# Learning the Hodgkin-Huxley Model
with Operator Learning Techniques

Edoardo Centofanti  Massimiliano Ghiotto  Luca Franco Pavarino

###### Abstract

We construct and compare three operator learning architectures, DeepONet,
Fourier Neural Operator, and Wavelet Neural Operator, in order to learn the
operator mapping a time-dependent applied current to the transmembrane
potential of the Hodgkin-Huxley ionic model. The underlying non-linearity of
the Hodgkin-Huxley dynamical system, the stiffness of its solutions, and the
threshold dynamics depending on the intensity of the applied current, are some
of the challenges to address when exploiting artificial neural networks to
learn this class of complex operators. By properly designing these operator
learning techniques, we demonstrate their ability to effectively address these
challenges, achieving a relative $L^{2}$ error as low as 1.4% in learning the
solutions of the Hodgkin-Huxley ionic model.

###### keywords:

Neural Operator , Hodgkin-Huxley model

††journal: Computer Methods in Applied Mechanics and Engineering

lstlisting

[label1]organization=Dipartimento di Matematica, Università di Pavia,
addressline=Via Adolfo Ferrata, 5, city=Pavia, postcode=27100, country=Italy

##  1 Introduction

Scientific Machine Learning is a rapidly developing field combining the
versatility of machine learning techniques, based on using data in order to
learn or recognize patterns, with more traditional modelling techniques based
on partial differential equations (PDE) [43, 15]. Scientific Machine Learning
has been recently categorized into three main areas [2]: PDE discovery, PDE
solvers through artificial neural networks and Operator Learning. The first
area is related to inverse problems [34], while the last two are related to
direct problems and they have been shown to be effective in reducing the
solution time of many traditional PDE solvers. Physics Informed Neural
Networks (PINNs) and their variants [10, 45, 3, 55, 37] are popular examples
of PDE solvers based on artificial neural networks: they have represented a
revolution in the field, privileging a physics-oriented approach in optimizing
the network parameters over a more traditional data-oriented approach.
However, PINNs are trained in order to solve efficiently a single problem
instead of a whole class of problems, where the solutions are the image of an
operator depending on an input function over a given domain. In order to
address this task, the third scientific machine learning branch, known as
operator learning, has been developed over the last few years [33, 30, 53, 39,
40]. Operator learning has been applied to several problems, especially PDE
systems depending on parameters or forcing terms, and also in industrial
applications [44, 4]. Among the emerging approaches are DeepONet [33] and
neural operator [27] based architectures, such as Fourier [30] and Wavelet
[53] Neural Operators, which have shown remarkable capabilities in learning
non-linear operators on regular or equispaced domains.

In this paper, we construct and study operator learning techniques to learn
the dynamics of ionic models, a very important class of dynamical systems that
describe the rich dynamics of excitable cells and biochemical reactions. These
models have significant applications in fields such as Neuroscience to
Computational Cardiology, see e.g. [9, 21, 46, 20, 35, 49]. The solutions of
ionic models are strongly dependent on time-dependent currents applied to the
system. In particular, they exhibits a threshold effect, with almost no
response from the system when the intensity of the applied current is below a
certain threshold, and a stiff nonlinear response for applied currents above
threshold. Furthermore, the complex dynamics of ionic models can include the
presence of limit cycles, bursting phenomena, and bifurcations. These
properties make this class of problems hard to learn by artificial neural
networks, despite their well known non-linear approximation capabilities [7,
12, 19, 51].

An interesting prototypical ionic model to study is undoubtedly the Hodgkin-
Huxley model [20]. It has been recently studied in the context of numerical
machine learning in its fractional version with PINNs [52] and with multi-
fidelity methods [50]. Other recent works have recovered the Hodgkin-Huxley
dynamics through recurrent maps built and trained as feed forward neural
networks [28], through an extension of the neural ODE architecture [5, 6] and
through a ResNet trained with short-term trajectories in the phase space [48].
Synchronization and oscillatory processes involving this model have also been
studied through data driven machine learning approaches [24, 47].

In this work, we exploit operator learning techniques in order to learn the
underlying non-linear operator, mapping time-dependent square-shaped applied
currents to the Hodgkin-Huxley transmembrane potential. In particular, we
construct three architectures, namely DeepONet, Fourier Neural Operator, and
Wavelet Neural Operator, and study their performance in learning the Hodgkin-
Huxley transmembrane potential dynamics.

This paper is organized as follows. In Section 2, we review the Hodgkin-Huxley
model, its dynamics and the response of the system when different currents are
applied. In Section 3, we present some mathematical details of operator
learning and the architectures employed in this work and we briefly review the
link between operator learning and differential equations theory. Finally, in
Section 4, we present our results and compare the learning strategies used,
studying in particular how the generalization error changes with different
architectures and their hyperparameters.

##  2 The Hodgkin-Huxley model

The Hodgkin-Huxeley (HH) model [20, 21] has been introduced in order to model
the propagation of electrical signals through the squid giant axon and it is
among the most studied biological dynamical systems in neuroscience and
computational biology [54, 16, 41]. It has been recently source of inspiration
for ANNs with neumorphic activations, such as the spiking neural networks [36,
23].

Electrical activity in excitable cells (e.g. neurons or cardiac cells) is
sustained and propagated via ionic currents through the cellular membrane.
Different concentration of ionic species between the inside and the outside of
the cell are responsible for electrochemical gradients, namely an electric
potential between the intra and the extracellular space. The HH model starts
from the hypothesis that the ionic currents are responsible for the generation
of this potential through ionic-specific channels (also known as gates). The
resulting currents across the membrane are strong enough to cause membrane
depolarization. In particular, Hodgkin and Huxley determined experimentally
that the current contributions are mainly three: a voltage-gated persistent
$\mathrm{K}^{+}$ current with four activation gates, a voltage-gated transient
$\mathrm{Na}^{+}$ current with three activation gates and one inactivation
gate, and an Ohmic leak current, $I_{L}$, mostly carried by $\mathrm{Cl}^{-}$
ions. During the depolarization phase the membrane ionic permeability, related
to sodium and potassium conductances, changes in order to restore the membrane
potential. This re-polarizes the membrane through a sudden increase in the
membrane potential (repolarization phase). These changes in the membrane
permeability are due to the activation and inactivation of the ionic channels
at a rate given by the so called gating variables.

The 0D version of this model, describes the variation in time of the membrane
voltage $V$ (in $mV$) through the axon membrane, when a current stimulus,
$I_{app}$ (in $\mu A/cm^{2}$), is applied to the system. $V$ also depends on
three gating variables, $m$, $h$ and $n$, which act as the probability for
each ionic species to pass through the cellular membrane.

Using the modern conventions employed in [8], the model can be written as the
following ODE system:

$\begin{cases}C_{m}\frac{dV}{dt}=-\left(g_{Na}m^{3}h(V-V_{Na})+g_{K}n^{4}(V-V_{K})+g_{L}(V-V_{L})\right)+I_{app}\\\\[6.0pt]
\displaystyle\frac{dm}{dt}=\alpha_{m}(V)(1-m)-\beta_{m}(V)m\\\\[8.0pt]
\displaystyle\frac{dh}{dt}=\alpha_{h}(V)(1-h)-\beta_{h}(V)h\\\\[8.0pt]
\displaystyle\frac{dn}{dt}=\alpha_{n}(V)(1-n)-\beta_{n}(V)n\\\ \end{cases}$
(1)

where we consider the following parameters and functions:

$\displaystyle\alpha_{m}(V)$
$\displaystyle=0.1(-V-50)\left[\exp\left(\frac{-V-50}{10}\right)-1\right]^{-1},$
$\displaystyle\\!\\!\beta_{m}(V)$
$\displaystyle\\!\\!=4\exp\left(\frac{-V-75}{18}\right)$
$\displaystyle\alpha_{h}(V)$
$\displaystyle=0.07\exp\left(\frac{-V-75}{20}\right),$
$\displaystyle\\!\\!\\!\\!\beta_{h}(V)$
$\displaystyle\\!\\!=\\!\\!\\!\left[\exp\left(\frac{-V-45}{10}\right)\\!+\\!1\right]^{-1}$
$\displaystyle\alpha_{n}(V)$
$\displaystyle\\!\\!=0.01(-V-65)\left[\exp{\left(\frac{-V-65}{10}\right)}-1\right]^{-1},$
$\displaystyle\\!\\!\beta_{n}(V)$
$\displaystyle\\!\\!=0.125\exp\left(\frac{-V-75}{80}\right)$ $\displaystyle
C_{m}=1\,\mu F/cm^{2},\,\,V_{Na}=40\,mV,\,\,V_{K}=-87\,mV,$ $\displaystyle
V_{L}=-64.387\,mV,\,\,g_{Na}=120\,mS/cm^{2},\,\,g_{K}=36\,mS/cm^{2},$
$\displaystyle g_{L}=0.3\,mS/cm^{2},$

and the initial conditions: $V_{0}=-75mV$, $n_{0}=0.317$, $m_{0}=0.05$ and
$h_{0}=0.595$.

Various current inputs $I_{app}$ can be applied, resulting in different
responses from the system. In our setup, we considered $I_{app}$ sampled from
the set of single square functions with different intensity and width. An
interesting property to point out is the translation invariance of the
response: two pulses with the same intensity and width, but different starting
time, will result in an identical solution up to a translation.

Figure 1: Different $I_{app}$ pulses lead to different responses from the
system. A pulse with an under-threshold height leads to zero peaks and a
negligible perturbation of the system from its equilibrium state. Once the
threshold value is crossed (around $2.21\,\mu A/cm^{2}$ with these
parameters), the system responds with a single peak or more peaks depending on
the input current.

We notice that the system (1) can be seen as an integral operator mapping the
time dependent applied current to the membrane potential $V$ [25] and exploits
the Volterra expansion for the reduced solution of an ODE system, which, in
the case of the Hodgkin-Huxley model, reduces the 4 equations ODE system to an
equivalent Integrate-And-Fire model for the membrane potential. Using the
notation introduced in this work:

$V(t)=\eta(t-t^{f})+\int_{0}^{\infty}\varepsilon^{(1)}(t-t^{f};s)I_{app}(t-s)\,ds+\mathcal{O}(t^{2})$
(2)

where the expansion is stopped after the first order term, giving a good
approximation for the solution $V(t)$ in (1). The kernels $\eta$ and
$\varepsilon^{(1)}$ describe the form of the spike and the postsynaptic
potential, respectively. Higher order terms depend on other kernels
$\varepsilon^{(i)}$, where $i$ is the order, and describe the nonlinear
interaction between several input pulses, but they are often negligible. We
notice that the information related to the gating variables $m$, $h$ and $n$
is recovered through the kernels $\eta$ and $\varepsilon^{i}$, which are
evaluated from the eigenvalues of the linearization of the system (1) we
started with.

##  3 Operator Learning

Operator learning aims to approximate an unknown operator $G$, which often
takes the form of the solution operator associated with a differential
equation [2]. Given pairs of data $(f,u)$, where $f\in\mathcal{A}$ and
$u\in\mathcal{U}$ are from function spaces on domains
$\Omega\in\mathbb{R}^{d}$ and $\Omega^{\prime}\in\mathbb{R}^{d^{\prime}}$
respectively, the goal is learning by mean of artificial neural networks, a
potentially nonlinear and not always known operator
$G:\mathcal{A}\rightarrow\mathcal{U}$ which maps $f$ to $u$.

