# Speeding up Computational Morphogenesis with
Online Neural Synthetic Gradients

Yuyu Zhang⋆§, Heng Chi†§, Binghong Chen⋆,
Tsz Ling Elaine Tang† Lucia Mirabella†, Le Song⋆ and Glaucio H. Paulino⋆
⋆Georgia Institute of Technology
†Siemens Corporate Technology
§Email: yuyu<EMAIL_ADDRESS>

###### Abstract

A wide range of modern science and engineering applications are formulated as
optimization problems with a system of partial differential equations (PDEs)
as constraints. These PDE-constrained optimization problems are typically
solved in a standard discretize-then-optimize approach. In many industry
applications that require high-resolution solutions, the discretized
constraints can easily have millions or even billions of variables, making it
very slow for the standard iterative optimizer to solve the exact gradients.
In this work, we propose a general framework to speed up PDE-constrained
optimization using online neural synthetic gradients (ONSG) with a novel two-
scale optimization scheme. We successfully apply our ONSG framework to
computational morphogenesis, a representative and challenging class of PDE-
constrained optimization problems. Extensive experiments have demonstrated
that our method can significantly speed up computational morphogenesis (also
known as topology optimization), and meanwhile maintain the quality of final
solution compared to the standard optimizer. On a large-scale 3D optimal
design problem with around 1,400,000 design variables, our method achieves up
to 7.5x speedup while producing optimized designs with comparable objectives.

###### Index Terms:

computational morphogenesis, PDE-constrained optimization, neural synthetic
gradients, deep learning

##  I Introduction

Many science and engineering applications in various domains are addressed by
turning them into optimization problems constrained by physical laws in forms
of a system of partial differential equations (PDEs). The resulting
mathematical problems are commonly known as the PDE-constrained optimization
[1, 2]. In practice, PDE-constrained optimization is widely used in optimal
design [3, 4] , optimal control [5, 6] , and inverse estimation [7, 8]
problems. More concretely, PDE-constrained optimization has been applied in
many scientific problems such as optical tomography in biomedical imaging [9],
radiation treatment planning in radiotherapy [10], magnetic drug delivery in
nanoscience [11] and architected materials design in material science [12]. It
has also been used in a variety of engineering problems, such as flow control
in fluid mechanics [13], structure design in aerospace, civil and
architectural engineering [14, 15, 16], and optimal design in automobile
industry and robotics [17, 18, 19].

PDE-constrained optimization problems are typically solved in a standard
discretize-then-optimize approach, which first discretizes the governing PDE
into finite-dimensional residual form, and then optimizes the design variables
using iterative gradient-based method. In many applications that require high-
resolution solution, such as optimal design, the discretized constraints are
high-dimensional, which can easily have millions or even billions of
variables. When solving such problems, each iteration can take hours or even
days to complete, and solving the entire problem typically takes hundreds of
iterations [20]. It is of important practical value to speed up large-scale
PDE-constrained optimization.

Recent works [21, 22, 23, 24] have attempted to accelerate the optimal
structure design by directly predicting the final design using deep learning
models. These iteration-free approaches require to collect plenty of training
data beforehand, and each training sample is obtained by completely solving a
problem with the standard optimizer. These previous works have severe
limitations in four aspects. First, collecting sufficient optimization results
as training samples is computationally prohibitive, especially for large-scale
3D design problems which can take days to generate a single training sample.
Second, due to the fixed dimensionality of training data, their trained models
are not able to be generalized to solve new design problems with different
design domains. Third, these approaches are not scalable due to the capacity
limit of deep learning models. These works only consider 2D or tiny-scale 3D
problems with less than 10,000 design variables. Lastly, these approaches are
not accurate enough and the predicted final designs tend to have flaws like
hanging or disconnected members. This happens because the above-mentioned
approaches fail to embed underlying physical laws during the offline learning.
Due to these limitations, a general framework for speeding up PDE-constrained
optimization that can be accurate, scalable, efficient and generalizable at
the same time is much needed.

To address the above challenges, we propose an online neural synthetic
gradients (ONSG) framework. Our method is involved in the optimization
process, and employs the optimization history to train a deep learning model
in online fashion. The deep learning model can generate synthetic gradients to
drive the optimization process. Synthetic gradients are much cheaper to
obtain, and can replace exact gradients to skip the expensive solve. During
the optimization, ONSG switches between applying exact and synthetic
gradients. Since the model is trained online, our method is generalizable and
can be directly applied to new problems without any pre-collected training
data. Moreover, in ONSG, we also propose a novel two-scale optimization
scheme, which decomposes the design variables into local subsets but still
incorporates global information. With the two-scale scheme, the ONSG can deal
with large-scale optimization problems with millions of design variables. We
experiment on challenging 3D optimal design problems with more than one
million design variables. Our method shows superior performance in terms of
accuracy and scalability, which achieves up to 7.5x speedup over the standard
optimizer with comparable or even better final objective achieved.

