disjoint essential supports,
$|\lambda_{k}^{b}|\leq\sqrt{\lambda_{k}^{2}}\leq 1$, and
$\boldsymbol{\phi}_{k}^{b}\in B_{R}^{d}(\Omega_{T})$, we have

(5.10)
$\begin{split}\|\boldsymbol{b}(t,\boldsymbol{x})\|_{2}&=\left\lVert\sum_{k=1}^{K}\lambda_{k}^{b}\boldsymbol{\phi}_{k}^{b}(t,\boldsymbol{x})\right\rVert_{2}=\max_{1\leq
k\leq
K}\left\lVert\lambda_{k}^{b}\boldsymbol{\phi}_{k}^{b}(t,\boldsymbol{x})\right\rVert_{2}\\\
&\leq\max_{1\leq k\leq
K}\left\lVert\boldsymbol{\phi}_{k}^{b}(t,\boldsymbol{x})\right\rVert_{2}\leq
R\end{split}$

for a.e. $(t,x)\in\Omega_{T}$, and therefore

(5.11)
$\|\boldsymbol{b}\|_{L^{\infty}(\Omega_{T};\ell^{2}(\mathbb{R}^{d}))}=\operatorname*{ess\,sup}_{t\in[0,T],\boldsymbol{x}\in\Omega}\|\boldsymbol{b}(t,\boldsymbol{x})\|_{2}\leq
R.$

Similar arguments show $\|c\|_{L^{\infty}(\Omega_{T})}\leq R$, and

$|\boldsymbol{a}^{ii}(t,\boldsymbol{x})-\tfrac{R+r}{2}|\leq\max_{1\leq k\leq
K}|\boldsymbol{\phi}_{k}^{a}(t,\boldsymbol{x})_{i}|\leq\max_{1\leq k\leq
K}\|\boldsymbol{\phi}_{k}^{a}(t,\boldsymbol{x})\|_{2}\leq\tfrac{R-r}{2}$

for $i=1,\dots,d$, a.e. $(t,\boldsymbol{x})\in\Omega_{T}$. Note that the
latter conclusion would imply $\boldsymbol{a}\in\Sigma_{r,R}^{d}(\Omega_{T})$
since its off-diagonal entries are zero. Furthermore, by Cauchy’s inequality
we have

(5.12)
$\begin{split}\|\mathbf{f}\|_{L^{2}([0,T];H^{-1}(\Omega))}^{2}&=\left\lVert\sum_{k=1}^{K}\lambda_{k}^{\mathbf{f}}\phi_{k}^{\mathbf{f}}(t)\right\rVert_{L^{2}([0,T];H^{-1}(\Omega))}^{2}\\\
&\leq\biggl{(}\sum_{k=1}^{K}|\lambda_{k}^{\mathbf{f}}|\cdot\|\phi_{k}^{\mathbf{f}}(t)\|_{L^{2}([0,T];H^{-1}(\Omega))}\biggr{)}^{2}\\\
&\leq\biggl{(}\sum_{k=1}^{K}(\lambda_{k}^{\mathbf{f}})^{2}\biggr{)}\biggl{(}\sum_{k=1}^{K}\left\lVert\phi_{k}^{\mathbf{f}}(t)\right\rVert_{L^{2}([0,T];H^{-1}(\Omega))}^{2}\biggr{)}\\\
&\leq\biggl{(}\sum_{k=1}^{K}\lambda_{k}^{2}\biggr{)}R^{2}\\\ &\leq
R^{2},\end{split}$

and similarly $\|g\|_{L^{2}(\Omega)}\leq R$. We conclude that
$(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)\in\mathcal{A}_{0}$.

(2) Let $\lambda,\bar{\lambda}\in\Lambda_{K}$,
$S_{\Lambda}(\lambda)=(\boldsymbol{a},\boldsymbol{b},c,\mathbf{f},g)$ and
$S_{\Lambda}(\bar{\lambda})=(\boldsymbol{\bar{a}},\boldsymbol{\bar{b}},\bar{c},\bar{\mathbf{f}},\bar{g})$.
Similar to (5.10) and (5.11), we have

(5.13)
$\begin{split}\|\boldsymbol{b}-\boldsymbol{\bar{b}}\|_{L^{\infty}(\Omega_{T};\ell^{2}(\mathbb{R}^{d}))}&=\max_{1\leq
k\leq
K}\left\lVert(\lambda_{k}^{b}-\bar{\lambda}_{k}^{b})\boldsymbol{\phi}_{k}^{b}(t,\boldsymbol{x})\right\rVert_{L^{\infty}(\Omega_{T};\ell^{2}(\mathbb{R}^{d}))}\\\
&\leq\max_{1\leq k\leq
K}\|\lambda-\bar{\lambda}\|\cdot\left\lVert\boldsymbol{\phi}_{k}^{b}(t,\boldsymbol{x})\right\rVert_{L^{\infty}(\Omega_{T};\ell^{2}(\mathbb{R}^{d}))}\\\
&\leq R\|\lambda-\bar{\lambda}\|,\end{split}$

and $\|c-\bar{c}\|_{L^{2}(\Omega_{T})}\leq R\|\lambda-\bar{\lambda}\|$. The
relation

$|\boldsymbol{a}^{ii}(t,\boldsymbol{x})-\boldsymbol{\bar{a}}^{ii}(t,\boldsymbol{x})|\leq\tfrac{R-r}{2}\|\lambda-\bar{\lambda}\|\leq
R\|\lambda-\bar{\lambda}\|$

holds for $i=1,\dots,d$, a.e. $(t,\boldsymbol{x})\in\Omega_{T}$, which implies
$\|\boldsymbol{a}-\boldsymbol{\bar{a}}\|_{L^{\infty}(\Omega_{T},\ell^{2}(\mathbb{R}^{d\times
d}))}\leq R\|\lambda-\bar{\lambda}\|$ since $\boldsymbol{a}$ and
$\boldsymbol{\bar{a}}$ are diagonal matrices. Furthermore, arguments analogous
to (5.12) would show
$\|\mathbf{f}-\bar{\mathbf{f}}\|_{L^{2}([0,T];H^{-1}(\Omega))}^{2}\leq
R\|\lambda-\bar{\lambda}\|$ and $\|g-\bar{g}\|_{L^{2}(\Omega)}\leq
R\|\lambda-\bar{\lambda}\|$. Combining the inequalities above, Proposition 4.4
indicates

$\|\mathbf{u}-\bar{\mathbf{u}}\|_{L^{2}([0,T];H_{0}^{1}(\Omega))}\leq
5CR\|\lambda-\bar{\lambda}\|,$

where $\mathbf{u}=S_{0}(S_{\Lambda}(\lambda))$ and
$\bar{\mathbf{u}}=S_{0}(S_{\Lambda}(\bar{\lambda}))$. We conclude that the
mapping $S_{0}\circ S_{\Lambda}$ is Lipschitz continuous with Lipschitz
constant $5CR$.

(3) According to the definition of the decoder width, it is easy to see that
$d_{5K,1}^{\text{rDeco}}(\Lambda_{K})=0$. In fact, we may view $\Lambda_{K}$
as the closed unit ball of $Z=\mathbb{R}^{5K}$, and take
$D=P_{\Lambda_{K}}:Z\to\Lambda_{K}$ to be the metric projection 5.3.
Proposition 3.1 then implies $d_{5K,5CR}^{\text{Deco}}(S_{0}\circ
S_{\Lambda}(\Lambda_{K}))=0$. ∎

###### Proof of the assertion in Example 4.7.

We have $d_{5n,1}^{\text{rDeco}}(\Lambda_{w})\leq\sqrt{w_{n}}$. In fact, given
$n>0$, we may take $D_{1}:Z=\mathbb{R}^{5n}\to\\{\lambda\\}$ to be an
inclusion that fills the entries $\lambda_{k}^{a},\dots,\lambda_{k}^{g}$ for
$k=1,\dots,n$. Denote $Z_{w}=D_{1}^{-1}(\Lambda_{w})$ to be the preimage of
$\Lambda_{w}$. It is easy to check that $Z_{w}$ is a convex subset of $Z$.
$Z_{w}\subseteq Z_{B}$ holds, since for any $z\in Z_{w}$,
$\lambda=D_{1}(z)\in\Lambda_{w}$ satisfies $\lambda_{k}^{2}=0$ for $k>n$, and
we have

$\|z\|_{2}^{2}=\sum_{k=1}^{n}\lambda_{k}^{2}\leq\sum_{k=1}^{n}\frac{\lambda_{k}^{2}}{w_{k}}=\sum_{k=1}^{\infty}\frac{\lambda_{k}^{2}}{w_{k}}\leq
1.$

For any $\lambda\in\Lambda_{w}$, we can find $z\in Z_{w}=P_{Z_{w}}(Z_{B})$
satisfying

$\|\lambda-D_{1}(z)\|^{2}=\sum_{k>n}\lambda_{k}^{2}\leq
w_{n}\sum_{k>n}\frac{\lambda_{k}^{2}}{w_{k}}\leq
w_{n}\sum_{k=1}^{\infty}\frac{\lambda_{k}^{2}}{w_{k}}\leq w_{n}.$

Then $D=D_{1}\circ P_{Z_{w}}:Z\to\Lambda_{w}$ can be chosen as the specific
decoder mapping, and $d_{5n,1}^{\text{rDeco}}(\Lambda_{w})\leq\sqrt{w_{n}}$
follows from the definition of the restricted decoder width.

Similar to Example 4.6, it can be shown that the mapping $S_{0}\circ
S_{\Lambda}:\Lambda_{w}\to L^{2}([0,T];H_{0}^{1}(\Omega))$ is $5CR$-Lipschitz
continuous. Proposition 3.1 then implies

$d_{5n,5CR}^{\text{Deco}}(S_{0}\circ S_{\Lambda}(\Lambda_{w}))\leq
5CR\sqrt{w_{n}}.\qed$

###  5.4. Elliptic Equations on a Variable Domain

####  5.4.1. Preparation: Solutions on Two Domains

We shall first focus on the simpler case where only two possible domains
$\Omega$ and $\Omega_{2}$ are considered, and estimate the distance between
the corresponding solutions $u,u_{2}$. This analysis forms the basis of the
subsequent proofs in Section 5.4, where we will investigate the more general
case involving a whole class of domains.