What is learned is an approximation $\hat{G}_{\theta}$ of the ground truth
operator $G$. The approximation due to the neural network architecture
employed, the optimizer used to minimize the network parameters and the number
and location of the collocation points for the elements in the dataset are all
potential sources of error [42].

Throughout this work, we will explore and compare the performance of operator
learning with three architectures, namely Deep Operator Networks (DeepONets)
[33], Fourier Neural Operators (FNOs) [30] and Wavelet Neural Operators (WNOs)
[53].

###  3.1 Deep Operator Networks

The first theoretical result regarding the approximation property of nonlinear
operators defined on infinite-dimensional spaces by shallow neural networks
was provided in [7]. This result has been recently extended to deep neural
networks (DNN) and a novel framework, called Deep Operator Network (DeepONet),
has been proposed in order to learn this class of operators [33]. We report
the following result:

######  Theorem 1 (Universal Approximation Theorem for Operator).

Let $\sigma$ be a continuous non-polynomial function, $\mathcal{X}$ is a
Banach space, $\mathcal{K}_{1}\subset\mathcal{X}$,
$\Omega\subset\mathbb{R}^{d}$ are two compact sets in $\mathcal{X}$ and
$\mathbb{R}^{d}$ respectively, $\mathcal{A}$ is a compact set in
$C(\mathcal{K}_{1})$, G a nonlinear continuous operator, mapping $\mathcal{A}$
into $\mathcal{U}\equiv C(\Omega)$. Then, for any $\varepsilon>0$, there exist
positive integers $n$,$p$ and $m$, and constants $c_{i}^{k}$, $\xi_{ij}^{k}$,
$\theta_{i}^{k}$, $\zeta_{k}$ $\in\mathbb{R}$, $w_{k}\in\mathbb{R}^{d}$,
$x_{j}\in\mathcal{K}_{1}$, $i=1,\cdots,n$, $k=1,\cdots,p$, and $j=1,\cdots,m$,
such that

$\left|G(f)(y)-\sum_{k=1}^{p}\sum_{i=1}^{n}c_{i}^{k}\sigma\left(\sum_{j=1}^{m}\xi_{ij}^{k}f(x_{j})+\theta_{i}^{k}\right)\sigma(w_{k}\cdot
y+\zeta_{k})\right|<\varepsilon$ (3)

holds for all $f\in\mathcal{A}$ and $y\in\Omega$. In this context,
$C(\mathcal{K})$ is the Banach space of continuous functions defined on
$\mathcal{K}$ with norm $||f||_{C(\mathcal{K})}=\max_{x\in K}|f(x)|$.

From (3) we can identify the structure of the DeepONet architecture. The
operator $G$ is approximated by a scalar product between two artificial neural
networks, called branch network and trunk network, with the same output
dimension $p$:

$\hat{G}_{\theta}(f)(y)=\sum_{k=1}^{p}\underbrace{b_{k}(f(x_{1}),...,f(x_{m}))}_{{}\text{branch
net}}\underbrace{t_{k}(y)}_{{}\text{trunk net}}$ (4)

Since our goal is learning operators taking functions as the input, we have to
represent the input functions to the branch discretely, so that network
training and consequent validation can be performed. This implies defining an
encoder [29]:

######  Definition 1 (Encoder).

Given a set of sensor points $x_{j}\in\mathcal{K}_{1}$, for $j=1,\cdots,m$ we
define the linear mapping

$\mathcal{E}:\mathcal{A}\rightarrow\mathbb{R}^{m},\qquad\mathcal{E}(f)=(f(x_{1}),\cdots,f(x_{m})),$

as the encoder mapping. The encoder $\mathcal{E}$ is well-defined as one can
evaluate continuous functions pointwise.

The only condition required for the encoding is that the sensor locations
$\left\\{x_{1},\dots,x_{m}\right\\}$ are the same but not necessarily on a
lattice for all input functions $f$, while we do not enforce any constraints
on the output locations $y$. In our numerical experiments, we actually
implement a different encoding strategy, which will be detailed in section 4.

Figure 2: Schematic DeepONet architecture. The outputs of the branch and the
trunk networks, here represented as forward neural networks, are multiplied
through a scalar product in order to approximate the operator $G$.

Finally, we define a reconstructor (or decoder) as the last operation
performed in a DeepONet.

######  Definition 2 (Reconstructor or Decoder).

Given the trunk as a neural network
$\tau:\mathbb{R}^{n}\rightarrow\mathbb{R}^{p+1}$,
$y=(y_{1},\cdots,y_{n})\mapsto\\{\tau_{k}(y)\\}^{p}_{k=0}$, with $y\in
U\subset\mathbb{R}^{n}$, we define a $\tau$-induced reconstructor (or decoder)
as the map

$D=D_{\tau}:\mathbb{R}^{p}\rightarrow C(U),\qquad
D_{\tau}(\alpha_{k}):=\tau_{0}(y)+\sum_{k=1}^{p}\alpha_{k}\tau_{k}(y).$ (5)

The definition above formalizes the scalar product in (4). We notice that each
$\tau_{k}$ is the composition of linear operators and activation functions.
$\tau_{0}$ is an extra bias often added in practical implementations, which in
(3) has been set to zero without loss of generality.

The architecture introduced in [33] is shown in Fig. 2. In their classical
formulation, DeepONets are composed of DNN, but also other architectures
involving Convolutional Neural Networks (CNNs), Residual Neural Networks
(ResNets), and Recurrent Neural Networks (RNNs) like Gated Recurrent Unit
(GRU) and Long Short Time Memory (LSTM) models are possible [33, 4, 17].

###  3.2 Neural Operator

The remaining two architectures which represent the object of the present work
are based on the Neural Operator architecture [27]. Again, in the following we
use $C(\Omega_{t}\times\Omega_{t},\,\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}})$
to denote the Banach space of continuous functions from
$\Omega_{t}\times\Omega_{t}$ to $\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}}$
equipped with the sup-norm
$||g||_{\infty}:=\sup\\{|g(\tau)|:\tau\in\Omega_{t}\times\Omega_{t}\\},$ $g\in
C(\Omega_{t}\times\Omega_{t},\,\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}})$.

######  Definition 3 (Neural Operator).

Assuming that the input functions $f\in\mathcal{A}$ are
$\mathbb{R}^{d}$-valued and defined on the bounded domain
$\Omega\subset\mathbb{R}^{d}$ while the output functions $u\in\mathcal{U}$ are
$\mathbb{R}^{d_{u}}$ valued and defined on the bounded domain
$\Omega^{\prime}\subset\mathbb{R}^{d^{\prime}}$, we define the Neural Operator
as the architecture $\hat{G}_{\theta}:\mathcal{A}\rightarrow\mathcal{U}$ with
the following structure:

  * 1.

Lifting: Through a pointwise function
$\mathbb{R}^{d}\rightarrow\mathbb{R}^{v_{0}}$, map the input
$\\{f:\Omega\rightarrow\mathbb{R}^{d}\\}\mapsto\\{v_{0}:\Omega\rightarrow\mathbb{R}^{d_{v_{0}}}\\}$
to its first hidden representation. A suitable choice is $d_{v_{0}}>d$, thus
this is called lifting operation performed by a fully local operator.

  * 2.

Iterative Kernel Integration: For $t=0,\cdots,T-1$, each hidden representation
is mapped to the next, namely
$\\{v_{t}:\Omega_{t}\rightarrow\mathbb{R}^{d_{v_{t}}}\\}\mapsto\\{v_{t+1}:\Omega_{t+1}\rightarrow\mathbb{R}^{d_{v_{t+1}}}\\}$,
through the action of the sum of a local linear operator, a non-local integral
kernel operator, and a bias function, composing the sum with a fixed pointwise
nonlinearity. Here we set $\Omega_{0}=\Omega$ and $\Omega_{T}=\Omega^{\prime}$
and impose that $\Omega_{t}\subset\mathbb{R}^{d_{t}}$ is a bounded domain.

  * 3.

Projection: Using a pointwise function
$\mathbb{R}^{d_{v_{T}}}\rightarrow\mathbb{R}^{d_{u}}$, map the last hidden
representation
$\\{v_{T}:\Omega^{\prime}\rightarrow\mathbb{R}^{d_{v_{T}}}\\}\mapsto\\{u:\Omega^{\prime}\rightarrow\mathbb{R}^{d_{u}}\\}$
to the output function. In this case the suitable choice of dimensionality is
$d_{v_{t}}>d_{u}$, thus this is a projection step performed by a fully local
operator.

From the definition above, we thus have

$\hat{G}_{\theta}:=Q\circ\sigma_{T}\left(W_{T-1}+K_{T-1}+b_{T-1}\right)\circ\cdots\circ\sigma_{1}\left(W_{0}+K_{0}+b_{0}\right)\circ
P$ (6)

with $P:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d_{v_{0}}}$,
$Q:\mathbb{R}^{d_{v_{T}}}\rightarrow\mathbb{R}^{d_{u}}$ are the local lifting
and projection operators, $W_{t}\in\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}}$
are local linear operators (matrices),
$K_{t}:\\{v_{t}:\Omega_{t}\rightarrow\mathbb{R}^{d_{v_{t}}}\\}\rightarrow\\{v_{t+1}:\Omega_{t+1}\rightarrow\mathbb{R}^{d_{v_{t+1}}}\\}$
are integral kernel operators,
$b_{t}:\Omega_{t+1}\rightarrow\mathbb{R}^{d_{v_{t+1}}}$ are bias functions and
$\sigma_{t}$ are fixed activation functions acting locally as maps
$\mathbb{R}^{v_{t+1}}\rightarrow\mathbb{R}^{v_{t+1}}$. Input and output
dimensions $d_{v_{0}},\cdots,d_{v_{T}}$ as well as the domains of definition
$\Omega_{1},\cdots,\Omega_{T-1}$ are hyperparameters of the architecture.
Furthermore, as local maps we consider their action as pointwise, namely for
lifting and projection we have respectively $(P(a))(x)=P(a(x))$ for $x\in D$
and $(Q(a))(x)=Q(a(x))$ for $x\in\Omega^{\prime}$. For the activation, we have
instead $(\sigma_{t}(v_{t+1}))(x)=\sigma(v_{t+1}(x))$ for any
$x\in\Omega_{t+1}$.

We now provide three admissible definitions for the integral kernel operators
employed in (6).

######  Definition 4 (Integral Kernel Operators).

The following definitions for the integral kernel operators $K_{t}$ are
admitted [27]:

  * 1.

Let $\kappa^{(t)}\in$
$C(\Omega_{t+1}\times\Omega_{t};\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}})$ and
let $\nu_{t}$ be a Borel measure on $\Omega_{t}$ (normally simply the Lebesgue
measure on $\mathbb{R}^{d_{t}}$). $K_{t}$ is defined by

$(K_{t}(v_{t}))(x)=\int_{\Omega_{t}}\kappa^{(t)}(x,y)v_{t}(y)\,d\nu_{t}(y),\qquad\forall
x\in\Omega_{t+1}.$ (7)

  * 2.