By designing ONSG, we make three key contributions:

  * •

We propose a general online learning framework named ONSG for speeding up PDE-
constrained optimization. By interacting with the optimization process, our
method can be directly applied to new problems without requiring to pre-
collect the training data.

  * •

We propose a two-scale optimization scheme to decompose the training data into
local subsets and incorporate global information. With this technique, the
dimensionality of training data can be constant, no matter how large-scale the
optimization problem is.

  * •

We successfully apply ONSG to computational morphogenesis, which is a
challenging and representative application in large-scale PDE-constrained
optimization. Our method achieves significant speedup over the standard
optimizer with comparable or even better final objective.

##  II Related Work

Solving large-scale PDE-constrained optimization is a computationally
intensive task as the computation of exact gradient at each iteration is
computationally expensive and, meanwhile, the complete optimization process
typically takes hundreds of iterations. Traditionally, to accelerate such
large-scale problems, one has to resort to parallel computing [25, 26, 20],
advanced iterative solvers [27, 28], or multi-scale and multi-resolution
approaches [29, 30, 31].

More recently, attempts have been made to apply machine learning to accelerate
PDE-constrained optimization problems. The surrogate-assisted optimization
approaches [32], which use various machine learning models such as Gaussian
process [33] and deep neural networks [34] to replace expensive function
evaluations, are popular to speed up gradient-free optimization problems.
However, those approaches are not suitable for topology optimization problems
which typically require gradient-based optimizers [35]. In topology
optimization, the dominant idea of the existing attempts roots in an
iteration-free and offline approach, that is, using machine learning to
replace the optimizer either partially or completely so that, once the models
are trained, one could directly employ them to the map initial guesses or
partially converged solutions to the final one. This idea has been widely
explored in optimal structure design problems. For example, Ulu et al. [36]
developed a data-driven approach for predicting optimized topologies under
various loading cases. In this approach, the optimized topologies obtained
under a wide range of loading cases are treated as training samples and the a
feed-forward neural network is adopted together with the Principal Component
Analysis (PCA) to establish a mapping for predicting the optimized topology
under a given loading scenario. Sosnovik and Oseledets [21] introduced a deep
learning-based framework which uses CNN to predict the final optimal design
based on two input images: a partially converged design and its change with
respect to the previous step. By doing so, the total number of optimization
steps are reduced. Later, Banga et al. [23] took a similar idea and extended
the work of Sosnovik and Oseledets to 3D design problems and to incorporate
additional inputs such as external loads and boundary conditions. More
recently, Yu et al. [22] proposed a two-stage prediction procedure to produce
nearly-optimal structural design without iterations. When given a design
domain, load and boundary condition, the procedure first uses a trained CNN-
based encoder and decoder to predict a low resolution structural design and,
then, the predicted low-resolution design is mapped to a high resolution one
using a trained Generative Adversarial Network (GAN). All the above-mentioned
frameworks are proposed in the context of density-based topology optimization
approach. With the same goal, Lei et al. [37] developed a machine learning-
based framework for the Moving Morphable Component (MMC) approach. The
framework first uses the MMC approach to generate a set of training samples
with different external loads locations and subsequently apply either
Supported Vector Regression (SVR) or K-Nearest Neighbors (KNN) to establish an
instantaneous mapping between the design parameters and optimized topology.
Although achieving partial successes on small- to medium-scale problems, these
iteration-free and offline approaches have severe limitations which prevent
their success in large-scale problems: 1) they are not efficient due to the
requirement of collecting a lot of optimization results as training samples;
2) they are not generalizable to solve new problems; 3) they are not scalable
due to the capacity limit of deep learning models; 4) they are not accurate
and the predicted solution tend to have flaws.

##  III Methodology

In this section, we first introduce the background of PDE-constrained
optimization problems. We then present the online neural synthetic gradients
(ONSG) framework to speed up PDE-constrained optimization, which includes an
online gradients learning method and a two-scale optimization scheme. As an
algorithmic overview, the procedure of speeding up PDE-constrained
optimization with ONSG is summarized in Algorithm 1.