Assume $\Omega\subseteq\Omega^{\text{m}}$ is a fixed subdomain, $\beta$ is a
$C^{1}$-diffeomorphism from $\bar{\Omega}^{\text{m}}$ onto itself. Denote
$\Omega_{2}=\beta(\Omega)$,

$\delta_{\beta}=\|\beta-\mathrm{id}\|_{C^{1}(\bar{\Omega}^{\text{m}};\mathbb{R}^{d})}=\max_{\boldsymbol{x}\in\bar{\Omega}^{\text{m}}}\max\left(\|\beta(\boldsymbol{x})-\boldsymbol{x}\|_{2},\left\|\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d}\right\|_{2}\right),$

where $\mathrm{id}$ is the identity mapping from $\Omega^{\text{m}}$ to
itself. The weak solution to (4.8) is denoted as $u\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega)$, and the solution with
the inner region $\Omega$ replaced by $\Omega_{2}$ is denoted as $u_{2}\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$. We state the main
theorem for this section (Section 5.4.1) as follows:

######  Theorem 5.4.

Under Assumption 4.8, we have $\|u-u_{2}\|_{L^{2}(\Omega^{\text{m}})}\leq
C\delta_{\beta}/2$ provided
$\delta_{\beta}\leq\delta_{0}\triangleq\min(1-(8/9)^{2/d},1/9)$, where the
constant $C$ is the same as in Proposition 4.10

We let $\tilde{u}(\boldsymbol{x})=u(\beta^{-1}(\boldsymbol{x}))$, and it is
easy to verify that $\tilde{u}\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$ holds. To prove
Theorem 5.4, we shall estimate $\|u-\tilde{u}\|_{L^{2}(\Omega^{\text{m}})}$
and $\|\tilde{u}-u_{2}\|_{L^{2}(\Omega^{\text{m}})}$ seperately. For the
former term, we construct a continuous transition between $u$ and $\tilde{u}$
in $L^{2}(\Omega^{\text{m}})$ by using an interpolation between $\beta$ and
the identity mapping on $\bar{\Omega}^{\text{m}}$ (see Proposition 5.8 and
Corollary 5.9). For the latter term, we derive the PDE that $\tilde{u}$
satisfies (Lemma 5.10), and then the upper bound can be given as in the fixed
domain case (Proposition 4.1).

The detailed proof will be presented in the following part of this section. We
assume Assumption 4.8 holds unless otherwise specified.

######  Lemma 5.5.

Let $Q\in\mathbb{R}^{d\times d}$ be a matrix. Then
$\det(I+Q)\leq(1+\alpha)^{d}$, where $\alpha=\|Q\|_{2}$. If $\alpha<1$, we
further have $\det(I+Q)\geq(1-\alpha)^{d}$.

###### Proof.

If $Q$ is symmetric, we may let $Q=P^{\mathrm{T}}\Lambda P$, where $P$ is
orthogonal and $\Lambda$ is diagonal. Then $\alpha=\|P^{\mathrm{T}}\Lambda
P\|_{2}=\|\Lambda\|_{2}$. By the definition of $\ell^{2}$-norm, we have
$\|\Lambda\|_{2}=\max_{i}|\Lambda_{ii}|$ since it is diagonal, and

$\begin{split}\det(I+Q)&=\det(I+P^{\mathrm{T}}\Lambda
P)=\det(P^{\mathrm{T}}(I+\Lambda)P)\\\
&=\det(I+\Lambda)=\prod_{i=1}^{d}(1+\Lambda_{ii})\\\
&\leq\prod_{i=1}^{d}|1+\Lambda_{ii}|\leq(\max_{i}|1+\Lambda_{ii}|)^{d}\\\
&\leq(1+\max_{i}|\Lambda_{ii}|)^{d}=(1+\|\Lambda\|_{2})^{d}\\\
&=(1+\alpha)^{d}.\end{split}$

For non-symmetric $Q$,

$\det(I+Q)^{2}=\det(I+Q)\det(I+Q^{\mathrm{T}})=\det(I+Q+Q^{\mathrm{T}}+QQ^{\mathrm{T}}).$

The matrix $Q_{1}=Q+Q^{\mathrm{T}}+QQ^{\mathrm{T}}$ is symmetric, and

$\|Q_{1}\|_{2}\leq\|Q\|_{2}+\|Q^{\mathrm{T}}\|_{2}+\|QQ^{\mathrm{T}}\|_{2}\leq
2\|Q\|_{2}+\|Q\|_{2}^{2}=2\alpha+\alpha^{2}.$

Then $\det(I+Q_{1})\leq(1+2\alpha+\alpha^{2})^{d}=(1+\alpha)^{2d}$ by what we
have proved. This shows $\det(I+Q)\leq(1+\alpha)^{d}$.

When $\alpha<1$, it is easy to verify that the sum

$I-Q+Q^{2}-Q^{3}+\cdots=(I+Q)^{-1}$

is convergent. We let $Q_{2}=(I+Q)^{-1}-I$, and

$\begin{split}\|Q_{2}\|_{2}&=\left\lVert-Q+Q^{2}-Q^{3}+\cdots\right\rVert_{2}\\\
&\leq\|Q\|_{2}+\|Q^{2}\|_{2}+\|Q^{3}\|_{2}+\cdots\\\
&\leq\alpha+\alpha^{2}+\alpha^{3}+\cdots=\frac{\alpha}{1-\alpha},\end{split}$
$\det(I+Q_{2})\leq(1+\|Q_{2}\|_{2})^{d}\leq\left(1+\frac{\alpha}{1-\alpha}\right)^{d}=\frac{1}{(1-\alpha)^{d}}.$

The conclusion follows from $\det(I+Q)=1/\det(I+Q_{2})$. ∎

######  Corollary 5.6.

Let
$J(\boldsymbol{x})=\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}$
be the Jacobian matrix of $\beta$ at $\boldsymbol{x}\in\Omega^{\text{m}}$.
When $\delta_{\beta}\leq\delta_{0}$, we have

$(1-\delta_{\beta})^{d}\leq\det J(\boldsymbol{x})\leq(1+\delta_{\beta})^{d}.$

###### Proof.

Note that
$\|J(\boldsymbol{x})-I_{d}\|_{2}=\left\|\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d}\right\|_{2}\leq\delta_{\beta}$.
By Lemma 5.5, we have

$\det
J(\boldsymbol{x})=\det(I_{d}+(J(\boldsymbol{x})-I_{d}))\leq(1+\|J(\boldsymbol{x})-I_{d}\|_{2})^{d}\leq(1+\delta_{\beta})^{d}.$

Since $\delta_{\beta}\leq\delta_{0}<1$, the other side of the inequality can
be derived from a similar argument. ∎

######  Proposition 5.7.

We have $\|u\|_{H_{0}^{1}(\Omega^{\text{m}})}\leq C_{\text{P}}R/r$ under
Assumption 4.8, where $C_{\text{P}}$ is the Poincaré constant of
$\Omega^{\text{m}}$. When $\delta_{\beta}\leq\delta_{0}$, we further have
$\|\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}\leq 2C_{\text{P}}R/r$.

###### Proof.

The proof of $\|u\|_{H_{0}^{1}(\Omega^{\text{m}})}\leq C_{\text{P}}R/r$ is the
same as Lemma 5.1 for the case of a fixed domain, in which we may take
$c(\boldsymbol{x})\equiv 0$. Now assume $\delta_{\beta}\leq\delta_{0}$. For
$\boldsymbol{x}\in\Omega^{\text{m}}$, we have
$u(\boldsymbol{x})=\tilde{u}(\beta(\boldsymbol{x}))$, and $\nabla
u(\boldsymbol{x})=J(\boldsymbol{x})^{\mathrm{T}}\nabla\tilde{u}(\beta(\boldsymbol{x}))$.
This gives

$\begin{split}\|\nabla
u(\boldsymbol{x})\|_{2}&\geq\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}-\|(I_{d}-J(\boldsymbol{x}))^{\mathrm{T}}\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}\\\
&\geq\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}-\|J(\boldsymbol{x})-I_{d}\|_{2}\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}\\\
&\geq\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}-\delta_{\beta}\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}\\\
&=(1-\delta_{\beta})\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2},\end{split}$

and

$\begin{split}(1-\delta_{\beta})^{-2}\|u\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2}&=(1-\delta_{\beta})^{-2}\int_{\Omega^{\text{m}}}\|\nabla
u(\boldsymbol{x})\|_{2}^{2}\,\mathrm{d}\boldsymbol{x}\\\
&\geq\int_{\Omega^{\text{m}}}\|\nabla\tilde{u}(\beta(\boldsymbol{x}))\|_{2}^{2}\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}\|\nabla\tilde{u}(\boldsymbol{y})\|_{2}^{2}\det
J(\beta^{-1}(\boldsymbol{y}))^{-1}\,\mathrm{d}\boldsymbol{y}\\\
&\geq(1+\delta_{\beta})^{-d}\int_{\Omega^{\text{m}}}\|\nabla\tilde{u}(\boldsymbol{y})\|_{2}^{2}\,\mathrm{d}\boldsymbol{y}\\\
&=(1+\delta_{\beta})^{-d}\|\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2}\end{split}$

by Corollary 5.6. Then

$\begin{split}\|\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}&\leq(1+\delta_{\beta})^{d/2}(1-\delta_{\beta})^{-1}\|u\|_{H_{0}^{1}(\Omega^{\text{m}})}\\\
&\leq\frac{9}{8}\cdot\left(1-\frac{1}{9}\right)^{-1}\frac{C_{\text{P}}R}{r}\\\
&\leq\frac{2C_{\text{P}}R}{r}.\end{split}$

Here we used the fact that $\delta_{\beta}\leq\delta_{0}\leq 1-(8/9)^{2/d}$
implies $(1+\delta_{\beta})^{d/2}\leq(1-\delta_{\beta})^{-d/2}\leq 9/8$. ∎

######  Proposition 5.8.

Assume $\delta_{\beta}\leq\delta_{0}$. Then for $v\in
H_{0}^{1}(\Omega^{\text{m}})$, we have

$\|v\circ\beta-v\|_{L^{2}(\Omega^{\text{m}})}\leq\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}(1-\delta_{\beta})^{-d/2}\delta_{\beta}\leq
2\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}\delta_{\beta}.$

###### Proof.

Since $C_{0}^{1}(\Omega^{\text{m}})$ is dense in
$H_{0}^{1}(\Omega^{\text{m}})$, we may assume without loss of generality that
$v\in C_{0}^{1}(\Omega^{\text{m}})$. Define
$\beta_{t}(\boldsymbol{x})=\boldsymbol{x}+t(\beta(\boldsymbol{x})-\boldsymbol{x})$
for $t\in[0,1]$. Then $\beta_{0}=\mathrm{id}$, $\beta_{1}=\beta$,
$\beta_{t}(\boldsymbol{x})\in\Omega^{\text{m}}$ by the convexity of
$\Omega^{\text{m}}$, and