Let $\kappa^{(t)}\in$
$C(\Omega_{t+1}\times\Omega_{t}\times\mathbb{R}^{d}\times\mathbb{R}^{d};\mathbb{R}^{d_{v_{t+1}}\times
d_{v_{t}}})$. $K_{t}$ is defined by

$(K_{t}(v_{t}))(x)=\int_{\Omega_{t}}\kappa^{(t)}(x,y,f\left(\Pi_{t+1}(x)\right),f\left(\Pi_{t}(y)\right))v_{t}(y)\,d\nu_{t}(y),\qquad\forall
x\in\Omega_{t+1}.$ (8)

where $\Pi_{t}:\Omega_{t}\rightarrow\Omega_{t}$ are fixed projection maps.

  * 3.

Let $\kappa^{(t)}\in$
$C(\Omega_{t+1}\times\Omega_{t}\times\mathbb{R}^{d_{v_{t}}}\times\mathbb{R}^{d_{v_{t}}};\mathbb{R}^{d_{v_{t+1}}\times
d_{v_{t}}})$. $K_{t}$ is defined by

$(K_{t}(v_{t}))(x)=\int_{\Omega_{t}}\kappa^{(t)}(x,y,v_{t}\left(\Pi_{t}(x)\right),v_{t}\left(y\right))v_{t}(y)\,d\nu_{t}(y),\qquad\forall
x\in\Omega_{t+1}.$ (9)

where $\Pi_{t}:\Omega_{t}\rightarrow\Omega_{t}$ are fixed projection maps,
often taken to be the identity.

From now on, we will consider $K_{t}$ as in the first definition. We now need
to define the single hidden layer update

######  Definition 5 (Single Hidden Layer Update).

Given $K_{t}$ as in (7), we define the single hidden layer update rule as

$v_{t+1}(x)=\sigma_{t+1}\left(W_{t}v_{t}\left(\Pi_{t}(x)\right)+\int_{\Omega_{t}}\kappa^{(t)}(x,y)v_{t}(y)\,d\nu_{t}(y)+b_{t}(x)\right)\qquad\forall
x\in\Omega_{t+1}.$ (10)

where $\Pi_{t}:\Omega_{t+1}\rightarrow\Omega_{t}$ are fixed projection maps.

Having defined the general structure for the Neural Operator, we can now
describe the FNO and the WNO by simply specializing the integral kernel (7)
for each architecture.

####  3.2.1 Fourier Neural Operator

The FNO works with the integral kernel parameterized in the Fourier space. We
thus need to define the Fourier transform and anti-transform.

######  Definition 6 (Fourier transform and anti-transform).

Let
$\mathcal{F}:L^{2}(\Omega,\mathbb{C}^{n})\to\ell^{2}(\mathbb{Z}^{d},\mathbb{C}^{n})$
denote the Fourier transform and $\mathcal{F}^{-1}$ its inverse. Given $u\in
L^{2}(\Omega,\mathbb{C}^{n})$ and
$\hat{u}\in\ell^{2}(\mathbb{Z}^{d},\mathbb{C}^{n})$, we define

$\begin{split}&\left(\mathcal{F}u\right)_{j}(k)=\hat{u}_{j}(k)=\langle
u_{j},\varphi_{k}\rangle_{L^{2}(\Omega,\mathbb{C}^{n})},\qquad
j\in\\{1,\cdots,n\\},\quad k\in\mathbb{Z}^{d},\\\
&\left(\mathcal{F}^{-1}\hat{u}\right)_{j}(x)=\sum_{k\in\mathbb{Z}^{d}}\hat{u}_{j}(k)\varphi^{-1}_{k}(x),\qquad
j\in\\{1,\cdots,n\\},\quad x\in\Omega\\\ &\varphi_{k}(x):=e^{-2\pi ik\cdot
x},\qquad x\in\Omega\end{split}$ (11)

For the FNO, $\Omega_{t}=\mathbb{T}^{d}$, namely the domains considered for
each layer are the unit torus. We also assume that the integral operator is in
the form (7), i.e. it does not depend on the input function $f$ or on the
output of each single hidden layer update $v_{t}$. In particular, let us
consider $\kappa^{(t)}\in
C(\Omega_{t}\times\Omega_{t},\,\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}})$ such
that

$(K_{t}(v_{t}))(x)=\int_{\Omega_{t}}\kappa^{(t)}(x,y)v_{t}(y)\ dy\quad\forall
x\in\Omega.$

Here, the integral kernel
$\kappa^{(t)}:\Omega_{t}\times\Omega_{t}\to\mathbb{R}^{d_{v_{t+1}}\times
d_{v_{t}}}$ is a function parameterized by some parameters $\theta_{t}$ in a
suitable parameters space $\Theta_{t}\subset\mathbb{R}^{d_{v_{t+1}}\times
d_{v_{t}}}$. We can thus also identify
$\kappa^{(t)}=\kappa^{(t)}_{\theta_{t}}$. Supposing that
$\kappa^{(t)}_{\theta_{t}}(x,y)=\kappa^{(t)}_{\theta_{t}}(x-y)$, we can apply
the convolution theorem and write

$(K_{t}(v_{t}))(x)=\int_{\Omega}\kappa^{(t)}_{\theta_{t}}(x-y)v_{t}(y)\,dy=\mathcal{F}^{-1}\left(\mathcal{F}\left(\kappa^{(t)}_{\theta_{t}}\right)\cdot\mathcal{F}(v_{t})\right)(x)\qquad\forall
x\in\Omega_{t}$ (12)

Where the dot operation is defined as

$\mathcal{F}\left(\kappa^{(t)}_{\theta_{t}}\right)\cdot\mathcal{F}(v_{t}):=\sum_{k\in\mathbb{Z}}\mathcal{F}\left(\kappa^{(t)}_{\theta_{t}}\right)(k)\cdot\mathcal{F}(v_{t})(k)=\sum_{k\in\mathbb{Z}}\hat{\kappa}^{(t)}_{\theta_{t}}(k)\cdot\hat{v}_{t}(k).$
(13)

Taken $k\in\mathbb{Z}^{d}$ as frequency mode, for $k$ fixed we have that
$\mathcal{F}\left(\kappa^{(t)}_{\theta_{t}}\right)(k)\in\mathbb{C}^{d_{v_{t+1}}\times
d_{v_{t}}}$ and $\mathcal{F}(v_{t})(k)\in\mathbb{C}^{d_{v_{t}}}$. This means
that $\kappa^{(t)}_{\theta_{t}}$ can be parameterized directly by its Fourier
coefficients, thus,

$(K_{t}(v_{t}))(x)=\mathcal{F}^{-1}\left(R_{\theta_{t}}\cdot\mathcal{F}(v_{t})\right)(x),\qquad\forall
x\in\Omega_{t}.$ (14)

where $R_{\theta_{t}}=\hat{\kappa}^{(t)}_{\theta_{t}}(k)$ for $k$ fixed. Fig.
3 reports a scheme relative to the FNO architecture considered in this work.

We must make some approximations from continuous space to discrete ones to
have a finite-dimensional parameterization. This approximation of the FNO is
called pseudo$(\Psi)$-Fourier Neural Operator ($\Psi$-FNO) [26]. In this case,
the domain $\Omega_{t}$ is discretized with $J\in\mathbb{N}$ points, thus
$v_{t}$ can be treated as a tensor in $\mathbb{C}^{J\times d_{v_{t}}}$. Since
integrals cannot be calculated exactly, this leads to considering the Fourier
series truncated at a maximal mode $k_{\max}$, such that

$k_{\max}=|Z_{k_{\max}}|=\big{|}\\{k\in\mathbb{Z}^{d}\ :\ |k_{j}|\leq
k_{\max,j},\text{ for }j=1,\dots,d\\}\big{|}.$

In practical implementation, the Fourier transform is replaced by the Fast
Fourier Transform (FFT), and the weight tensor $R_{\theta_{t}}$ is
parameterized as a complex-valued tensor
$R_{\theta_{t}}\in\mathbb{C}^{k_{\max}\times d_{v_{t+1}}\times d_{v_{t}}}$ for
$i=1,\dots,L$. From now on we represent $R_{\theta_{t}}$ with only $R$ to
simplify the notation. Finally, we note that if $v_{t}$ is real-valued and we
want that $v_{t+1}$ to be real-valued too, we can enforce conjugate symmetry
in tensor $R$, namely $R(-k)_{j,l}=\overline{R}(k)_{j,l}$.

Figure 3: Schematic FNO architecture.

####  3.2.2 Wavelet Neural Operator

The Wavelet Neural Operator [53] learns the network parameters in the wavelet
space, which is both frequency and spatial localized. The WNO has a similar
structure to FNO but exploits wavelet integrals instead of Fourier integrals.

Again, we impose translation invariance for $\kappa^{(t)}_{\theta_{t}}$ and we
can rewrite our integral kernel (7) as

$(K_{t}(v_{t}))(x)=\int_{\Omega}\kappa^{(t)}_{\theta_{t}}(x-y)v_{t}(y)\,dy\qquad\forall
x\in\Omega.$ (15)

As in the previous case, we exploit the convolution theorem, but we apply it
in order to parameterize $\kappa^{(t)}_{\theta_{t}}$ in the wavelet domain.

A function $\psi\in L^{2}(\Omega,\mathbb{R})$ is called orthonormal mother
wavelet if it can be used to define a complete orthonormal basis for
$L^{2}(\Omega,\mathbb{R})$. A set of orthonormal functions for this space in
this setting is built from the mother wavelet through scaling and shifting
operations, obtaining a family of functions
$\\{\psi_{k,l}:k,l\in\mathbb{Z}\\}$ with the following form:

$\psi_{k,l}(x)=2^{\frac{k}{2}}\psi\left(2^{k}x-l\right)$ (16)

Figure 4: Schematic WNO architecture.

We can now define $\mathcal{W}$ and $\mathcal{W}^{-1}$, the forward and
inverse wavelet transform.

######  Definition 7.

Given $g\in L^{2}(\Omega,\mathbb{R}^{n})$, $h\in
L^{2}(\mathbb{R}^{+}_{*}\times\mathbb{R},\mathbb{R}^{n})$,
$\alpha\in\mathbb{R}^{+}_{*}$ (i.e. $\mathbb{R}^{+}$ including 0) and
$\beta\in\mathbb{R}$, we define the forward wavelet transform $\mathcal{W}(g)$
as

$(\mathcal{W}g)_{j}(\alpha,\beta)=\int_{\Omega}g_{j}(x)\frac{1}{|\alpha|^{\frac{1}{2}}}\psi\left(\frac{x-\beta}{\alpha}\right)d\,x$
(17)

and the inverse wavelet transform $\mathcal{W}^{-1}(h)$ as

$(\mathcal{W}^{-1}h)_{j}(x)=\frac{1}{C_{\psi}}\int_{0}^{\infty}\int_{\Omega}h_{j}(\alpha,\beta)\frac{1}{|\alpha|^{\frac{1}{2}}}\tilde{\psi}\left(\frac{x-\beta}{\alpha}\right)d\beta\frac{d\alpha}{\alpha^{2}}$
(18)

where $\psi\in L^{2}(\Omega,\mathbb{R})$ is the mother wavelet, $\tilde{\psi}$
is called dual mother wavelet (for our case $\psi=\tilde{\psi}$) and
$C_{\psi}$ is defined as

$C_{\psi}=2\pi\int_{\Omega}\frac{|\psi(k)|^{2}}{|k|}\,dk$ (19)

In the previous definition, $\alpha$ and $\beta$ are scaling and translation
parameters respectively. From (16), $\alpha=2^{-k}$ and $\beta=2^{-k}l$, thus
for each choice of the pair $(k,l)$ exists a pair $(\alpha,\beta)$ and we can
map
$\psi_{k,l}\mapsto\psi_{\alpha,\beta}:=\psi\left(\frac{x-\beta}{\alpha}\right)$.