###  III-A Problem Background

The general form of PDE-constrained optimization can be expressed as

$\displaystyle\min_{\bm{z},\bm{u}}$ $\displaystyle\mathcal{J}(\bm{z},\bm{u})$
(1) subject to $\displaystyle\bm{r}(\bm{z},\bm{u})=\bm{0},$

where $\bm{z}\in\mathbb{R}^{N_{d}}$ and $\bm{u}\in\mathbb{R}^{N_{s}}$ are the
design variables and the state variables, respectively; $\mathcal{J}(\cdot)$
is the objective function; $\bm{r}(\bm{z},\bm{u})=\bm{0}$ are the discretized
PDE constraints. Due to the infinite-dimensional nature of the governing PDE,
the standard discretize-then-optimize approach first discretizes it into
finite-dimensional residual form using the finite element method [38, 39]. The
discretization results in a system of equations denoted as
$\bm{r}(\bm{z},\bm{u})=\bm{0}$, which can be either linear or nonlinear in
terms of the state variables $\bm{u}$ depending on the problem.

To solve the optimization problem above, the Nested Analysis and Design (NAND)
approach is typically used [2, 3], which treats the state variables $\bm{u}$
as an implicit function of the design variables $\bm{z}$, namely
$\bm{u}(\bm{z})$. Thus the original optimization problem in Eq. (1) is recast
as

$\displaystyle\min_{\bm{z}}$
$\displaystyle\mathcal{J}(\bm{z},\bm{u}(\bm{z})).$ (2)

The NAND approach requires to solve the discretized PDE constraints
$\bm{r}(\bm{z},\bm{u})=\bm{0}$ for the design variables $\bm{z}$ at every
iteration in the optimization process. In many industry-scale applications,
high-resolution solutions are required, which means that the state variables
$\bm{u}$ are usually of high dimensionality, e.g., in the order of million or
even higher. If we denote the design and state variables at optimization
iteration $k$ as $\bm{z}^{(k)}$ and $\bm{u}^{(k)}$, respectively, the
gradients of the objective function $\mathcal{J}(\cdot)$ at that iteration is
given by

$\displaystyle\bm{g}^{(k)}=\frac{\partial\mathcal{J}}{\partial\bm{z}}\big{(}\bm{z},\bm{u}(\bm{z})\big{)}\Bigg{|}_{\bm{z}=\bm{z}^{(k)}}=\frac{\partial\mathcal{J}}{\partial\bm{z}}(\bm{z},\bm{u})\Bigg{|}_{\begin{subarray}{c}\bm{z}=\bm{z}^{(k)}\\\
\bm{u}=\bm{u}^{(k)}\end{subarray}}$
$\displaystyle-\Bigg{[}\bigg{(}\frac{\partial\bm{r}}{\partial\bm{u}}(\bm{z},\bm{u})\Bigg{|}_{\begin{subarray}{c}\bm{z}=\bm{z}^{(k)}\\\
\bm{u}=\bm{u}^{(k)}\end{subarray}}\bigg{)}^{-\top}\frac{\partial\mathcal{J}}{\partial\bm{u}}(\bm{z},\bm{u})\Bigg{|}_{\begin{subarray}{c}\bm{z}=\bm{z}^{(k)}\\\
\bm{u}=\bm{u}^{(k)}\end{subarray}}\Bigg{]}\frac{\partial\bm{r}}{\partial\bm{z}}(\bm{z},\bm{u})\Bigg{|}_{\begin{subarray}{c}\bm{z}=\bm{z}^{(k)}\\\
\bm{u}=\bm{u}^{(k)}\end{subarray}}.$ (3)

As shown in the equation above, to compute the gradients $\bm{g}^{(k)}$, one
needs to solve $\bm{r}(\bm{z}^{(k)},\bm{u})=\bm{0}$ for $\bm{u}^{(k)}$, and
compute
$(\partial\bm{r}/\partial\bm{u})^{-\top}(\partial\mathcal{J}/\partial\bm{u})$,
both of which are very computationally expensive in large-scale optimization
problems.

input : Initialized design variables $\bm{z}^{(0)}$; original discretized PDE
constraints $\bm{r}(\bm{z},\bm{u})=\bm{0}$; coarse-scale discretized PDE
constraints $\bm{r}_{C}(\bm{z}_{C},\bm{u}_{C})=\bm{0}$; objective function
$\mathcal{J}(\bm{z},\bm{u})$; total number of iterations $K$; iterations
applying synthetic gradients $\mathcal{I}_{s}$; deep learning model
$\mathcal{M}$

output : Final solution of design variables $\bm{z}^{(K)}$

1 $\mathcal{D}_{train}\leftarrow\text{\O}$

2 for  _$k=0$ to $K$_ do

3 $\bm{z}_{C}^{(k)}\leftarrow$ coarse-grain design variables $\bm{z}^{(k)}$

4 $\bm{u}_{C}^{(k)}\leftarrow$ solve
$\bm{r}_{C}(\bm{z}_{C}^{(k)},\bm{u}_{C})=\bm{0}$