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}v(\beta_{t}(\boldsymbol{x}))&=\nabla
v(\beta_{t}(\boldsymbol{x}))\cdot(\beta(\boldsymbol{x})-\boldsymbol{x}),\\\
\left\lvert\frac{\mathrm{d}}{\mathrm{d}t}v(\beta_{t}(\boldsymbol{x}))\right\rvert&\leq|\nabla
v(\beta_{t}(\boldsymbol{x}))|\cdot\|\beta-\mathrm{id}\|_{L^{\infty}(\Omega^{\text{m}};\mathbb{R}^{d})}\\\
&\leq\delta_{\beta}|\nabla v(\beta_{t}(\boldsymbol{x}))|,\\\
(v(\beta(\boldsymbol{x}))-v(\boldsymbol{x}))^{2}&=\left(\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}t}v(\beta_{t}(\boldsymbol{x}))\,\mathrm{d}t\right)^{2}\\\
&\leq\int_{0}^{1}\left(\frac{\mathrm{d}}{\mathrm{d}t}v(\beta_{t}(\boldsymbol{x}))\right)^{2}\,\mathrm{d}t\cdot\int_{0}^{1}1^{2}\,\mathrm{d}t\\\
&\leq\delta_{\beta}^{2}\int_{0}^{1}|\nabla
v(\beta_{t}(\boldsymbol{x}))|^{2}\,\mathrm{d}t.\end{split}$

As
$\|\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d}\|_{2}\leq\delta_{\beta}$
for any $\boldsymbol{x}\in\Omega^{\text{m}}$, the mapping
$\boldsymbol{x}\mapsto\beta(\boldsymbol{x})-\boldsymbol{x}$ is
$\delta_{\beta}$-Lipschitz continuous, and $\beta_{t}(\boldsymbol{x})$ is an
injection. In fact, if
$\beta_{t}(\boldsymbol{x}_{1})=\beta_{t}(\boldsymbol{x}_{2})$ for some
$t\in(0,1)$, then
$\boldsymbol{x}_{2}-\boldsymbol{x}_{1}=t(\beta(\boldsymbol{x}_{1})-\boldsymbol{x}_{1})-t(\beta(\boldsymbol{x}_{2})-\boldsymbol{x}_{2})$,
and $|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|\leq
t\delta_{\beta}|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|$, thereby
$|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}|=0$ since
$t\delta_{\beta}<\delta_{\beta}\leq 1$.

Now for any $\boldsymbol{x}\in\Omega^{\text{m}}$,
$\|\frac{\partial\beta_{t}(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d}\|_{2}=\|t(\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d})\|_{2}\leq
t\delta_{\beta}$, and Lemma 5.5 would imply
$\det\frac{\partial\beta_{t}(\boldsymbol{x})}{\partial\boldsymbol{x}}\geq(1-t\delta_{\beta})^{d}\geq(1-\delta_{\beta})^{d}$.
We have

$\begin{split}\int_{\Omega^{\text{m}}}|\nabla
v(\beta_{t}(\boldsymbol{x}))|^{2}\,\mathrm{d}\boldsymbol{x}&=\int_{\beta_{t}(\Omega^{\text{m}})}|\nabla
v(\boldsymbol{y})|^{2}\left(\det\frac{\partial\beta_{t}}{\partial\boldsymbol{x}}\biggm{|}_{\boldsymbol{x}=\beta_{t}^{-1}(\boldsymbol{y})}\right)^{-1}\,\mathrm{d}\boldsymbol{y}\\\
&\leq\int_{\Omega^{\text{m}}}|\nabla
v(\boldsymbol{y})|^{2}(1-\delta_{\beta})^{-d}\,\mathrm{d}\boldsymbol{y}\\\
&=(1-\delta_{\beta})^{-d}\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2},\\\
\|v\circ\beta-v\|_{L^{2}(\Omega^{\text{m}})}^{2}&=\int_{\Omega^{\text{m}}}(v(\beta(\boldsymbol{x}))-v(\boldsymbol{x}))^{2}\,\mathrm{d}\boldsymbol{x}\\\
&\leq\int_{\Omega^{\text{m}}}\delta_{\beta}^{2}\int_{0}^{1}|\nabla
v(\beta_{t}(\boldsymbol{x}))|^{2}\,\mathrm{d}t\,\mathrm{d}\boldsymbol{x}\\\
&\leq\delta_{\beta}^{2}\int_{0}^{1}(1-\delta_{\beta})^{-d}\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2}\,\mathrm{d}t\\\
&=\delta_{\beta}^{2}(1-\delta_{\beta})^{-d}\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2}.\end{split}$

We use $(1-\delta_{\beta})^{-d/2}\leq 9/8<2$ to conclude the proof. ∎

Taking $v=\tilde{u}$ in Proposition 5.8, a direct combination with Proposition
5.7 gives

######  Corollary 5.9.

Assuming $\delta_{\beta}\leq\delta_{0}$ and Assumption 4.8 holds. Then
$\|u-\tilde{u}\|_{L^{2}(\Omega^{\text{m}})}\leq\frac{4}{r}C_{\text{P}}R\delta_{\beta}$.

######  Lemma 5.10.

$\tilde{u}\in H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$ is the
weak solution to $\tilde{L}\tilde{u}=\tilde{f}$ in
$\Omega^{\text{m}}\setminus\partial\Omega_{2}$, where

$\begin{split}\tilde{L}u(\boldsymbol{y})&=-\nabla_{y}\cdot(\boldsymbol{\tilde{a}}(\boldsymbol{y})\nabla_{y}u(\boldsymbol{y})),\\\
\boldsymbol{\tilde{a}}(\boldsymbol{y})&=\boldsymbol{\hat{a}}(\beta^{-1}(\boldsymbol{y})),\\\
\boldsymbol{\hat{a}}(\boldsymbol{x})&=\frac{J(\boldsymbol{x})\boldsymbol{a}(\boldsymbol{x})J(\boldsymbol{x})^{\mathrm{T}}}{\det
J(\boldsymbol{x})},\\\
\tilde{f}(\boldsymbol{y})&=\frac{f(\beta^{-1}(\boldsymbol{y}))}{\det
J(\beta^{-1}(\boldsymbol{y}))}\end{split}$

for all $\boldsymbol{y}\in\Omega^{\text{m}}$,
$\boldsymbol{x}=\beta^{-1}(\boldsymbol{y})$.

###### Proof.

For any $\tilde{v}\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$, we let
$v(\boldsymbol{x})=\tilde{v}(\beta(\boldsymbol{x}))$, and then

$\nabla_{x}v(\boldsymbol{x})=J(\boldsymbol{x})^{\mathrm{T}}\nabla_{y}\tilde{v}(\beta(\boldsymbol{x})).$

Since $\beta(\boldsymbol{x})$ is a $C^{1}$-diffeomorphism from
$\bar{\Omega}^{\text{m}}$ to itself, $J(\boldsymbol{x})$ is continuous on
$\bar{\Omega}^{\text{m}}$, and then $\|J(\boldsymbol{x})\|_{2}$ must be
bounded. This implies $v\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega)$. We have

$\begin{split}\int_{\Omega^{\text{m}}}f(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}&=\int_{\Omega^{\text{m}}}(\nabla_{x}u(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\nabla_{x}v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}(J(\boldsymbol{x})^{\mathrm{T}}\nabla_{y}\tilde{u}(\beta(\boldsymbol{x})))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})J(\boldsymbol{x})^{\mathrm{T}}\nabla_{y}\tilde{v}(\beta(\boldsymbol{x}))\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}(\nabla_{y}\tilde{u}(\beta(\boldsymbol{x})))^{\mathrm{T}}\boldsymbol{\hat{a}}(\boldsymbol{x})\nabla_{y}\tilde{v}(\beta(\boldsymbol{x}))\det
J(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}(\nabla_{y}\tilde{u}(\boldsymbol{y}))^{\mathrm{T}}\boldsymbol{\hat{a}}(\beta^{-1}(\boldsymbol{y}))\nabla_{y}\tilde{v}(\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}\\\
&=\int_{\Omega^{\text{m}}}(\nabla_{y}\tilde{u}(\boldsymbol{y}))^{\mathrm{T}}\boldsymbol{\tilde{a}}(\boldsymbol{y})\nabla_{y}\tilde{v}(\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}.\end{split}$

Furthermore,

(5.14)
$\begin{split}\int_{\Omega^{\text{m}}}f(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}&=\int_{\Omega^{\text{m}}}\tilde{f}(\beta(\boldsymbol{x}))\tilde{v}(\beta(\boldsymbol{x}))\det
J(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}\tilde{f}(\boldsymbol{y})\tilde{v}(\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}.\end{split}$

Since these equalities hold for any $\tilde{v}\in
H^{1}_{0}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$, the conclusion
follows from the definition of a weak solution. ∎

######  Proposition 5.11.

If $\delta_{\beta}\leq\delta_{0}$, we have
$\|\tilde{f}-f\|_{H^{-1}(\Omega^{\text{m}})}\leq
R\delta_{\beta}(1-\delta_{\beta})^{-d/2}\leq 2R\delta_{\beta}$.

###### Proof.

Assume $v\in H^{1}_{0}(\Omega^{\text{m}})$. Similar to (5.14),

$\langle\tilde{f},v\rangle\triangleq\int_{\Omega^{\text{m}}}\tilde{f}(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}=\int_{\Omega^{\text{m}}}f(\boldsymbol{x})v(\beta(\boldsymbol{x}))\,\mathrm{d}\boldsymbol{x},$

and therefore

$\begin{split}\langle\tilde{f}-f,v\rangle&=\langle f,v\circ\beta-v\rangle\\\
&\leq\|f\|_{L^{2}(\Omega^{\text{m}})}\|v\circ\beta-v\|_{L^{2}(\Omega^{\text{m}})}\\\
&\leq
R\cdot\|v\|_{H_{0}^{1}(\Omega^{\text{m}})}(1-\delta_{\beta})^{-d/2}\delta_{\beta},\end{split}$

where the last inequality makes use of Assumption 4.8(3) and Proposition 5.8.
The conclusion follows immediately by the arbitrariness of $v$. ∎

######  Proposition 5.12.

Assume $\delta_{\beta}<1$. For all $\boldsymbol{x}\in\Omega^{\text{m}}$, we
have

$\left\lVert\frac{J(\boldsymbol{x})}{\sqrt{\det
J(\boldsymbol{x})}}-I\right\rVert_{2}\leq\delta_{\beta}(1-\delta_{\beta})^{-d/2}+(1-\delta_{\beta})^{-d/2}-1.$

Furthermore, if $\delta_{\beta}\leq\delta_{0}$, then

$\left\lVert\frac{J(\boldsymbol{x})}{\sqrt{\det
J(\boldsymbol{x})}}-I\right\rVert_{2}\leq C_{J}\delta_{\beta}\leq\frac{1}{4},$

where $C_{J}=\bigl{(}9+\delta_{0}^{-1}\bigr{)}/8$.