From the convolution theorem, one can prove also for the wavelet transform
case, as in (12), that

$K_{t}(x)=\mathcal{W}^{-1}\left(R_{\theta_{t}}\cdot\mathcal{W}(v_{t})\right)(x);\quad
x\in\Omega,$ (20)

where $R_{\theta_{t}}$ in this case is the wavelet transform of
$\kappa^{(t)}_{\theta_{t}}$ parameterized by some in a suitable parameters
space $\Theta_{t}\subset\mathbb{R}^{d_{v_{t+1}}\times d_{v_{t}}}$.

Analogously to the $\Psi$-FNO case, in order to implement the WNO, one has to
consider the discrete wavelet transform (DWT). Moreover, $R_{\theta_{t}}$ is
parameterized as a real-valued $(\zeta\times d_{v_{t+1}}\times d_{v_{t}})$
tensor, where $\zeta=n_{s}/2^{k}+(2n_{w}-2)$, with $n_{s}$ length of the
discrete input of the DWT and $n_{w}$ length of the vanishing moment of the
undertaken wavelet, after having applied recursively $k$ times the DWT. In
practice, we fix the level $k$ in (16) such that the corresponding
$\psi_{k,l}$ contain the most important features of the transformed function.
This usually happens when $\psi_{k,l}$ is related to the low frequency
information, corresponding to coefficients of (18) with higher scales $2^{k}$.
Figure 4 shows schematically the WNO architecture described in this work. For
more information about signal processing and wavelet transform we suggest [11,
38, 13].

##  4 Results

In this section we present the results of our numerical tests. We have
considered the architectures introduced in Section 3, and ran experiments
changing the hyperparameters setup, the choices of activation function and the
network complexity, namely the number of neurons and trainable parameters. All
the tests presented in this work have been performed exploiting the PyTorch
library running on an Nvidia Quadro RTX $5000$ GPU, with $16$ GB of memory.

###  4.1 Dataset, loss and optimizer

For our dataset, we have considered 2000 numerical solutions of the problem
(1), divided between 1600 training data and 400 test data. During the training
phase the dataset has been divided into mini batches of size 100. The
numerical solutions in the dataset have been obtained from the same number of
$I_{app}$ applied to the system, randomly chosen. In particular, the square
pulses have been extracted with a uniformely distributed duration over a
timespan of 100 $ms$ and with a uniformely distributed height from 0 to 10
$\mu A/cm^{2}$. The solutions have been sampled on an equispaced 500 points
grid and solved numerically using the MATLAB routine ode15s.

Since our input functions are square pulses in time, the encoder for DeepONet
can be reconstructed in the following way:

$f(x)=\hat{I}\,H(x-t_{i})\,H(t_{f}-x)$ (21)

where $H$ is the Heaviside step function, $\hat{I}$ is the intensity of the
pulse and $t_{i}$ and $t_{f}$ are the starting and the ending time of the
pulse. Considering as encoding strategy the function
$\hat{\mathcal{E}}:\mathcal{A}\rightarrow\mathbb{R}^{3}$ defined as

$\hat{\mathcal{E}}(f)=(t_{i},t_{f},\hat{I})$ (22)

it is possible to save memory resources without losing information, since the
triple $(t_{i},t_{f},\hat{I})$ completely characterizes a single input
function. For FNO and WNO we have instead considered as input the whole signal
discretized in time. As a decoder for the DeepONet we have considered the dot
product between the trunk and the branch network outputs. As loss for all the
architectures explored, we have chosen the relative $L^{2}$ error, namely

$L_{2}=\frac{1}{N}\sum_{i=1}^{N}\frac{||\hat{G}_{\theta}(f_{i})-u_{i}||_{L^{2}(\Omega)}}{||u_{i}||_{L^{2}(\Omega)}}.$
(23)

where $\hat{G}_{\theta}$ is the operator approximated by the architecture of
our choice, while $f_{i},u_{i}$ are respectively the $i$-th input function and
solution of the dataset.

Regarding the parameters optimization, we considered the PyTorch
ReduceOnPlateau policy as learning rate strategy, where the starting learning
rate ($1\cdot 10^{-3}$ for DeepONet and WNO, $7\cdot 10^{-4}$ for FNO) is
reduced of 10% when the loss is non-decreasing for 10 epochs and we have
considered the AdamW algorithm as optimizer.

###  4.2 DeepONet results

In all the DeepONet tests performed, as input for the branch, we have
considered the output of the encoder discussed in the previous section, while
for the trunk, we have considered the 500 equispaced domain points.

For both branch and trunk networks, we have considered feed-forward neural
networks with layer normalization [1] and adaptive activation function [22].
Both branch and trunk networks employ the tanh activation function.

In Table 1 we report our best results for this architecture. The best mean
test loss, evaluated from three trained networks each one with different
randomly initialized parameters is around 2.2% and has been obtained with a
branch and trunk network width of 700 neurons, for a total of 3.4 million
parameters.

In Fig. 5(a) we study the behavior of the train and test losses as a function
of the width of the networks, which is the same among all the layers, from 400
up to 1000 neurons. After an absolute minimum in the test error corresponding
to a network width of 700 neurons, the error increases again up to 2.4%.

In Fig. 6(a) we report four examples of solutions approximated by DeepONet,
for our best model. The four corresponding inputs and the numerical solutions
are taken from the test dataset. Single or double peak solutions are in
general very well approximated, with the $L^{2}$ relative error below the mean
test error. However, solutions with many peaks (e.g. the six-peaks solution
shown in figure) tend to accumulate the error and the approximation
deteriorates in time.

DeepONet | Test Error |  Configuration |  | Total Parameters | Epochs
---|---|---|---|---|---
|  |  Trunk | Branch |  |
| 0.0274$\pm$0.0006 |  [400]*3 | [400]*4 | 1.1M | $10000$
| 0.0274$\pm$0.0007 |  [500]*3 | [500]*4 | 1.7M |
| 0.0243$\pm$0.0014 |  [600]*3 | [600]*4 | 2.5M |
| 0.0220$\pm$0.0010 |  [700]*3 | [700]*4 | 3.4M |
| 0.0231$\pm$0.0004 |  [800]*3 | [800]*4 | 4.5M |
| 0.0231$\pm$0.0002 |  [900]*3 | [900]*4 | 5.6M |
| 0.0235$\pm$0.0005 |  [1000]*3 | [1000]*4 | 7.0M |
FNO |  |  Fourier modes |  |  |
| 0.0180$\pm$0.0018 |  8 |  | 34k | $2000$
| 0.0143$\pm$0.0023 |  16 |  | 58k |
| 0.0149$\pm$0.0039 |  32 |  | 108k |
| 0.0235$\pm$0.0075 |  64 |  | 206k |
| 0.0211$\pm$0.0023 |  128 |  | 402k |
WNO |  |  $R_{\theta_{t}}$ width |  |  |
| 0.0505$\pm$0.0080 |  8 |  | 206k | $2000$
| 0.0413$\pm$0.0040 |  16 |  | 821k |
| 0.0403$\pm$0.0018 |  32 |  | 3.2M |
| 0.0330$\pm$0.0005 |  64 |  | 13M |
| 0.0345$\pm$0.0010 |  128 |  | 52M |
| 0.0345$\pm$0.0018 |  256 |  | 209M |
Table 1: Our best results obtained from the implementation of DeepONet, FNO
and WNO for the prediction of the membrane potential in the Hodgkin-Huxley
model. For this problem the best relative validation error for DeepONet is
around $2.1\%$, penalized by solutions resulting in more than 3 peaks, which
have been particularly challenging to learn. The notation $[w]*L$ indicates a
FNN composed by $L$ layer of width $w$. For FNO the best error is around
$1.4\%$, obtained with 16 Fourier modes. For WNO this value is $3.3\%$ and the
parameterized kernel has been discretized into a tensor $R_{\theta_{t}}$ with
64 as first dimension (width).

(a)

(b)

(c)

Figure 5:  (5(a)) $L^{2}$ train and test loss vs network width for DeepONet.
All the branch nets considered have a depth of 4, while all the trunks
considered have a depth of 3. The width is indicated on the $x$ axis and it is
the same for all the layers and the output layer. A width of 700 neurons is
the best among the ones explored for our problem. The training error is
indicated with a blue line, while the test error is indicated with an orange
line. (5(b)) $L^{2}$ train and test loss vs Fourier modes for the Fourier
transform employed in the FNO. 16 and 32 modes resulted in a better performing
trained network for our problem with our configuration. The training error is
indicated with a blue line, while the test error is indicated with an orange
line. (5(c)) $L^{2}$ train and test loss vs $R_{\theta_{t}}$ width for WNO.
$R_{\theta_{t}}$ is a tensor of trainable parameters in the wavelet space. The
width is indicated on the $x$ axis and it is the same for all the levels of
the wavelet transform. In our numerical tests, a width of 64 has resulted in
the best approximation error. The training error seems to improve with a
superlinear rate when the width is increased. The training error is indicated
with a blue line, while the test error is indicated with an orange line.

###  4.3 FNO results

In all the FNO tests performed, we considered all the $500$ discretization
points as input to the network, without further encoding.

In Table 1 we report our best results for this architecture. Among the hyper-
parameters of the best performing architecture we highlight: lifting dimension
$d_{1}=32$, Maximum number of Fourier modes considered for the FFT
$k_{max}=16$, number of Fourier integral layers $L=3$, Rectified Linear Unit
(ReLU) activation function. The best mean test loss, evaluated from three
trained networks each one with different randomly initialized parameters is
around 1.4% and has been obtained considering 16 Fourier modes in the integral
kernel, for a total of 58k parameters.

In Fig. 5(b) we study how the relative $L^{2}$ error varies when we change the
number of Fourier modes for the FFT employed in the approximation of the FNO
integral kernel. After observing an initial decline, extending beyond 32 modes
resulted in a rise in both test and training errors.

In Fig. 6(b) we report four examples of solutions approximated by FNO, for our
best model. The four corresponding inputs and the numerical solutions are
taken from the test dataset. In this case, we observe that both single and
many peaks solutions are approximated properly and there is little variance
around the mean test error, as also reported in Fig. 8.

(a) Deep Operator Network approximations.

(b) Fourier Neural Operator approximations.

(c) Wavelet Neural Operator approximations.

(d) Input functions.

Figure 6:  (6(a)) DeepONets approximations compared to ode15s solutions.
(6(b)) FNO approximations compared to ode15s solutions. (6(c)) WNO
approximations compared to ode15s solutions. (6(d)) Plot of four samples from
the test set of the input applied currents functions. In the plots above the
numerical solution obtained with ode15s is plotted in blue and the
approximation obtained with neural operators is in red dashed line.