###### Proof.

Making use of Corollary 5.6 and the fact that the function
$\psi(s)=|\frac{1}{\sqrt{s}}-1|$ achieves its maximum at endpoints, we have

$\begin{split}\left\lVert\frac{J(\boldsymbol{x})}{\sqrt{\det
J(\boldsymbol{x})}}-I\right\rVert_{2}&\leq\left\lVert\frac{J(\boldsymbol{x})-I}{\sqrt{\det
J(\boldsymbol{x})}}\right\rVert_{2}+\left\lVert\biggl{(}\frac{1}{\sqrt{\det
J(\boldsymbol{x})}}-1\biggr{)}I\right\rVert_{2}\\\
&=\frac{\|J(\boldsymbol{x})-I\|_{2}}{\sqrt{\det
J(\boldsymbol{x})}}+\left|\frac{1}{\sqrt{\det J(\boldsymbol{x})}}-1\right|\\\
&\leq\delta_{\beta}(1-\delta_{\beta})^{-d/2}+\max((1-\delta_{\beta})^{-d/2}-1,1-(1+\delta_{\beta})^{-d/2})\\\
&=\delta_{\beta}(1-\delta_{\beta})^{-d/2}+(1-\delta_{\beta})^{-d/2}-1.\end{split}$

When $\delta_{\beta}\leq\delta_{0}$, the condition $\delta_{\beta}\leq
1-(8/9)^{2/d}$ implies $(1-\delta_{\beta})^{-d/2}\leq 9/8$. The convexity of
the function $\phi(t)=(1-t)^{-d/2}$ on $0\leq t<1$ ensures
$\frac{\phi(\delta_{\beta})-\phi(0)}{\delta_{\beta}}\leq\frac{\phi(\delta_{0})-\phi(0)}{\delta_{0}}$,
hence

$(1-\delta_{\beta})^{-d/2}-1\leq((1-\delta_{0})^{-d/2}-1)\frac{\delta_{\beta}}{\delta_{0}}\leq\left(\frac{9}{8}-1\right)\frac{\delta_{\beta}}{\delta_{0}}=\frac{\delta_{\beta}}{8\delta_{0}}.$

This gives

$\delta_{\beta}(1-\delta_{\beta})^{-d/2}+(1-\delta_{\beta})^{-d/2}-1\leq\frac{9}{8}\delta_{\beta}+\frac{\delta_{\beta}}{8\delta_{0}}=C_{J}\delta_{\beta}.$

Since $\delta_{\beta}\leq\frac{1}{9}$ holds as well, we further have

$\frac{9}{8}\delta_{\beta}+\frac{1}{8}\cdot\frac{\delta_{\beta}}{\delta_{0}}\leq\frac{9}{8}\cdot\frac{1}{9}+\frac{1}{8}\cdot
1=\frac{1}{4}.$

∎

######  Proposition 5.13.

If $\delta_{\beta}\leq\delta_{0}$, then
$\boldsymbol{\tilde{a}}\in\Sigma_{r/2,2R}^{d}(\Omega^{\text{m}})$.

###### Proof.

It suffices to prove
$\boldsymbol{\hat{a}}\in\Sigma_{r/2,2R}^{d}(\Omega^{\text{m}})$. For
$\boldsymbol{x}\in\Omega^{\text{m}}$, we denote for simplicity
$B=\frac{J(\boldsymbol{x})}{\sqrt{\det J(\boldsymbol{x})}}$. Proposition 5.12
shows $\|B-I\|_{2}\leq 1/4$. Then

$\|B\xi\|_{2}\geq\|\xi\|_{2}-\|(B-I)\xi\|_{2}\geq\|\xi\|_{2}-\|B-I\|_{2}\|\xi\|_{2}\geq\frac{3}{4}\|\xi\|_{2},$

and
$\|B\xi\|_{2}^{2}\geq\frac{9}{16}\|\xi\|_{2}^{2}\geq\frac{1}{2}\|\xi\|_{2}^{2}$.
Likewise, we have $\|B\xi\|_{2}^{2}\leq(1+\frac{1}{4})^{2}\|\xi\|_{2}^{2}\leq
2\|\xi\|_{2}^{2}$. Since $a\in\Sigma_{r,R}^{d}(\Omega^{\text{m}})$, for any
$\xi\in\mathbb{R}^{d}$,

(5.15)
$\xi^{\mathrm{T}}\boldsymbol{\hat{a}}(\boldsymbol{x})\xi=(B\xi)^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})B\xi\in[r\|B\xi\|_{2}^{2},R\|B\xi\|_{2}^{2}]\subseteq\left[\frac{r}{2}\|\xi\|_{2}^{2},2R\|\xi\|_{2}^{2}\right]$

holds, and the proposition is proved. ∎

######  Proposition 5.14.

If $\delta_{\beta}\leq\delta_{0}$, we have

$\|\boldsymbol{a}-\boldsymbol{\tilde{a}}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}\leq R(3C_{J}+1)\delta_{\beta}.$

###### Proof.

For $\boldsymbol{x}\in\Omega^{\text{m}}$, by Assumption 4.8(1), we have

$\displaystyle\|\boldsymbol{a}(\boldsymbol{x})-\boldsymbol{a}(\beta^{-1}(\boldsymbol{x}))\|_{2}$
$\displaystyle\leq R\|\boldsymbol{x}-\beta^{-1}(\boldsymbol{x})\|_{2}$
$\displaystyle=R\|\beta(\beta^{-1}(\boldsymbol{x}))-\beta^{-1}(\boldsymbol{x})\|_{2}$
$\displaystyle\leq
R\max_{\boldsymbol{y}\in\Omega^{\text{m}}}\|\beta(\boldsymbol{y})-\boldsymbol{y}\|_{2}$
$\displaystyle\leq R\delta_{\beta}.$

Denote for simplicity $B=\frac{J(\beta^{-1}(\boldsymbol{x}))}{\sqrt{\det
J(\beta^{-1}(\boldsymbol{x}))}}$, and
$A=\boldsymbol{a}(\beta^{-1}(\boldsymbol{x}))$. Then
$\boldsymbol{\tilde{a}}(\boldsymbol{x})=B^{\mathrm{T}}AB$, and

$\begin{split}\|\boldsymbol{\tilde{a}}(\boldsymbol{x})-\boldsymbol{a}(\beta^{-1}(\boldsymbol{x}))\|_{2}&=\|B^{\mathrm{T}}AB-A\|_{2}\\\
&\leq\|B^{\mathrm{T}}AB-AB\|_{2}+\|AB-A\|_{2}\\\
&=\|(B-I)^{\mathrm{T}}AB\|_{2}+\|A(B-I)\|_{2}\\\
&\leq\|B-I\|_{2}\|A\|_{2}\bigl{(}\|B\|_{2}+1\bigr{)}.\end{split}$

By Proposition 5.12 we have $\|B-I\|_{2}\leq C_{J}\delta_{\beta}$, and

$\|B\|_{2}\leq\|B-I\|_{2}+\|I\|_{2}\leq\frac{1}{4}+1\leq 2.$

Since $\boldsymbol{a}\in\Sigma_{r,R}^{d}(\Omega^{\text{m}})$, the matrix $A$
is symmetric and positive definite, which indicates

$\|A\|_{2}=\lambda_{\max}(A)=\max_{\xi\in\mathbb{R}^{d},\|\xi\|_{2}=1}\xi^{\mathrm{T}}A\xi\leq
R\cdot 1^{2}.$

The conclusion follows by

$\begin{split}\|\boldsymbol{a}(\boldsymbol{x})-\boldsymbol{\tilde{a}}(\boldsymbol{x})\|_{2}&\leq\|\boldsymbol{a}(\boldsymbol{x})-\boldsymbol{a}(\beta^{-1}(\boldsymbol{x}))\|_{2}+\|\boldsymbol{\tilde{a}}(\boldsymbol{x})-\boldsymbol{a}(\beta^{-1}(\boldsymbol{x}))\|_{2}\\\
&\leq R\delta_{\beta}+C_{J}\delta_{\beta}\cdot R\cdot(2+1)\\\
&=R(3C_{J}+1)\delta_{\beta}.\qed\end{split}$

We are now ready to prove the main theorem of Section 5.4.1.

###### Proof of Theorem 5.4.

We first estimate
$\|u_{2}-\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}}\setminus\partial\Omega_{2})}$.
According to Lemma 5.10 and the definition of $u_{2}$, we have

$\begin{split}\int_{\Omega^{\text{m}}}(\nabla\tilde{u}(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{\tilde{a}}(\boldsymbol{x})\nabla
v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}&=\int_{\Omega^{\text{m}}}\tilde{f}(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x},\\\
\int_{\Omega^{\text{m}}}(\nabla
u_{2}(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\nabla
v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}&=\int_{\Omega^{\text{m}}}f(\boldsymbol{x})v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\end{split}$

for any $v\in H_{0}^{1}(\Omega^{\text{m}}\setminus\partial\Omega_{2})$, and
subtraction gives

$\begin{split}&\quad\int_{\Omega^{\text{m}}}(\nabla
u_{2}(\boldsymbol{x})-\nabla\tilde{u}(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\nabla
v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&=\int_{\Omega^{\text{m}}}(\nabla\tilde{u}(\boldsymbol{x}))^{\mathrm{T}}(\boldsymbol{\tilde{a}}(\boldsymbol{x})-\boldsymbol{a}(\boldsymbol{x}))\nabla
v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&\quad+\int_{\Omega^{\text{m}}}(f(\boldsymbol{x})-\tilde{f}(\boldsymbol{x}))v(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}.\end{split}$

Denote $w=u_{2}-\tilde{u}$, and take $v=w$, inequalities analogous to (5.2)
gives

$\begin{split}&\quad\int_{\Omega^{\text{m}}}(\nabla
w(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\nabla
w(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\\\
&\leq\|\boldsymbol{\tilde{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}\|\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}\|w\|_{H_{0}^{1}(\Omega^{\text{m}})}\\\
&\quad+\|f-\tilde{f}\|_{H^{-1}(\Omega^{\text{m}})}\|w\|_{H_{0}^{1}(\Omega^{\text{m}})}.\end{split}$

Assumption 4.8(2) guarantees

$\int_{\Omega^{\text{m}}}(\nabla
w(\boldsymbol{x}))^{\mathrm{T}}\boldsymbol{a}(\boldsymbol{x})\nabla
w(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}\geq\int_{\Omega^{\text{m}}}r(\nabla
w(\boldsymbol{x}))^{\mathrm{T}}\nabla
w(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x}=r\|w\|_{H_{0}^{1}(\Omega^{\text{m}})}^{2},$

and thus

$\begin{split}\|w\|_{H_{0}^{1}(\Omega^{\text{m}})}&\leq\frac{1}{r}\left(\|\boldsymbol{\tilde{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}\|\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}+\|f-\tilde{f}\|_{H^{-1}(\Omega^{\text{m}})}\right)\\\
&\leq\frac{1}{r}\left(R(3C_{J}+1)\delta_{\beta}\cdot\frac{2C_{\text{P}}R}{r}+2R\delta_{\beta}\right)\\\
&=\frac{2R}{r^{2}}((3C_{J}+1)RC_{\text{P}}+r)\delta_{\beta},\end{split}$

where we have used Proposition 5.14, 5.7 and 5.11. Combining with Corollary
5.9 gives

$\begin{split}\|u-u_{2}\|_{L^{2}(\Omega^{\text{m}})}&\leq\|u-\tilde{u}\|_{L^{2}(\Omega^{\text{m}})}+\|\tilde{u}-u_{2}\|_{L^{2}(\Omega^{\text{m}})}\\\
&\leq\|u-\tilde{u}\|_{L^{2}(\Omega^{\text{m}})}+C_{\text{P}}\|u_{2}-\tilde{u}\|_{H_{0}^{1}(\Omega^{\text{m}})}\\\
&\leq\frac{2RC_{\text{P}}}{r^{2}}(3r+(3C_{J}+1)RC_{\text{P}})\delta_{\beta}.\qed\end{split}$

####  5.4.2. Proofs for the Fixed Parameter Case

###### Proof of Proposition 4.10.

Given $\lambda,\bar{\lambda}\in\Lambda$, $\lambda\neq\bar{\lambda}$, the aim
is to show $\|u^{\lambda}-u^{\bar{\lambda}}\|_{L^{2}(\Omega^{\text{m}})}\leq
C\|\lambda-\bar{\lambda}\|_{2}$. We devide the interval $[0,1]$ into $K$
segments with endpoints $0=t_{0}<t_{1}<\cdots<t_{K}=1$, satisfying
$2\|\lambda-\bar{\lambda}\|_{2}(t_{k}-t_{k-1})\leq\delta_{0}$ for
$k=1,\dots,K$. Denote for simplicity
$\Omega_{k}=\Omega_{\lambda+t_{k}(\bar{\lambda}-\lambda)}$,
$T_{k}=T_{\lambda+t_{k}(\bar{\lambda}-\lambda)}$,
$u_{k}=u^{\lambda+t_{k}(\bar{\lambda}-\lambda)}$ for $k=1,\dots,K$. Given $k$,
we take $\beta=T_{k}\circ T_{k-1}^{-1}$. Then $\beta$ is a
$C^{1}$-diffeomorphism from $\bar{\Omega}^{\text{m}}$ onto itself,
$\Omega_{k}=\beta(\Omega_{k-1})$. For any
$\boldsymbol{x}\in\bar{\Omega}^{\text{m}}$, denote
$\boldsymbol{y}=T_{k-1}^{-1}(\boldsymbol{x})$, we have

$\|\beta(\boldsymbol{x})-\boldsymbol{x}\|_{2}=\|T_{k}(\boldsymbol{y})-T_{k-1}(\boldsymbol{y})\|_{2}\leq(t_{k}-t_{k-1})\|\lambda-\bar{\lambda}\|_{2}$

by Assumption 4.9(3), and

$\begin{split}\left\lVert\frac{\partial\beta(\boldsymbol{x})}{\partial\boldsymbol{x}}-I_{d}\right\rVert_{2}&=\left\lVert\frac{\partial
T_{k}(\boldsymbol{y})}{\partial\boldsymbol{y}}\left(\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right)^{-1}-I_{d}\right\rVert_{2}\\\
&\leq\left\lVert\frac{\partial
T_{k}(\boldsymbol{y})}{\partial\boldsymbol{y}}-\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right\rVert_{2}\cdot\left\lVert\left(\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right)^{-1}\right\rVert_{2}\\\
&=\sigma_{\min}\left(\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right)^{-1}\left\lVert\frac{\partial
T_{k}(\boldsymbol{y})}{\partial\boldsymbol{y}}-\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right\rVert_{2}\\\ &\leq
2\left\lVert\frac{\partial
T_{k}(\boldsymbol{y})}{\partial\boldsymbol{y}}-\frac{\partial
T_{k-1}(\boldsymbol{y})}{\partial\boldsymbol{y}}\right\rVert_{2}\\\ &\leq
2(t_{k}-t_{k-1})\|\lambda-\bar{\lambda}\|_{2},\end{split}$

where we have used (2), (3) of Assumption 4.9. This assures
$\delta_{\beta}\leq
2(t_{k}-t_{k-1})\|\lambda-\bar{\lambda}\|_{2}\leq\delta_{0}$, and Theorem 5.4
asserts

$\|u_{k-1}-u_{k}\|_{L^{2}(\Omega^{\text{m}})}\leq C\delta_{\beta}/2\leq
C(t_{k}-t_{k-1})\|\lambda-\bar{\lambda}\|_{2}.$

The conclusion follows by

$\begin{split}\|u^{\lambda}-u^{\bar{\lambda}}\|_{L^{2}(\Omega^{\text{m}})}&\leq\sum_{k=1}^{K}\|u_{k-1}-u_{k}\|_{L^{2}(\Omega^{\text{m}})}\\\
&\leq\sum_{k=1}^{K}C(t_{k}-t_{k-1})\|\lambda-\bar{\lambda}\|_{2}\\\
&=C\|\lambda-\bar{\lambda}\|_{2}.\qed\end{split}$

###### Proof of the assertion in Example 4.12.

For this specific setting, we may take the parametrized domain transformation
to be the rotation

$T_{\lambda}:\boldsymbol{x}\mapsto
R_{\lambda}\boldsymbol{x}=\begin{bmatrix}\cos\lambda&-\sin\lambda\\\
\sin\lambda&\cos\lambda\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix},$

which satisfies $\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}=R_{\lambda}$ for any
$\boldsymbol{x}\in\bar{\Omega}^{\text{m}}$. Then Assumption 4.9 is satisfied,
since we have $\sigma_{\min}(R_{\lambda})=1$,
$\|T_{\lambda}(\boldsymbol{x})-T_{\bar{\lambda}}(\boldsymbol{x})\|_{2}=\|R_{\lambda}\boldsymbol{x}-R_{\bar{\lambda}}\boldsymbol{x}\|_{2}=2\sin(\frac{|\lambda-\bar{\lambda}|}{2})\|\boldsymbol{x}\|_{2}\leq|\lambda-\bar{\lambda}|$,
and

$\left\lVert\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}-\frac{\partial
T_{\bar{\lambda}}(\boldsymbol{x})}{\partial\boldsymbol{x}}\right\rVert_{2}=\|R_{\lambda}-R_{\bar{\lambda}}\|_{2}=\max_{\|\boldsymbol{x}\|_{2}\leq
1}\|R_{\lambda}\boldsymbol{x}-R_{\bar{\lambda}}\boldsymbol{x}\|_{2}\leq|\lambda-\bar{\lambda}|.$

Theorem 4.11 then implies

$d_{1,C\pi}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})\leq
Cd_{1,\pi}^{\text{rDeco}}(\Lambda).$

By taking the decoder mapping to be $D=D_{1}\circ
P_{[-1,1]}:Z=\mathbb{R}^{1}\to\Lambda$, where $D_{1}:Z_{B}=[-1,1]\to\Lambda$
is given by $z\mapsto\pi z$, we see that $D(Z_{B})=\Lambda$, and
$d_{1,\pi}^{\text{rDeco}}(\Lambda)=0$. The assertion is proved. ∎

###### Proof of the assertion in Example 4.13.

The proof is the same as Example 4.12. ∎

###### Proof of the assertion in Example 4.14.

The domain transformations
$T_{\lambda}:\bar{\Omega}^{\text{m}}\to\bar{\Omega}^{\text{m}}$ can be
constructed as

$T_{\lambda}(\boldsymbol{x})=(x_{1}+\lambda_{1}\sin
x_{1},x_{2}+\lambda_{2}\sin
x_{2}),\quad\boldsymbol{x}=(x_{1},x_{2})\in\bar{\Omega}^{\text{m}},$

which satisfies

$\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}=\mathrm{diag}(1+\lambda_{1}\cos
x_{1},1+\lambda_{2}\cos x_{2}),$

and it is easy to check that Assumption 4.9 holds. Theorem 4.11 implies

$d_{2,C/\sqrt{2}}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})\leq
Cd_{2,1/\sqrt{2}}^{\text{rDeco}}(\Lambda).$

We take $Z=\mathbb{R}^{2}$, and $Z_{\Lambda}=[-1/\sqrt{2},1/\sqrt{2}]^{2}$
which is a convex subset of $Z_{B}$. The mapping
$D_{1}:Z\to\mathbb{R}^{2},z\mapsto z/\sqrt{2}$ satisfies
$D_{1}(Z_{\Lambda})=\Lambda$. By taking the decoder mapping to be
$D=D_{1}\circ P_{Z_{\Lambda}}$, we conclude that
$d_{2,1/\sqrt{2}}^{\text{rDeco}}(\Lambda)=0$, and hence
$d_{2,C/\sqrt{2}}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})=0$. ∎

###### Proof of the assertion in Example 4.15.

In this setting, the domain transformation can be taken as
$T_{\lambda}(\boldsymbol{x})=(x_{1},x_{2}+\rho_{\lambda}(x_{1})\sin x_{2})$
for $\boldsymbol{x}\in\bar{\Omega}^{\text{m}}$, with Jacobian matrix