Figure 7: Plot of the train and test relative $L^{2}$ loss selected from our
best training for each case. While DON and FNO show similar train and test
errors, WNO trained with the described configuration has shown about an order
of magnitude gap between the train and the test error during the training.

Figure 8: Distribution of the relative $L^{2}$ validation error on the test
dataset each one selected from our best training for each case. FNO has shown
less variance around the median validation error value. Despite showing a
similar behaviour, WNO trained with the described configuration has 14% of
outliers over 10% of relative error. DON shows more variance between 0% and
10% of relative test error, but not many outliers, which is also the case of
FNO. The median value is the same for all the architectures studied.

###  4.4 WNO results

In all the WNO tests performed, we considered all the $500$ discretization
points as input to the network, without further encoding.

In Table 1 we have reported our best results for this architecture. Among the
hyper-parameters of the best performing trained model we highlight: Gaussian
Error Linear Unit (GELU) [18] activation function, Daubechies wavelet basis
db24 for the DWT implemented in the integral kernel, number of integral layers
$L=3$ and width of the integral kernel parameterized in the wavelet space
$R_{\theta_{t}}$ equal to $256$. The best mean test loss, evaluated from three
trained networks each one with different randomly initialized parameters is
around 3.3% and has been obtained considering a $R_{\theta_{t}}$ width of 64,
for a total of 13 million parameters, the highest number among the
architectures considered.

In Fig. 5(c) we study how the relative $L^{2}$ error varies when we change the
width of the integral kernel parameterized in the wavelet space,
$R_{\theta_{t}}$. We observe a general superlinear reduction of the training
error with respect to the width of $R_{\theta_{t}}$, while the test error
appears to saturate after a width of 64. We also observe that, despite having
obtained good results in terms of training and test losses (also see Fig. 7),
our best case has worse generalization capabilities and employs a much larger
set of parameters, compared to DON and FNO.

In Fig. 6(c) we report four examples of solutions approximated by WNO, for our
best model. The four corresponding inputs and the numerical solutions are
taken from the test dataset. In this case, we observe that both single and
many peaks solutions can be approximated properly, but the trained network
appears more unstable in keeping the test error around the mean value if
compared to DON and FNO, as shown in Fig. 8.

###  4.5 Training Comparison

In this section we compare the training of the three different architectures
considered.

In Fig. 7 the evolution of the loss function through the training epochs,
namely plots of the relative $L^{2}$ error (23) are reported for each one of
the architectures object of our study. The FNO has shown less difference
between the train and the test error. The DON and the WNO have presented
higher loss values in approximating the solutions to our problem, but while
the training error in the DON case is similar to the validation one, in the
WNO after 50 epochs the training error decays significantly, while the test
error has a lower decay rate and saturates quickly. This effect is
particularly evident when the width of the $R_{\theta_{t}}$ operator is
increased.

In Fig. 8 histograms showing the relative $L^{2}$ error distributions over all
test cases for our dataset are reported. An effective training is expected to
have an high percentage of the total samples with a relative error around the
mean value. For the DON and the FNO all the samples are contained in an
interval between $<1\%$ and $16\%$ of validation error. The WNO instead
contains all the samples in a wider interval, with a single outlier greater
than $25\%$ of relative error.

##  5 Conclusions

In this work, we have constructed and studied three different operator
learning architectures, based on DeepONet, Fourier and Wavelet Neural Operator
techniques, to learn the membrane potential dynamics of the Hodgkin-Huxley
ionic model.

FNOs have shown promising results in predicting the membrane potential when
samples are taken from the test dataset, with a mean validation error of 1.4%.
However, it is well known that FNOs need an equispaced domain in order to work
properly, which means a potentially large number of domain points when facing
stiff problems. DeepONets are more flexible in this sense, allowing adaptive
and irregular domains. However, for the Hodgkin-Huxley model they have shown
less capability to learn many-peaks solutions, compared to the other
architectures. Their mean validation error of 2.2% with our best configuration
is still an important result. For further studies it can be useful to explore
for this class of problems also different DeepONet based architectures, e.g.
S-DeepONets [17]. WNOs have shown a very good capability of generalizing
solutions with many peaks over the test dataset, but they required more tuning
in the number of integral layers and in the wavelet type employed, as well as
a larger number of trainable parameters and a consequent larger training time.
Also, the mean validation error of 3.3% is the highest among the architectures
studied.

As a final remark, while at this level there is no real need of employing
these methods in order to get a more efficient solver in terms of time or
complexity for the Hodgkin-Huxley model if compared to the classical ones
(e.g. Runge-Kutta), it is important to study systems which can be challenging
to learn by artificial neural networks due to their properties, in particular
the stiffness of the solutions and the threshold-dependent dynamics. This work
paves the way for employing the same techniques for a whole family of similar
systems, with way more equations involved or even with more dimensions:
further studies in this directions can be useful to design architectures
capable of solving more physiological models in computational cardiology or
neuroscience, making properly trained operator learning models a valid aid for
personalized medicine.

##  6 Acknowledgements

The authors wish to thank Carlo Marcati for helpful discussions and
suggestions. Edoardo Centofanti and Luca F. Pavarino have been supported by
grants of INdAM–GNCS and MUR (PRIN 202232A8AN_002 and PRIN P2022B38NR_001),
funded by European Union - Next Generation EU.

##  7 Code Availability

The codes for the numerical tests presented in this work will be made publicly
available on GitHub.

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[1]organization=St. Petersburg State University, addressline=7-9
Universitetskaya Embankment, postcode=199034, city=Saint Petersburg,
country=Russia [2]organization=Kazan Federal University, addressline=18
Kremlyovskaya street, postcode=420008, city=Kazan, country=Russia
[3]organization=Moscow Polytechnic University, addressline=38 Bolshaya
Semyonovskaya street, postcode=107023, city=Moscow, country=Russia

# Weissler and Bernoulli type inequalities in Bergman spaces

Anton D. Baranov<EMAIL_ADDRESS>Ilgiz R. Kayumov
<EMAIL_ADDRESS>Diana M. Khammatova<EMAIL_ADDRESS>Ramis Sh.
Khasyanov<EMAIL_ADDRESS>
###### Abstract

We consider Weissler type inequalities for Bergman spaces with general radial
weights and give conditions on the weight $w$ in terms of its moments ensuring
that $\|f_{r}\|_{A^{2n}(w)}\leq\|f\|_{A^{2}(w)}$ whenever $n\in\mathbb{N}$ and
$0<r\leq 1/\sqrt{n}$. For noninteger exponents a special case of this
inequality is proved which can be considered as a certain analog of the
Bernoulli inequality. An example of a monotonic weight is constructed for
which these inequalities are no longer true.

###### keywords:

Bergman space , Weissler inequality , Bernoulli inequality

##  1 Introduction

We consider Weissler type inequalities for Bergman spaces in the disc with
general radial weights. While such questions were studied extensively for the
classical weights, the case of general weights apparently was not previously
addressed. The aim of this note is to propose regularity/convexity conditions
on the weight, expresses in terms of its moments, ensuring that some special
cases of the Weissler type inequality hold true. We show that certain
regularity is required since there exist monotonic weights for which even
these special cases of the Weissler type inequality are false.

To describe our results, let us recall some definitions and classical
inequalities. A function $f$, analytic in the unit disk $\mathbb{D}$, belongs
to the Hardy space $H^{p}$, $0<p<\infty$, if

$\|f\|_{H^{p}}:=\sup\limits_{0<r<1}\left(\frac{1}{2\pi}\int\limits_{0}^{2\pi}\lvert
f(re^{i\theta})\rvert^{p}\,d\theta\right)^{\frac{1}{p}}<\infty.$

For $r\in(0,1)$ let $f_{r}(z)=f(rz)$. A well-known result of F. B. Weissler
[9] states that for the Hardy spaces $H^{p}$ and $H^{q}$ ($0<p<q$)

$\|f_{r}\|_{H^{q}}\leq\|f\|_{H^{p}}\ \ \text{for any}\ f\in
H^{p}\qquad\Longleftrightarrow\qquad r\leq\sqrt{\frac{p}{q}}\leq 1.$

Given a summable nonnegative function $w$ on $[0,1)$, we say that a function
$f$, analytic in the unit disk $\mathbb{D}$, belongs to the weighted Bergman
space $A^{p}(w)$ if

$\|f\|_{A^{p}(w)}:=\left(\frac{1}{2\pi}\int\limits_{\mathbb{D}}\lvert
f(z)\rvert^{p}w(\lvert z\rvert)\,dA(z)\right)^{1/p}<\infty,$

where $A$ denotes the planar Lebesgue measure. We define the moments of the
weight $w$ as

$h_{m}=\int\limits_{0}^{1}\rho^{m+1}w(\rho)\,d\rho,\qquad m\geq 0.$

In what follows we always assume the normalization $h_{0}=1$.

The weights $w_{\alpha}(r)=2(\alpha-1)(1-r^{2})^{\alpha-2}$, where $\alpha>1$,
are called classical weights. We will denote Bergman spaces with the weight
$w_{\alpha}$ as $A_{\alpha}^{p}$. For the theory of Bergman spaces see, e.g.,
[4].

The classical Carleman inequality

$\left(\sum\limits_{n=0}^{\infty}\frac{\lvert
a_{n}\rvert^{2}}{n+1}\right)^{\frac{1}{2}}\leq\|f\|_{H^{1}}$

for $f(z)=\sum\limits_{n=0}^{\infty}a_{n}z^{n}\in H^{1}$ implies that
$\|f\|_{A_{2}^{2}}\leq\|f\|_{H^{1}}$ whence it is easy to deduce that

$\|f\|_{A_{2}^{2p}}\leq\|f\|_{H^{p}},\qquad 0<p<\infty.$

The following generalization of the Carleman inequality was proved by J.
Burbea [3]. Let $k\in\mathbb{N}$, $k\geq 2$, and $p=\frac{2}{k}$. Then, for
$f(z)=\sum\limits_{n=0}^{\infty}a_{n}z^{n}\in H^{p}$,

$\|f\|_{A_{2/p}^{2}}=\left(\sum\limits_{n=0}^{\infty}\frac{\lvert
a_{n}\rvert^{2}}{c_{2/p}(n)}\right)^{\frac{1}{2}}\leq\|f\|_{H^{p}},\qquad
c_{\beta}(n)=C_{n+\beta-1}^{n}.$ (1)

It was conjectured in [2] that inequality (1) holds for all $0<p\leq 2$. An
inequality similar to (1), but with a constant slightly worse than 1 on the
right, was proved in [2, 6].

Contractive inequalities for Bergman spaces $A_{\alpha}^{p}$ were also studied
by many authors. In particular, F. Bayart, O. F. Brevig, A. Haimi, J. Ortega-
Cerdà and K.-M. Perfekt [1] proved the following counterpart of the Weissler
inequality. Let $0<p\leq q<\infty$ and $\alpha=\frac{n+1}{2}$ for some
$n\in\mathbb{N}$. Then for $f\in A_{\alpha}^{p}$

$\|f_{r}\|_{A_{\alpha}^{q}}\leq\|f\|_{A_{\alpha}^{p}}\ \ \text{for any}\ f\in
A_{\alpha}^{p}\quad\Longleftrightarrow\quad r\leq\sqrt{\frac{p}{q}}\leq 1.$
(2)

A remarkable progress in these problems was achieved in 2022 (after the first
version of the present note was finished). A. Kulikov [5] proved the
conjectured inequality (1) for all $0<p\leq 2$, he also showed that it is
sharp in every coefficient. Using the results and methods of the paper by
Kulikov, P. Melentijević [7] proved that (2) is true for any $0<p<q<\infty$
such that $q\geq 2$ and for any $\alpha>1$. Moreover, (2) holds for $q<2$ as
well if we assume that $f$ is zero-free, while in general only a lower bound
for $r$ was found in this case.