$\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}=\begin{bmatrix}1\\\
\rho_{\lambda}^{\prime}(x_{1})\sin x_{2}&1+\rho_{\lambda}(x_{1})\cos
x_{2}\end{bmatrix}.$

Validation of Assumption 4.9(1) is straightforward. As for (2) and (3), we
note that

$\begin{split}\rho_{\lambda}^{\prime}(x_{1})^{2}&=\Bigl{(}\sum_{k=1}^{\infty}\frac{\sqrt{6}}{k\pi}(-a_{k}\sin
kx_{1}+b_{k}\cos kx_{1})\Bigr{)}^{2}\\\
&\leq\Bigl{(}\sum_{k=1}^{\infty}\frac{6}{k^{2}\pi^{2}}\Bigr{)}\Bigl{(}\sum_{k=1}^{\infty}(-a_{k}\sin
kx_{1}+b_{k}\cos kx_{1})^{2}\Bigr{)}\\\ &\qquad\text{(by Cauchy's
inequality)}\\\ &=\sum_{k=1}^{\infty}|-a_{k}\sin kx_{1}+b_{k}\cos
kx_{1}|^{2}\\\
&\leq\sum_{k=1}^{\infty}\left(\sqrt{a_{k}^{2}+b_{k}^{2}}\right)^{2}=\|\lambda\|^{2},\end{split}$

and similarly $|\rho_{\lambda}(x_{1})|\leq\|\lambda\|$ holds as well. Then

$T_{\lambda}(\boldsymbol{x})-T_{\bar{\lambda}}(\boldsymbol{x})=(0,(\rho_{\lambda}(x_{1})-\rho_{\bar{\lambda}}(x_{1}))\sin
x_{2})=(0,\rho_{\lambda-\bar{\lambda}}(x_{1})\sin x_{2}),$
$\|T_{\lambda}(\boldsymbol{x})-T_{\bar{\lambda}}(\boldsymbol{x})\|_{2}\leq|\rho_{\lambda-\bar{\lambda}}(x_{1})|\cdot|\sin
x_{2}|\leq\|\lambda-\bar{\lambda}\|,$

and

$\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}-\frac{\partial
T_{\bar{\lambda}}(\boldsymbol{x})}{\partial\boldsymbol{x}}=\begin{bmatrix}0&0\\\
\rho_{\lambda-\bar{\lambda}}^{\prime}(x_{1})\sin
x_{2}&\rho_{\lambda-\bar{\lambda}}(x_{1})\cos x_{2}\end{bmatrix},$
$\begin{split}\left\lVert\frac{\partial
T_{\lambda}(\boldsymbol{x})}{\partial\boldsymbol{x}}-\frac{\partial
T_{\bar{\lambda}}(\boldsymbol{x})}{\partial\boldsymbol{x}}\right\rVert_{2}^{2}&=\bigl{(}\rho_{\lambda-\bar{\lambda}}^{\prime}(x_{1})\sin
x_{2}\bigr{)}^{2}+\bigl{(}\rho_{\lambda-\bar{\lambda}}(x_{1})\cos
x_{2}\bigr{)}^{2}\\\
&\leq\|\lambda-\bar{\lambda}\|^{2}(\sin^{2}x_{2}+\cos^{2}x_{2})\\\
&=\|\lambda-\bar{\lambda}\|^{2},\end{split}$

indicating that Assumption 4.9(3) holds. Furthermore, note that $w_{k}\leq
w_{1}=1$ implies

$\|\lambda\|^{2}=\sum_{k=1}^{\infty}(a_{k}^{2}+b_{k}^{2})\leq\sum_{k=1}^{\infty}\frac{a_{k}^{2}+b_{k}^{2}}{w_{k}}\leq(3/8)^{2}.$

Given $p,q\in\mathbb{R}$,

$\begin{split}\sigma_{\min}([\begin{smallmatrix}1\\\
q&p\end{smallmatrix}])^{2}&=\lambda_{\min}([\begin{smallmatrix}1\\\
q&p\end{smallmatrix}][\begin{smallmatrix}1&q\\\
&p\end{smallmatrix}])=\lambda_{\min}([\begin{smallmatrix}1&q\\\
q&p^{2}+q^{2}\end{smallmatrix}])\\\
&=\frac{1}{2}\bigl{(}p^{2}+q^{2}+1-\sqrt{(p^{2}+q^{2}+1)^{2}-4p^{2}}\bigr{)},\end{split}$

and $\sigma_{\min}([\begin{smallmatrix}1\\\ q&p\end{smallmatrix}])\geq 1/2$ if
and only if $p^{2}\geq q^{2}/3+1/4$. For any $x\in\bar{\Omega}^{\text{m}}$, we
let $p=1+\rho_{\lambda}(x_{1})\cos x_{2}$,
$q=\rho_{\lambda}^{\prime}(x_{1})\sin x_{2}$. Then

$p\geq 1-|\rho_{\lambda}(x_{1})|\geq 1-\|\lambda\|\geq 5/8,$
$|q|\leq|\rho_{\lambda}^{\prime}(x_{1})|\leq\|\lambda\|\leq 3/8,$

and $p^{2}\geq q^{2}/3+1/4$ indeed holds. This validates Assumption 4.9(2) as
$\sigma_{\min}(\frac{\partial T_{\lambda}(x)}{\partial
x})=\sigma_{\min}([\begin{smallmatrix}1\\\ q&p\end{smallmatrix}])\geq 1/2$.

Theorem 4.11 then implies

$d_{2n,3C/8}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})\leq
Cd_{2n,3/8}^{\text{rDeco}}(\Lambda).$

To estimate the latter term, we take $Z=\mathbb{R}^{2n}$, and let
$D_{1}:z\mapsto\lambda=(a_{1},b_{1},a_{2},\dots)$ be given as
$a_{k}=3z_{2k-1}/8$, $b_{k}=3z_{2k}/8$ for $k=1,\dots,n$, $a_{k}=b_{k}=0$ for
$k>n$. Then the mapping $D_{1}$ is $3/8$-Lipschitz. Each
$\lambda=(a_{1},b_{1},a_{2},\dots)\in\Lambda$ can be associated with a vector
$z$ such that $z_{2k-1}=8a_{k}/3$, $z_{2k}=8b_{k}/3$, $k=1,\dots,n$. Then
$z\in Z_{B}$ since $\|z\|\leq 8\|\lambda\|/3\leq 1$, and

$\begin{split}\lambda-D_{1}(z)&=(0,\dots,0,a_{n+1},b_{n+1},a_{n+2},\dots),\\\
\|\lambda-D_{1}(z)\|^{2}&=\sum_{k>n}(a_{k}^{2}+b_{k}^{2})\leq
w_{n}\sum_{k>n}\frac{a_{k}^{2}+b_{k}^{2}}{w_{k}}\\\ &\leq
w_{n}\sum_{k=1}^{\infty}\frac{a_{k}^{2}+b_{k}^{2}}{w_{k}}\leq\frac{9w_{n}}{64}.\end{split}$

Now $Z_{\Lambda}=D_{1}^{-1}(\Lambda)$ is a convex subset of $Z_{B}$. Taking
$D=D_{1}\circ P_{Z_{\Lambda}}$, we have

$\sup_{\lambda\in\Lambda}\inf_{z\in Z_{B}}\|\lambda-D(z)\|\leq
3\sqrt{w_{n}}/8,$

and $D(Z)\subset\Lambda$ holds. This indicates
$d_{2n,3/8}^{\text{rDeco}}(\Lambda)\leq 3\sqrt{w_{n}}/8$ by the definition of
the restricted decoder width, and we conclude that
$d_{2n,3C/8}^{\text{Deco}}(\\{u^{\lambda}|\lambda\in\Lambda\\})\leq
3C\sqrt{w_{n}}/8$. ∎

####  5.4.3. Proofs for the Variable Parameter Case

###### Proof of Proposition 4.16.

By Proposition 4.1, we have

$\begin{split}\|u^{(\boldsymbol{a},f,\bar{\lambda})}-u^{(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})}\|_{L^{2}(\Omega^{\text{m}})}&\leq
C_{\text{P}}\|u^{(\boldsymbol{a},f,\bar{\lambda})}-u^{(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})}\|_{H_{0}^{1}(\Omega^{\text{m}})}\\\
&\leq\frac{C_{\text{P}}^{2}}{r^{2}}\Bigl{(}R\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}+r\|\bar{f}-f\|_{L^{2}}\Bigr{)}\\\ &\leq
C_{1}\Bigl{(}\|\boldsymbol{\bar{a}}-\boldsymbol{a}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d\times
d}))}+\|\bar{f}-f\|_{L^{2}}\Bigr{)}.\end{split}$

By Proposition 4.10, we have

$\begin{split}\|u^{(\boldsymbol{a},f,\lambda)}-u^{(\boldsymbol{a},f,\bar{\lambda})}\|_{L^{2}(\Omega^{\text{m}})}\leq
C\|\lambda-\bar{\lambda}\|_{2}\leq
C_{1}\|\lambda-\bar{\lambda}\|_{2}.\end{split}$

A direct combination of these two inequalities shall give

$\begin{split}\|u^{(\boldsymbol{a},f,\lambda)}-u^{(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})}\|_{L^{2}(\Omega^{\text{m}})}&\leq\|u^{(\boldsymbol{a},f,\lambda)}-u^{(\boldsymbol{a},f,\bar{\lambda})}\|_{L^{2}(\Omega^{\text{m}})}+\|u^{(\boldsymbol{a},f,\bar{\lambda})}-u^{(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})}\|_{L^{2}(\Omega^{\text{m}})}\\\
&\leq
C_{1}d_{\mathcal{A}_{0}}((\boldsymbol{a},f,\lambda),(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})).\qed\end{split}$

###### Proof of the assertion in Example 4.18.

The proof is similar to that of Example 4.6. The PDE coefficients are now
parametrized by the mapping
$S_{\Lambda}:\Lambda_{\mu}\times\Lambda\to\mathcal{A}_{0}$,
$(\mu,\lambda)\mapsto(\boldsymbol{a},f,\lambda)$. In assumption (c), the
condition $\sum_{k=1}^{K}(\mu_{k}^{a})^{2}\leq\frac{1}{4}$ implies
$|\mu_{k}^{a}|\leq\frac{1}{2}$ for $k=1,\dots,K$. By assumption (a), it is
easy to see that all possible $\boldsymbol{a}$’s are 2-Lipschitz continuous,
and $\boldsymbol{a}\in\Sigma_{1,2}^{d}(\Omega^{\text{m}})$, thereby
$\boldsymbol{a}\in\mathsf{D}_{1,2}^{a}$. For a different set of coefficients
$(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda})=S_{\Lambda}(\bar{\mu},\bar{\lambda})$,
an argument similar to (5.13) shows