In the present note we are interested in analogs of (2) for Bergman spaces
with general radial weights. Our goal is to find conditions on the weight for
which a Weissler type inequality is true. We give such conditions in terms of
the moments $h_{2m}$ of the weight $w$. While our present results deal with
very special situations (even integer exponents or a specific choice of a
function) we conjecture that under these or similar conditions on the weight
the results can be extended to the case of general exponents.

Note that by the Cauchy inequality, one always has $h^{2}_{2m}\leq
h_{2(m-1)}h_{2(m+1)}$. Our condition in the next theorem is a converse (in a
sense) inequality.

######  Theorem 1.1.

Let $w$ be a Bergman weight satisfying

$\frac{h_{2m}}{h_{2(m-1)}}\geq\frac{m}{m+1}\frac{h_{2(m+1)}}{h_{2m}}$ (3)

all $m\geq 1$. Then for any $n\in\mathbb{N}$ we have

$\|f_{r}\|_{A^{2n}(w)}\leq\|f\|_{A^{2}(w)}\ \ \text{for any}\ f\in
A^{2}(w)\quad\Longleftrightarrow\quad r\leq\frac{1}{\sqrt{n}}\leq 1.$ (4)

It is easy to see that (3) holds for all classical weights $w_{\alpha}$.

We now turn to Weissler type inequalities between $A^{2}(w)$ and $A^{2q}(w)$
where $q>1$ is an arbitrary real number. This problem is much more complex,
because now one cannot use combinatorics. In fact, a natural conjecture arises
that the inequality $\|f_{r}\|_{A^{2q}(w)}\leq\|f\|_{A^{2}(w)}$ holds for
$r=\frac{1}{\sqrt{q}}$. In other words,

$\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left\lvert
f^{q}\left(\frac{\rho e^{i\theta}}{\sqrt{q}}\right)\right\rvert^{2}\rho
w(\rho)\,d\theta\,d\rho\leq\left(\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\lvert
f(\rho e^{i\theta})\rvert^{2}\rho w(\rho)\,d\theta\,d\rho\right)^{q}.$ (5)

To have the possibility of taking the powers we consider functions
nonvanishing in $\mathbb{D}$. They can be represented as
$f(z)=e^{\varphi(z)}$. Let $\varphi(z)=\sum_{n=0}^{\infty}a_{k}z^{k}$. Since
the norms in $A_{\alpha}^{n}$ are expressed in terms of modulus of
coefficients, we can assume that all $a_{k}\geq 0$. For such functions

$f^{q}\left(z\right)=e^{q\varphi(z)}=\sum_{n=0}^{\infty}\frac{q^{n}}{n!}\left(\sum_{k=0}^{\infty}a_{k}z^{k}\right)^{n}=\sum_{n=0}^{\infty}\frac{q^{n}}{n!}\sum_{k=0}^{\infty}\sum_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}\cdot\ldots\cdot
a_{j_{n}}z^{k}.$

Changing the order of summation and substituting the value of the argument, we
have

$f^{q}\left(\frac{\rho
e^{i\theta}}{\sqrt{q}}\right)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{q^{k}}{k!}\sum_{j_{1}+\ldots+j_{k}=n}a_{j_{1}}\cdot\ldots\cdot
a_{j_{k}}\right)\frac{\rho^{n}e^{i\theta n}}{q^{n/2}}.$

We introduce the following notation.

$g_{nk}=\frac{1}{k!}\sum_{j_{1}+\ldots+j_{k}=n}a_{j_{1}}\cdot\ldots\cdot
a_{j_{k}}.$

Then the left-hand side of (5) can be represented as

$\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left\lvert
f^{q}\left(\frac{\rho e^{i\theta}}{\sqrt{q}}\right)\right\rvert^{2}\rho
w(\rho)\,d\theta\,d\rho=\sum_{n=0}^{\infty}\frac{1}{q^{n}}\left(\sum_{k=0}^{\infty}q^{k}g_{nk}\right)^{2}h_{2n}.$

Similarly, for the right-hand side of (5) we have

$\left(\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\lvert f(\rho
e^{i\theta})\rvert^{2}\rho
w(\rho)\,d\theta\,d\rho\right)^{q}=\left(\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}g_{nk}\right)^{2}h_{2n}\right)^{q}.$

Therefore, the desired inequality takes the following form:

$\sum_{n=0}^{\infty}\frac{1}{q^{n}}\left(\sum_{k=0}^{\infty}q^{k}g_{nk}\right)^{2}h_{2n}\leq\left(\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}g_{nk}\right)^{2}h_{2n}\right)^{q}.$

This inequality looks, in general, quite complicated, so it seems logical to
consider some particular cases. For example, one can take the function
$f(z)=e^{z}$. In this case $g_{nk}=\frac{\delta_{nk}}{k!}$ and the desired
inequality is

$\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}\leq\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)^{q}.$
(6)

Although this is only a particular case, this inequality seems to be of
independent interest because it can be considered as an analog of the
classical Bernoulli inequality for the moment sequences. It should be
emphasized that it is essential here that $h_{2n}$ is a sequence of moments of
some weight and also that this inequality need not be true for an arbitrary
(even monotonic) weight.

The following theorem gives a sufficient condition for the inequality (6). We
have to replace condition (3) from Theorem 1.1 by a stronger condition on the
moments.

######  Theorem 1.2.

Let $w$ be a Bergman weight. If the inequality

$\frac{h_{2m}}{h_{2(m-1)}}\geq\frac{h_{2(m+1)}}{(m+1)h_{2(m-1)}}+\frac{m}{m+1}\frac{h_{2(m+1)}}{h_{2m}}$
(7)

holds for all $m\geq 1$, then

$\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}\leq\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)^{q},\qquad
q\geq 1.$ (8)

In particular, the estimate (7) holds for all standard Bergman weights
$w_{\alpha}$; for them it turns into equality. Thus, it looks plausible that
the weights satisfying (7) are a correct class for generalizations of (2) to
general weights.

However, (4) and (8) do not hold for arbitrary weights and we indeed need to
impose some regularity conditions.

######  Theorem 1.3.

There exists a monotonically decreasing weight $w$ such that the inequality

$\|f_{r}\|_{A^{2q}(w)}\leq\|f\|_{A^{2}(w)}$

does not hold for $f(z)=e^{z}$, $r=\frac{1}{\sqrt{q}}$ and $q=2$ as well as
for $q\in(1,1+\varepsilon)$ for some $\varepsilon>0$.

The paper is organized as follows. In Section 2 we prove Theorem 1.1 which
gives the counterpart of Weissler inequality for Bergman spaces when $q$ is
even and $p=2$. In Section 3 two auxiliary lemmas are proved, while Section 4
is devoted to the proof of Theorems 1.2 and 1.3.

##  2 Proof of Theorem 1.1

We need to prove the following inequality.

$\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\left\lvert
f^{n}\left(\frac{\rho e^{i\theta}}{\sqrt{n}}\right)\right\rvert^{2}\rho
w(\rho)\,d\theta\,d\rho\leq\left(\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\lvert
f(\rho e^{i\theta})\rvert^{2}\rho w(\rho)\,d\theta\,d\rho\right)^{n}.$ (9)

Let $f(z)=\sum\limits_{k=0}^{\infty}a_{k}z^{k}$. Then $f^{n}(z)$ can be
represented as

$f^{n}(z)=\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}\cdot\ldots\cdot
a_{j_{n}}\right)z^{k}.$

The integrals in (9) can be calculated using Parseval identity. Thus, the
left-hand side of (9) is

$\int\limits_{0}^{1}\rho
w(\rho)\sum\limits_{k=0}^{\infty}\left(\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}\cdot\ldots\cdot
a_{j_{n}}\right)^{2}\frac{\rho^{2k}}{n^{k}}\,d\rho=\sum\limits_{k=0}^{\infty}\frac{1}{n^{k}}\left(\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}\cdot\ldots\cdot
a_{j_{n}}\right)^{2}h_{2k}$

and the right-hand side of (9) is

$\left(\int\limits_{0}^{1}\rho
w(\rho)\sum\limits_{k=0}^{\infty}a_{k}^{2}\rho^{2k}\,d\rho\right)^{n}=\left(\sum\limits_{k=0}^{\infty}a_{k}^{2}\int\limits_{0}^{1}\rho^{2k+1}w(\rho)\,d\rho\right)^{n}=\\\
=\sum\limits_{k=0}^{\infty}\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}^{2}\cdot\ldots\cdot
a_{j_{n}}^{2}h_{2j_{1}}\cdot\ldots\cdot h_{2j_{n}}.$

From the Cauchy inequality

$\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{n^{k}}\left(\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}\cdot\ldots\cdot
a_{j_{n}}\right)^{2}\leq\\\
\leq\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{n^{k}}\sum\limits_{j_{1}+\ldots+j_{n}=k}a_{j_{1}}^{2}\cdot\ldots\cdot
a_{j_{n}}^{2}h_{2j_{1}}\cdot\ldots\cdot
h_{2j_{n}}\cdot\sum\limits_{j_{1}+\ldots+j_{n}=k}\frac{1}{h_{2j_{1}}\cdot\ldots\cdot
h_{2j_{n}}}.$

Now it is sufficient to prove that

$\sum\limits_{j_{1}+\ldots+j_{n}=k}\frac{1}{h_{2j_{1}}\cdot\ldots\cdot
h_{2j_{n}}}\leq\frac{n^{k}}{h_{2k}},$

and then the desired inequality will hold term by term.

Inequality (3) implies that for $m\geq n$

$h_{2n}h_{2m}\geq\frac{n}{m+1}h_{2(n-1)}h_{2(m+1)}.$

We need to estimate the product $h_{2j_{1}}\cdot\ldots\cdot h_{2j_{n}}$, where
$j_{1}+\ldots+j_{n}=k$. Without loss of generality we assume that $j_{1}\leq
j_{2}$. Then

$h_{2j_{1}}h_{2j_{2}}\geq\frac{j_{1}}{j_{2}+1}h_{2(j_{1}-1)}h_{2(j_{2}+1)}\geq\frac{j_{1}(j_{1}-1)}{(j_{2}+1)(j_{2}+2)}h_{2(j_{1}-2)}h_{2(j_{2}+2)}\geq\\\
\geq\frac{j_{1}!}{(j_{2}+1)\ldots(j_{2}+j_{1})}h_{0}h_{2(j_{2}+j_{1})}=\frac{j_{1}!j_{2}!}{(j_{1}+j_{2})!}h_{0}h_{2(j_{2}+j_{1})}$

Next, assume, without loss of generality, that $j_{1}+j_{2}\geq j_{3}$. Then

$h_{2j_{1}}h_{2j_{2}}h_{2j_{3}}\geq\frac{j_{1}!j_{2}!}{(j_{1}+j_{2})!}h_{2(j_{1}+j_{2})}h_{2j_{3}}\geq\frac{j_{1}!j_{2}!j_{3}!}{(j_{1}+j_{2}+j_{3})!}h_{2(j_{1}+j_{2}+j_{3})}.$

Proceeding in this way we finally get

$h_{2j_{1}}\cdot\ldots\cdot h_{2j_{n}}\geq\frac{j_{1}!j_{2}!\ldots
j_{n}!}{(j_{1}+j_{2}+\ldots+j_{n})!}h_{2(j_{1}+j_{2}+\ldots+j_{n})}=\frac{j_{1}!j_{2}!\ldots
j_{n}!}{k!}h_{2k}$

Therefore,

$\sum\limits_{j_{1}+\ldots+j_{n}=k}\frac{1}{h_{2j_{1}}\cdot\ldots\cdot
h_{2j_{n}}}\leq\frac{k!}{h_{2k}}\sum\limits_{j_{1}+\ldots+j_{n}=k}\frac{1}{j_{1}!\cdot\ldots\cdot
j_{n}!}=\frac{k!}{h_{2k}}\cdot\frac{n^{k}}{k!}=\frac{n^{k}}{h_{2k}}.$

The equality
$\sum\limits_{j_{1}+\ldots+j_{n}=k}\frac{1}{j_{1}!\cdot\ldots\cdot
j_{n}!}=\frac{n^{k}}{k!}$ is obtained by comparing the coefficients of

$e^{nz}=\sum\limits_{k=0}^{\infty}\frac{n^{k}z^{k}}{k!}=\sum\limits_{k=0}^{\infty}\sum\limits_{j_{1}+\ldots+j_{m}=k}\frac{z^{k}}{j_{1}!\cdot\ldots\cdot
j_{n}!}.$

This finishes the proof of (9).

It remains to show the sharpness of the value $r=\frac{1}{\sqrt{n}}$. For this
we use the standard test function $f(z)=1+\varepsilon z$ where $\varepsilon$
can be taken arbitrarily small. Then the left-hand side of the desired
inequality (i.e., $\|f_{r}\|^{2n}_{A^{2n}(w)}$) will be equal to

$\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\big{\lvert}1+\varepsilon
r\rho e^{i\theta}\big{\rvert}^{2n}\rho
w(\rho)\,d\theta\,d\rho=1+n^{2}\varepsilon^{2}r^{2}\int\limits_{0}^{1}\rho^{3}w(\rho)\,d\rho+O(\varepsilon^{4})$

and the right-hand side (i.e., $\|f\|^{2n}_{A^{2}(w)}$) will be given by

$\left(\frac{1}{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\big{\lvert}1+\varepsilon\rho
e^{i\theta}\big{\rvert}^{2}\rho
w(\rho)\,d\theta\,d\rho\right)^{n}=1+n\varepsilon^{2}\int\limits_{0}^{1}\rho^{3}w(\rho)\,d\rho+O(\varepsilon^{4}).$

Comparing these values with $\varepsilon\to 0$, we see that if the desired
inequality holds then $nr^{2}\leq 1$, so $r\leq\frac{1}{\sqrt{n}}$.

Let us show that the theorem applies to all classical weights $w_{\alpha}$. In
this case

$h_{2n}=\frac{\Gamma(\alpha)\Gamma(n+1)}{\Gamma(\alpha+n)},$

whence

$\frac{h_{2(n+1)}}{(n+1)h_{2(n-1)}}+\frac{n}{n+1}\frac{h_{2(n+1)}}{h_{2n}}=\frac{n}{n+\alpha-1}=\frac{h_{2n}}{h_{2(n-1)}}.$

Thus, inequality (7) turn into equality and, in particular, the weaker
condition (3) is satisfied. Theorem 1.1 is proved.

##  3 Two lemmas

######  Lemma 3.1.

Let $g_{n}$ be a sequence of real numbers, such that $0<g_{n}\leq g_{n-1}$ for
$n\geq 1$, $g_{0}=1$ and

$\frac{g_{n}}{g_{n-1}}\geq\frac{n}{n+1}\frac{g_{n+1}}{g_{n}},\qquad n\geq 1.$
(10)

Then the inequality

$\sum\limits_{k=0}^{n}\frac{g_{n-k}\cdot
g_{k+2}}{((n-k)!)^{2}(k!)^{2}(k+1)(k+2)}\leq\sum\limits_{k=0}^{n}\frac{g_{n-k+1}\cdot
g_{k+1}}{((n-k)!)^{2}(k!)^{2}(n-k+1)(k+1)}.$

holds for all $n\geq 0$.

###### Proof.

We can write

$T_{n}:=\sum\limits_{k=0}^{n}\frac{g_{n-k}\cdot
g_{k+2}}{((n-k)!)^{2}(k!)^{2}(k+1)(k+2)}-\\\
-\sum\limits_{k=0}^{n}\frac{g_{n-k+1}\cdot
g_{k+1}}{((n-k)!)^{2}(k!)^{2}(n-k+1)(k+1)}=\\\
=\frac{g_{0}\,g_{n+2}}{(n!)^{2}(n+1)(n+2)}-\frac{g_{1}\,g_{n+1}}{(n!)^{2}(n+1)}+\sum\limits_{k=0}^{n-1}\frac{g_{n-k}\,g_{k+2}(2k-n+1)}{((n-k)!)^{2}((k+1)!)^{2}(k+2)}.$

Case 1. Let $n$ be even, i.e. $n=2m$. Then it can be represented as

$T_{2m}:=\frac{g_{0}\,g_{2m+2}}{((2m)!)^{2}(2m+1)(2m+2)}-\frac{g_{m+1}^{2}}{\left(m!\right)^{2}\left(\left(m+1\right)!\right)^{2}\left(m+1\right)}+\\\
+\sum\limits_{k=0}^{m-1}\frac{g_{m+k+2}\,g_{m-k}}{\left(\left(m+k+2\right)!\right)^{2}\left(\left(m-k\right)!\right)^{2}}(-2m+4k^{2}+8k+2).$

We denote

$t_{k}=\frac{g_{m+k+2}\,g_{m-k}}{\left(\left(m+k+2\right)!\right)^{2}\left(\left(m-k\right)!\right)^{2}}(-2m+4k^{2}+8k+2)$

and introduce the sequence $s_{k}$ the following way.

$s_{0}=-\frac{g_{m+1}^{2}}{\left(m!\right)^{2}\left(\left(m+1\right)!\right)^{2}\left(m+1\right)},$
$s_{k}=s_{k-1}\cdot\frac{\left(m-k+2\right)}{\left(m+k+1\right)}\frac{g_{m-k+1}\,g_{m+k+1}}{g_{m-k+2}\,g_{m+k}}+t_{k-1},\qquad
0\leq k\leq m.$

It is easy to show by induction that

$s_{k}=-\frac{(2k+1)g_{m+k+1}\,g_{m-k+1}}{\left(\left(m-k\right)!\right)^{2}\left(\left(m+k+1\right)!\right)^{2}\left(m-k+1\right)}.$

Therefore, $s_{k}<0$. One can note that at each iteration, the previously
obtained result is multiplied by some coefficient and added to $t_{k-1}$,
starting with the term $s_{0}$, which participates in $T_{2m}$.

From the condition (10) one has

$\frac{p}{q+1}g_{p-1}\,g_{q+1}\leq
g_{p}\,g_{q}\leq\frac{q}{p+1}g_{p+1}g_{q-1},\qquad p<q.$ (11)

Hence,

$g_{m-k+1}\,g_{m+k+1}\leq\frac{\left(m+k+1\right)}{\left(m-k+2\right)}g_{m-k+2}\,g_{m-k}.$

We conclude that

$s_{k}\geq s_{k-1}+t_{k-1}\geq s_{k-2}+t_{k-2}+t_{k-1}\geq\ldots\geq
s_{0}+\sum_{q=0}^{k-1}t_{k}.$

Therefore,

$T_{2m}\leq\frac{g_{0}\,g_{2m+2}}{((2m)!)^{2}(2m+1)(2m+2)}+s_{m}.$

From the property (11)

$s_{m}=-\frac{(2m+1)g_{2}\,g_{2m+1}}{((2m+1)!)^{2}}\leq-\frac{g_{0}\,g_{2m+2}}{((2m)!)^{2}(2m+1)(2m+2)}$

and $T_{n}=T_{2m}\leq 0$.
Case 2. Now let $n$ be odd. In this case

$T_{n}:=\frac{g_{0}\,g_{n+2}}{(n!)^{2}(n+1)(n+2)}+\sum\limits_{k=0}^{\frac{n-1}{2}}\frac{g_{\frac{n+3}{2}+k}\,g_{\frac{n+1}{2}-k}}{\left(\left(\frac{n+3}{2}+k\right)!\right)^{2}\left(\left(\frac{n+1}{2}-k\right)!\right)^{2}}(-n+4k^{2}+4k-1).$

As before, we denote

$t_{k}=\frac{g_{\frac{n+3}{2}+k}\,g_{\frac{n+1}{2}-k}}{\left(\left(\frac{n+3}{2}+k\right)!\right)^{2}\left(\left(\frac{n+1}{2}-k\right)!\right)^{2}}(-n+4k^{2}+4k-1)$

and introduce the sequence

$s_{0}=t_{0}=-\frac{2g_{\frac{n+3}{2}}\,g_{\frac{n+1}{2}}}{\left(\left(\frac{n+3}{2}\right)!\right)^{2}\left(\left(\frac{n-1}{2}\right)!\right)^{2}\left(\frac{n+1}{2}\right)},$
$s_{k}=s_{k-1}\cdot\frac{\left(\frac{n+3}{2}-k\right)}{\left(\frac{n+3}{2}+k\right)}\cdot\frac{g_{\frac{n+3}{2}+k}\,g_{\frac{n+1}{2}-k}}{g_{\frac{n+1}{2}+k}\,g_{\frac{n+3}{2}-k}}+t_{k}.$

It can be shown by induction that

$s_{k}=-\frac{2(k+1)g_{\frac{n+3}{2}+k}\,g_{\frac{n+1}{2}-k}}{\left(\left(\frac{n+3}{2}+k\right)!\right)^{2}\left(\left(\frac{n-1}{2}-k\right)!\right)^{2}\left(\frac{n+1}{2}-k\right)}.$

Using the same considerations as for the even case, we get

$T_{n}\leq\frac{g_{0}\,g_{n+2}}{(n!)^{2}(n+1)(n+2)}+s_{\frac{n-1}{2}}.$

The property (11) gives

$s_{\frac{n-1}{2}}=-\frac{(n+1)g_{n+1}\,g_{1}}{((n+1)!)^{2}}\leq-\frac{g_{0}\,g_{n+2}}{(n!)^{2}(n+1)(n+2)}.$

Therefore, $T_{n}\leq 0$ for both cases. This completes the proof. ∎

######  Lemma 3.2.

If the Bergman weight $w$ satisfies inequality (7) for all $n\geq 1$, then

$\frac{h_{2(n+1)}}{n+1}\leq
h_{2n}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right),\qquad
n\geq 1.$ (12)

###### Proof.

Let $n=1$. From the Hölder inequality, we get the estimate

$h_{2}^{n}=\left(\int\limits_{0}^{1}\rho^{3}w(\rho)\,d\rho\right)^{n}\leq\int\limits_{0}^{1}\rho^{2n+1}w(\rho)\,d\rho\cdot\left(\int\limits_{0}^{1}\rho
w(\rho)\,d\rho\right)^{n-1}=h_{2n}$

(recall that $h_{0}=1$). It also follows from condition (7) with $m=1$, that

$h_{4}\leq\frac{2h_{2}^{2}}{h_{2}+1}.$ (13)

Hence,

$2h_{2}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right)-h_{4}\geq
2h_{2}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2}^{k}}{(k!)^{2}}\right)-\frac{2h_{2}^{2}}{h_{2}+1}\geq\\\
\geq 2h_{2}\ln I_{0}(2\sqrt{h_{2}})-2h_{2}^{2}\left(1-\frac{h_{2}}{4}\right).$

Here $I_{0}$ is the modified Bessel function. Recall that the modified Bessel
functions $I_{n}$, defined as

$I_{n}(z)=\sum\limits_{m=0}^{\infty}\frac{(z/2)^{2m+n}}{m!(m+n)!},$

are connected with the classical Bessel functions $J_{n}$ by the equality
$I_{n}(z)=e^{-\frac{in\pi}{2}}J_{n}(ze^{\frac{i\pi}{2}})$.