$\begin{split}\|\boldsymbol{a}-\boldsymbol{\bar{a}}\|_{L^{\infty}(\Omega^{\text{m}};\ell^{2}(\mathbb{R}^{d}\times\mathbb{R}^{d}))}&=\max_{1\leq
k\leq
K}\left\lVert(\mu_{k}^{a}-\bar{\mu}_{k}^{a})\phi_{k}^{a}\right\rVert_{L^{\infty}(\Omega^{\text{m}})}\\\
&\leq\max_{1\leq k\leq
K}\|\mu-\bar{\mu}\|_{2}\cdot\left\lVert\phi_{k}^{a}\right\rVert_{L^{\infty}(\Omega^{\text{m}})}\\\
&\leq\|\mu-\bar{\mu}\|_{2}.\end{split}$

As for $f$, we have $f\in\mathsf{D}_{2}^{f}$ since
$\|f\|_{L^{2}}^{2}\leq\left(\sum_{k=1}^{K}(\mu_{k}^{f})^{2}\right)\left(\sum_{k=1}^{K}\|\phi_{k}^{f}\|_{L^{2}}^{2}\right)\leq\frac{1}{4}\cdot
1\leq 4$ (similar to (5.12)), and

$\|f-\bar{f}\|_{L^{2}}^{2}\leq\left(\sum_{k=1}^{K}(\mu_{k}^{f}-\bar{\mu}_{k}^{f})^{2}\right)\left(\sum_{k=1}^{K}\|\phi_{k}^{f}\|_{L^{2}}^{2}\right)\leq\|\mu-\bar{\mu}\|_{2}^{2}\cdot
1.$

Viewing $(\mu,\lambda)$ as a vector in $\mathbb{R}^{2K+2}$, we apply the
definition of the metric on $\mathcal{A}_{0}$ (as given in (4.9)) to get

$d_{\mathcal{A}_{0}}((\boldsymbol{a},f,\lambda),(\boldsymbol{\bar{a}},\bar{f},\bar{\lambda}))\leq\|\bar{\mu}-\mu\|_{2}+\|\bar{\mu}-\mu\|_{2}+\|\bar{\lambda}-\lambda\|_{2}\leq
3\|(\bar{\mu},\bar{\lambda})-(\mu,\lambda)\|_{2},$

and the mapping $S_{\Lambda}:(\mu,\lambda)\mapsto(\boldsymbol{a},f,\lambda)$
is therefore 3-Lipschitz continuous. Furthermore, assumption (c) as well as
the definition of $\Lambda$ implies

$\|(\mu,\lambda)\|_{2}^{2}=\sum_{k=1}^{K}(\mu_{k}^{a})^{2}+\sum_{k=1}^{K}(\mu_{k}^{f})^{2}+(\lambda_{1}^{2}+\lambda_{2}^{2})\leq\frac{1}{4}+\frac{1}{4}+\left(\frac{1}{4}+\frac{1}{4}\right)=1$

for any $(\mu,\lambda)\in\Lambda_{\mu}\times\Lambda$, and hence
$d_{2K+2,1}^{\text{rDeco}}(\Lambda_{\mu}\times\Lambda)=0$ since we can take
$Z=\mathbb{R}^{2K+2}$, $D=P_{\Lambda_{\mu}\times\Lambda}$. Proposition 3.1
then implies
$d_{2K+2,3}^{\text{Deco}}(S_{\Lambda}(\Lambda_{\mu}\times\Lambda))=0$, and we
may use Theorem 4.17 to conclude the proof (note that
$\\{u^{\mu,\lambda}|\mu\in\Lambda_{\mu},\lambda\in\Lambda\\}=S_{0}(S_{\Lambda}(\Lambda_{\mu}\times\Lambda))$
in this example). ∎

###  5.5. Advection Equations on a Fixed Domain

###### Proof of Proposition 4.19.

We have

$\begin{split}\left\|u-\bar{u}\right\|_{L^{p}}^{p}&=\int_{-1}^{1}\mathrm{d}x\int_{0}^{1}\left|u(t,x)-\bar{u}(t,x)\right|^{p}\,\mathrm{d}t\\\
&=\int_{-1}^{1}\mathrm{d}x\int_{0}^{1}\left|\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\}}-\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{\bar{c}(s)}\\}}\right|^{p}\,\mathrm{d}t\\\
&\leq\int_{-1}^{1}\mathrm{d}x\int_{-\infty}^{+\infty}\left|\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\}}-\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{\bar{c}(s)}\\}}\right|^{p}\,\mathrm{d}t\\\
&=\int_{0}^{1}\mathrm{d}x\int_{-\infty}^{+\infty}\left|\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\}}-\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{\bar{c}(s)}\\}}\right|\,\mathrm{d}t\\\
&=\int_{0}^{1}\left|\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}-\int_{0}^{x}\frac{\mathrm{d}s}{\bar{c}(s)}\right|\,\mathrm{d}x\\\
&\leq\int_{0}^{1}\mathrm{d}x\int_{0}^{x}\left|\frac{1}{c(s)}-\frac{1}{\bar{c}(s)}\right|\mathrm{d}s\\\
&\leq\int_{0}^{1}\mathrm{d}x\int_{-1}^{1}\left|\frac{1}{c(s)}-\frac{1}{\bar{c}(s)}\right|\mathrm{d}s\\\
&\leq\int_{-1}^{1}\frac{|\bar{c}(s)-c(s)|}{r^{2}}\mathrm{d}s\\\
&=r^{-2}\|c-\bar{c}\|_{L^{1}([-1,1])}.\qed\end{split}$

####  5.5.1. The Case of Using $L^{1}$-norm

######  Lemma 5.15.

Assume $\mathcal{V}$ is a Banach space. Given
$\phi_{0},\dots,\phi_{K}\in\mathcal{V}$, we let

$\mathcal{K}_{\mathcal{V}}=\left\\{\phi_{0}+\sum_{k=1}^{K}\lambda_{k}\phi_{k}\
\middle|\ \lambda\in[-1,1]^{K}\right\\}\subset\mathcal{V}.$

Then $d_{K,l}^{\text{rDeco}}(\mathcal{K}_{\mathcal{V}})=0$ holds, where
$l=\sqrt{K\sum_{k=1}^{K}\mu_{k}^{2}}$ ,
$\mu_{k}=\|\phi_{k}\|_{\mathcal{V}}>0$.

###### Proof.

The proof has much in common with that of Example 4.3. For any
$v=\phi_{0}+\sum_{k=1}^{K}\lambda_{k}\phi_{k}\in\mathcal{K}_{\mathcal{V}}$,
$\bar{v}=\phi_{0}+\sum_{k=1}^{K}\bar{\lambda}_{k}\phi_{k}\in\mathcal{K}_{\mathcal{V}}$,
we have

(5.16)
$\begin{split}\|v-\bar{v}\|_{\mathcal{V}}&=\left\lVert\sum_{k=1}^{K}(\lambda_{k}-\bar{\lambda}_{k})\phi_{k}\right\rVert_{\mathcal{V}}\leq\sum_{k=1}^{K}|\lambda_{k}-\bar{\lambda}_{k}|\cdot\|\phi_{k}\|_{\mathcal{V}}\\\
&=\sum_{k=1}^{K}|\lambda_{k}-\bar{\lambda}_{k}|\mu_{k}\leq\sqrt{\sum_{k=1}^{K}|\lambda_{k}-\bar{\lambda}_{k}|^{2}}\sqrt{\sum_{k=1}^{K}\mu_{k}^{2}}\\\
&=\|\lambda-\bar{\lambda}\|_{2}\sqrt{\sum_{k=1}^{K}\mu_{k}^{2}}.\end{split}$

We consider the mapping

$D_{1}:Z=\mathbb{R}^{K}\to\mathcal{V},\quad z\mapsto
v=\phi_{0}+\sqrt{K}\sum_{k=1}^{K}z_{k}\phi_{k}.$

By (5.16), $D_{1}$ is $\sqrt{K\sum_{k=1}^{K}\mu_{k}^{2}}$-Lipschitz
continuous. Now $Z_{1}=[-1/\sqrt{K},1/\sqrt{K}]^{K}$ is a convex subset of
$Z_{B}$, and $D_{1}(Z_{1})=\mathcal{K}_{\mathcal{V}}$ holds. We take
$D=D_{1}\circ P_{Z_{1}}$, whose Lipschitz constant is also
$l=\sqrt{K\sum_{k=1}^{K}\mu_{k}^{2}}$. The lemma then follows directly by the
definition of the restricted decoder width. ∎

###### Proof of the assertion in Example 4.21.

Denote $\mathcal{A}=\\{c^{\lambda}\mid\lambda\in\Lambda\\}$. According to
Lemma 5.15 and the definition of $c^{\lambda}$, we see that

$d_{K,l}^{\text{rDeco}}(\mathcal{A})_{L^{1}([-1,1])}=0$

holds with $l=\sqrt{K\sum_{k=1}^{K}\|\phi_{k}\|_{L^{1}([-1,1])}^{2}}$. We may
apply Theorem 4.20 to conclude that
$d_{K,l/r^{2}}^{\text{Deco}}\left(\left\\{u^{\lambda}\ \middle|\
\lambda\in\Lambda\right\\}\right)_{L^{1}}=0.$ ∎

####  5.5.2. The Case of Using $L^{2}$-norm

Similar to Proposition 3.1, we have the following lemma for the entropy
number:

######  Lemma 5.16.

Let $\mathcal{V,U}$ be two Banach spaces, $\mathcal{K_{V}}\subset\mathcal{V}$
be a compact subset, and $\mathcal{F}:\mathcal{K_{V}}\to\mathcal{U}$ be a
continuous mapping satisfying

$\|\mathcal{F}(v)-\mathcal{F}(\bar{v})\|_{\mathcal{U}}\leq
w(\|v-\bar{v}\|_{\mathcal{V}})$

for all $v,\bar{v}\in\mathcal{K_{V}}$, where $w:[0,+\infty)\to[0,+\infty)$ is
a non-decreasing continuous function. Then, denoting $\mathcal{K=F(K_{V})}$,
we shall have $\tilde{\epsilon}_{n}(\mathcal{K})\leq
w(\tilde{\epsilon}_{n}(\mathcal{K_{V}})).$ The obvious corollary is
$\epsilon_{n}(\mathcal{K})\leq w(\tilde{\epsilon}_{n}(\mathcal{K_{V}})).$

###### Proof.

For $V_{n}\subset\mathcal{K_{V}}$, we have

$\begin{split}\sup_{u\in\mathcal{F}(\mathcal{K_{V}})}\inf_{\bar{u}\in\mathcal{F}(V_{n})}\|u-\bar{u}\|_{\mathcal{U}}&=\sup_{v\in\mathcal{K_{V}}}\inf_{\bar{v}\in
V_{n}}\|\mathcal{F}(v)-\mathcal{F}(\bar{v})\|_{\mathcal{U}}\\\
&\leq\sup_{v\in\mathcal{K_{V}}}\inf_{\bar{v}\in
V_{n}}w(\|v-\bar{v}\|_{\mathcal{V}})\\\
&=w\Bigl{(}\sup_{v\in\mathcal{K_{V}}}\inf_{\bar{v}\in
V_{n}}\|v-\bar{v}\|_{\mathcal{V}}\Bigr{)},\end{split}$

where the last equality is due to the monotonicity and the continuity of
$w(\cdot)$. Since
$V_{n}\subset\mathcal{K_{V}},\mathcal{F}(V_{n})\subset\mathcal{K}$ and
$\\#\mathcal{F}(V_{n})\leq\\#V_{n}\leq 2^{n}$, the lemma follows directly from
the definition of the inner entropy number. ∎

###### Proof of Theorem 4.22.

Taking $p=2$ in Proposition 4.19, we see that the solution mapping
$S_{0}|_{\mathcal{A}}:\mathcal{A}\to\mathcal{U}=L^{2}([0,1]\times[-1,1])$
satisfies $\|S_{0}(c)-S_{0}(\bar{c})\|_{\mathcal{U}}\leq
w(\|c-\bar{c}\|_{L^{1}([-1,1])})$, where $w(\epsilon)=r^{-1}\sqrt{\epsilon}$.
Lemma 5.16 then implies $\epsilon_{n}(S_{0}(\mathcal{A}))\leq
r^{-1}\sqrt{\tilde{\epsilon}_{n}(\mathcal{A})}$

If $c,\bar{c}\in\mathcal{A}$, $u=S_{0}(c),\bar{u}=S_{0}(\bar{c})$, we have

$\begin{split}\left\|u-\bar{u}\right\|_{\mathcal{U}}^{2}&=\int_{-1}^{1}\mathrm{d}x\int_{0}^{1}\left|\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{c(s)}\\}}-\chi_{\\{t\leq\int_{0}^{x}\frac{\mathrm{d}s}{\bar{c}(s)}\\}}\right|^{2}\,\mathrm{d}t\\\
&\leq\int_{-1}^{1}\mathrm{d}x\int_{0}^{1}1\,\mathrm{d}t\\\ &=2,\end{split}$

indicating $\mathrm{diam}(S_{0}(\mathcal{A}))\leq\sqrt{2}<2$. Corollary 3.3
then gives $d_{26n,4\cdot 2}^{\mathrm{Deco}}(S_{0}(\mathcal{A}))\leq
3\epsilon_{n}(S_{0}(\mathcal{A})),$ and we conclude that
$d_{26n,8}^{\mathrm{Deco}}(S_{0}(\mathcal{A}))\leq
3r^{-1}\sqrt{\tilde{\epsilon}_{n}(\mathcal{A})}$ ∎

######  Lemma 5.17.

Let $\mathcal{V}$ be a Banach space. Given
$\phi_{0},\dots,\phi_{K}\in\mathcal{V}$, we let

$\mathcal{K_{V}}=\left\\{\phi_{0}+\sum_{k=1}^{K}\lambda_{k}\phi_{k}\ \middle|\
\lambda\in[-1,1]^{K}\right\\}\subset\mathcal{V}$

as in Lemma 5.15. Then $\tilde{\epsilon}_{n}(\mathcal{K_{V}})\leq
C_{1}2^{-n/K}$ holds for sufficiently large $n$, where
$C_{1}=4K\left(\prod_{k=1}^{K}\mu_{k}\right)^{1/K}$,
$\mu_{k}=\|\phi_{k}\|_{\mathcal{V}}>0$.

###### Proof.

Given $n$, we take
$\delta_{n}=2^{-n/K}\left(\prod_{k=1}^{K}\mu_{k}\right)^{1/K}$, and assume $n$
is large enough such that $\delta_{n}<\frac{1}{2}\mu_{k}$ holds for
$k=1,\dots,K$. Let $m_{k}=\left\lfloor\mu_{k}/\delta_{n}\right\rfloor$. The
interval $[-1,1]$ is split into $m_{k}$ disjoint sub-intervals with equal
length, and the midpoints of these sub-intervals form the set

$A_{k}=\left\\{\frac{2i}{m_{k}}-1-\frac{1}{m_{k}}\ \middle|\
i=1,\dots,m_{k}\right\\}.$

The number of points contained in $A=\prod_{k=1}^{K}A_{k}\subset[-1,1]^{K}$ is

$\\#A=\prod_{k=1}^{K}m_{k}\leq\prod_{k=1}^{K}\frac{\mu_{k}}{\delta_{n}}=2^{n}.$

Taking $V_{n}=\left\\{\phi_{0}+\sum_{k=1}^{K}\lambda_{k}\phi_{k}\mid\lambda\in
A\right\\}\subset\mathcal{K_{V}}$, it can be shown that

(5.17) $\sup_{v\in\mathcal{K_{V}}}\inf_{\bar{v}\in
V_{n}}\|v-\bar{v}\|_{\mathcal{V}}\leq\sum_{k=1}^{K}\frac{\mu_{k}}{m_{k}}.$

In fact, for any
$v=\phi_{0}+\sum_{k=1}^{K}\lambda_{k}\phi_{k}\in\mathcal{K_{V}}$, we may find
$\bar{\lambda}_{k}\in A_{k}$ with
$|\bar{\lambda}_{k}-\lambda_{k}|\leq\frac{1}{m_{k}}$, $k=1,\dots,K$ by the
definition of each $A_{k}$. Then
$\bar{v}=\phi_{0}+\sum_{k=1}^{K}\bar{\lambda}_{k}\phi_{k}$ lies in $V_{n}$,
and

$\|v-\bar{v}\|_{\mathcal{V}}\leq\sum_{k=1}^{K}|\lambda_{k}-\bar{\lambda}_{k}|\cdot\|\phi_{k}\|_{\mathcal{V}}\leq\sum_{k=1}^{K}\frac{\mu_{k}}{m_{k}}.$

We get (5.17) since $v$ is arbitrary.

Now
$\tilde{\epsilon}_{n}(\mathcal{K_{V}})\leq\sum_{k=1}^{K}\frac{\mu_{k}}{m_{k}}$
by the definition of the inner entropy number. Since
$m_{k}=\left\lfloor\mu_{k}/\delta_{n}\right\rfloor\geq\frac{\mu_{k}}{\delta_{n}}-1$,
we have

$\tilde{\epsilon}_{n}(\mathcal{K_{V}})\leq\delta_{n}\sum_{k=1}^{K}\frac{\mu_{k}}{\mu_{k}-\delta_{n}}.$

Note that

$\lim_{n\to\infty}\delta_{n}=0,\quad\lim_{n\to\infty}\sum_{k=1}^{K}\frac{\mu_{k}}{\mu_{k}-\delta_{n}}=\sum_{k=1}^{K}\frac{\mu_{k}}{\mu_{k}}=K,$

and thus $\tilde{\epsilon}_{n}(\mathcal{K_{V}})\leq 4K\delta_{n}$ holds for
large enough $n$. This proves the lemma. ∎

###### Proof of the assertion in Example 4.23.

Again, we denote $\mathcal{A}=\\{c^{\lambda}\mid\lambda\in\Lambda\\}$.
According to Lemma 5.15 and the definition of $c^{\lambda}$, we see that
$\tilde{\epsilon}_{n}(\mathcal{A})\leq C_{1}2^{-n/K}$ for sufficiently large
$n$, where
$C_{1}=4K\left(\prod_{k=1}^{K}\|\phi_{k}\|_{L^{1}([-1,1])}\right)^{1/K}.$ The
assertion then follows directly by applying Theorem 4.22. ∎

##  6\. Discussions and Conclusions

Figure 4. If we do not impose the $l$-Lipschitz continuity constraint on the
decoder mapping $D$, the resulting trial manifold $D(Z_{B})$ may resemble a
space-filling curve.

In the definition of the decoder width, we impose the $l$-Lipschitz continuity
condition as well as the restriction $\|\boldsymbol{z}\|_{2}\leq 1$. Both of
them are required to give a reasonable notion of width. If either of the
conditions is dropped, the width could become identically zero even for latent
dimension $n=1$, since $D(Z_{B})$ or $D(Z)$ may resemble the space-filling
curve as illustrated in Figure 4. In such cases, the corresponding ROM would
indeed have a strong representational power, and be capable of approximating
any desired solutions to the parametric PDEs. However, it leaves all the
challenges to the online process of finding the latent vector
$\boldsymbol{z}$, as the corresponding optimization problem has become highly
non-convex. This makes the ROM unuseful in the realistic settings.

We also note that, although these two conditions are theoretically essential,
they might not be introduced explicitly in the numerical implementation of the
MAD method. As we have explained in [35], the hard constraint
$\|\boldsymbol{z}\|_{2}\leq 1$ can be relaxed to a soft constraint, i.e. an
additional penalty term in the objective function during the offline pre-
training and the online fine-tuning stages. Furthermore, we represent the
decoder mapping $D:Z\to\mathcal{U}$ as a neural network trained via stochastic
gradient descent (SGD), which is typically known to exhibit the implicit
regularization phenomenon [29, 28]. Even if the explicit Lipschitz continuity
constraint is not imposed on the decoder mapping during the offline pre-
training stage, we would expect to obtain a $D$ without the ill-posedness that
is theoretically possible. Further numerical verification may be helpful in
this respect, and is left for future work.

The current work admits many possible extensions, especially when we focus on
the parametric PDEs on a variable domain. While the analysis in this paper is
limited to the elliptic case, it may be extended to a broader range of PDEs,