In what follows we will use the following three properties of the functions
$I_{n}$ which can be found in [8, §2.12]:

$I_{n}^{\prime}(x)=I_{n+1}(x)+\frac{n}{x}I_{n}(x),$ (14)
$\frac{2n}{x}I_{n}(x)=I_{n-1}(x)-I_{n+1}(x),$ (15)
$2I_{n}^{\prime}(x)=I_{n-1}(x)+I_{n+1}(x).$ (16)

We denote $2\sqrt{h_{2}}=t$ and consider the function

$u_{1}(t)=16\ln I_{0}(t)-4t^{2}+\frac{t^{4}}{4}.$

Since $0<h_{2}\leq 1$, our goal is to show that $u_{1}(t)\geq 0$ for $0\leq
t\leq 2$. By the property (14) for $n=0$, we have

$u_{1}^{\prime}(t)=\frac{16I_{1}(t)}{I_{0}(t)}-8t+t^{3}.$

Since $I_{0}(t)>0$ for $t\geq 0$, we need to consider

$16I_{1}(t)-8tI_{0}(t)+t^{3}I_{0}(t).$

Using the property (15) with $n=1$, we get the expression

$t^{3}I_{0}(t)-8tI_{2}(t)=:tu_{2}(t)$

We take the derivative of $u_{2}(t)$ and use the properties (14)–(16) to
obtain that

$u_{2}^{\prime}(t)=2tI_{0}(t)+t^{2}I_{1}(t)-4(I_{1}(t)+I_{3}(t))\geq
2tI_{0}(t)-4I_{1}(t)-4I_{3}(t)=\\\
=2tI_{2}(t)-4I_{3}(t)=2tI_{2}(t)-\frac{2}{3}tI_{2}(t)+\frac{2}{3}tI_{4}(t)=\frac{4}{3}tI_{2}(t)+\frac{2}{3}tI_{4}(t).$

Since $I_{n}(x)>0$, $x>0$, it follows that

$u_{2}(t)\geq u_{2}(0)=0$

and so $u_{1}$ also increases. Then

$u_{1}(t)\geq u_{1}(0)=0.$

Thus, we proved that

$\frac{h_{4}}{2}\leq
h_{2}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right).$

Let us suppose that the inequality (12) is proved for $n-1$. Obviously, (7)
implies

$\frac{h_{2n}}{h_{2(n-1)}}\geq\frac{n}{n+1}\frac{h_{2(n+1)}}{h_{2n}},\qquad
n\geq 1.$

Hence,

$\frac{h_{2(n+1)}}{n+1}\leq\frac{h_{2n}^{2}}{nh_{2(n-1)}}\leq\frac{h_{2n}}{h_{2(n-1)}}\cdot
h_{2(n-1)}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right),$

which is exactly the desired inequality. ∎

##  4 Proofs of Theorems 1.2 and 1.3

###### Proof of Theorem 1.2.

We consider the function

$\varphi(q)=\ln\left(\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}\right)-q\ln\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right).$

Its derivative is

$\varphi^{\prime}(q)=\frac{\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}(n+1)}h_{2(n+1)}}{\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}}-\ln\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right).$

We also take the second derivative

$\varphi^{\prime\prime}(q)=\frac{\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}(n+1)(n+2)}h_{2(n+2)}\cdot\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}-\left(\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}(n+1)}h_{2(n+1)}\right)^{2}}{\left(\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}\right)^{2}}.$

Multiplying the series, we see that the first term in the numerator is

$\sum\limits_{n=0}^{\infty}\left(\sum\limits_{k=0}^{n}\frac{h_{2(n-k)}\cdot
h_{2(k+2)}}{((n-k)!)^{2}(k!)^{2}(k+1)(k+2)}\right)q^{n},$

and the second is equal to

$\sum\limits_{n=0}^{\infty}\left(\sum\limits_{k=0}^{n}\frac{h_{2(n-k+1)}\cdot
h_{2(k+1)}}{((n-k)!)^{2}(k!)^{2}(n-k+1)(k+1)}\right)q^{n}.$

Since (7) implies (10) for $g_{n}=h_{2n}$, we can apply Lemma 3.1 and see that
for all $n\geq 0$

$\sum\limits_{k=0}^{n}\frac{h_{2(n-k)}\cdot
h_{2(k+2)}}{((n-k)!)^{2}(k!)^{2}(k+1)(k+2)}\leq\sum\limits_{k=0}^{n}\frac{h_{2(n-k+1)}\cdot
h_{2(k+1)}}{((n-k)!)^{2}(k!)^{2}(n-k+1)(k+1)}.$

It follows that $\varphi^{\prime\prime}(q)\leq 0$. Therefore,
$\varphi^{\prime}(q)\leq\varphi^{\prime}(1)$ when $q\geq 1$. We denote

$\psi=\sum\limits_{n=0}^{\infty}\frac{h_{2(n+1)}}{(n!)^{2}(n+1)}-\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\cdot\ln\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right).$

From Lemma 3.2 we know that

$\frac{h_{2(n+1)}}{n+1}\leq
h_{2n}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right),\qquad
n\geq 1.$

We proved that all the terms of the series

$\psi=\sum\limits_{n=0}^{\infty}\frac{1}{(n!)^{2}}\left(\frac{h_{2(n+1)}}{n+1}-h_{2n}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right)\right),$

are negative except the case $n=0$. To get rid of this exceptional term, we
add it to the term with $n=1$.

$h_{2}-\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right)+\frac{h_{4}}{2}-h_{2}\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right)=\\\
=h_{2}+\frac{h_{4}}{2}-(1+h_{2})\ln\left(\sum\limits_{k=0}^{\infty}\frac{h_{2k}}{(k!)^{2}}\right)\leq\\\
\leq
h_{2}+\frac{h_{4}}{2}-(1+h_{2})\ln\left(1+h_{2}+\frac{h_{4}}{4}\right)=:y(h_{2},h_{4}).$

The derivative of this expression with respect to $h_{4}$

$\frac{\partial y}{\partial h_{4}}=\frac{2+2h_{2}+h_{4}}{8+8h_{2}+2h_{4}}>0,$

so $y$ increases with respect to $h_{4}$. Recall that we have
$h_{4}\leq\frac{2h_{2}^{2}}{1+h_{2}}$ (see (13)). Therefore, it is sufficient
to substitute this value into $y$:

$y\left(h_{2},\frac{2h_{2}^{2}}{1+h_{2}}\right)=h_{2}+\frac{h_{2}^{2}}{1+h_{2}}-(1+h_{2})\ln\left(1+h_{2}+\frac{h_{2}^{2}}{2+2h_{2}}\right).$

The function
$v(h_{2})=y\left(h_{2},\frac{2h_{2}^{2}}{1+h_{2}}\right)/(1+h_{2})$ has the
same sign as the expression $y\left(h_{2},\frac{2h_{2}^{2}}{1+h_{2}}\right)$.
Calculations show that its derivative equals to

$v^{\prime}(h_{2})=-\frac{h_{2}^{2}(2+3h_{2}+3h_{2}^{2})}{(1+h_{2})^{3}(2+4h_{2}+3h_{2}^{2})}\leq
0.$

Then

$y\left(h_{2},\frac{2h_{2}^{2}}{1+h_{2}}\right)\leq y(0,0)=0$

for all $h_{2}\in(0,1)$. Thus, the term with $n=0$ does not affect the sign
and $\psi<0$.

We proved that $\varphi^{\prime}(q)<0$, and so

$\varphi(q)\leq\varphi(1)=\ln\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)-\ln\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)=0.$

Since logarithm is an increasing function, we finally obtain the desired
inequality

$\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}\leq\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)^{q}.$

∎

###### Proof of Theorem 1.3.

Consider the weight

$w^{*}(\rho)=\left\\{\begin{aligned} \frac{3}{2\rho},\qquad
0\leq\rho\leq\frac{1}{2},\\\ \frac{1}{2\rho},\qquad\frac{1}{2}<\rho\leq
1.\end{aligned}\right.$

Then $h_{n}=\frac{1+2^{-n}}{2(1+n)}$. One can note that $h_{0}=1$, so this
weight is admissble. It is not continuous but we always can approximate it
with a continuous or even smooth monotonically decreasing weight with
arbitrarily close moments $h_{n}$.

However, one can check numerically that the derivative of the function

$\psi(q)=\sum\limits_{n=0}^{\infty}\frac{q^{n}}{(n!)^{2}}h_{2n}-\left(\sum\limits_{n=0}^{\infty}\frac{h_{2n}}{(n!)^{2}}\right)^{q}$

at $q=1$ is positive ($\approx 0.0048$). So, the function $\psi$ is equal to
zero when $q=1$ and then increases. It means that this value will be positive
at least on some interval $(1,1+\varepsilon)$, $\varepsilon>0$, and (6) does
not hold. Also numerical calculations show that $\psi(2)\approx 0.0105>0$. ∎

######  Remark 4.1.

As we have shown at the end of Section 2, for the classical weights
$w_{\alpha}$ inequality (7) turns into equality. Let us show that all power
weights $w(\rho)=(m+2)\rho^{m}$, $m\geq 0$ (not necessarily integer), also
satisfy (7). Indeed, we have $h_{2n}=\frac{2+m}{2+m+2n}$ and

$\frac{h_{2n}}{h_{2(n-1)}}-\left(\frac{h_{2(n+1)}}{(n+1)h_{2(n-1)}}+\frac{n}{n+1}\frac{h_{2(n+1)}}{h_{2n}}\right)=\frac{2m}{(1+n)(2+m+2n)(4+m+2n)}\geq
0.$

## Funding

The results of Sections 3 and 4 (Theorem 1.2) were obtained with the support
of Russian Science Foundation grant 22-71-10094. Other results of the paper
were obtained with the support of Ministry of Science and Higher Education of
the Russian Federation, agreement No 075-15-2021-602.

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