5 if  _$k\in\mathcal{I}_{s}$_ then /* apply synthetic gradients */

6 $\widetilde{\bm{g}}^{(k)}\leftarrow$ get synthetic gradients predicted by
$\mathcal{M}$

7

8 else /* apply exact gradients */

9 $\bm{u}^{(k)}\leftarrow$ solve $\bm{r}(\bm{z}^{(k)},\bm{u})=\bm{0}$

10 $\bm{g}^{(k)}\leftarrow$ get exact gradients with $\bm{u}^{(k)}$ using
(III-A)

11
$\mathcal{D}_{train}\leftarrow\mathcal{D}_{train}\cup\\{\langle\bm{z}^{(k)},\bm{u}_{C}^{(k)},\bm{g}^{(k)}\rangle\\}$

12 update model $\mathcal{M}$ with training data $\mathcal{D}_{train}$

13

14 $\bm{z}^{(k+1)}\leftarrow$ update $\bm{z}^{(k)}$ with $\bm{g}^{(k)}$ or
$\widetilde{\bm{g}}^{(k)}$

15

Algorithm 1 Online Neural Synthetic Gradients

###  III-B Online Gradients Learning

To address all limitations of the iteration-free and offline approaches
dominated in the literature, we take a completely different path from previous
methods. Instead of collecting data beforehand and training model offline, we
employ the optimization history as data, and train the deep learning model
online. The deep learning model can quickly generate synthetic gradients on-
the-fly, and drive the optimization forward. As illustrated in Algorithm 1,
ONSG switches between applying exact and synthetic gradients. For iterations
using exact gradients, ONSG collects data as training samples and updates the
deep learning model. For iterations using synthetic gradients as a shortcut,
ONSG skips the expensive solve of exact gradients. There are various ways to
design $\mathcal{I}_{s}$, i.e., which iterations are to apply synthetic
gradients. We adopt a relatively simple strategy: use exact gradients for the
initial $N_{I}$ iterations to train a decent deep learning model, and
afterwards use exact gradients every $N_{F}$ iterations. The exact gradients
serve as the supervision signal to train the deep learning model. Since the
model is trained online, the inaccurate prediction at the beginning can be
corrected in later iterations. Also, our framework can be directly plugged
into new optimization problems, and has no need of any pre-collected training
data. Therefore, ONSG is both efficient and generalizable. We will explain why
ONSG is scalable in the next section, and show that ONSG is accurate in the
section of experiments.

###  III-C Two-scale Optimization

As discussed before, large-scale PDE-constrained optimization problems can
easily have millions or even billions of variables in many industry
applications. It is very challenging to build a scalable online deep learning
model for such problems. The reasons are three folds: 1) as the number of
variables increases, the dimensionality of the input and output grows, so that
learning the mapping from input to output becomes much more difficult due to
the limited model capacity; 2) such increase of learning difficulty may be
alleviated in small-scale problems by providing more supervision, i.e.,
collecting exact gradients in more iterations for model training. However, for
large-scale problems, evaluating exact gradients is very time-consuming, and
it can significantly hurt the speedup performance if more supervision is
provided; 3) the size of deep learning model grows together with the
dimensionality of input and output, which costs more GPU memory for training
and predicting. Therefore, the maximum GPU memory available bounds the scale
of the optimization problem that can be handled.

To address the challenges above in the single-scale deep learning model, we
propose a novel two-scale optimization scheme, which decomposes the training
data into local subsets but still incorporates global information. More
specifically, we segment the design variables $\bm{z}$ into small subsets by
grouping them based on their physical locations in the design space, and each
subset serves as one training sample. Based on the segmentation, we also
introduce the corresponding coarse-grained design variable vector $\bm{z}_{C}$
so that each of its component is the averaged value of $\bm{z}$ in each
subset. Since we are optimizing all design variables together, it is not
feasible to build a deep learning model using only local information within
each subset. To incorporate global information, we introduce coarse-scale
discretized PDE constraints $\bm{r}_{C}(\bm{z}_{C},\bm{u}_{C})=\bm{0}$ and
solve it for $\bm{u}_{C}$, which is used as additional features for each
subset of $\bm{z}$. In this way, we maintain a constant dimensionality of the
input and output spaces, thus a constant number of model parameters, no matter
how large-scale the optimization problem is. In our experiments, we
quantitatively show the scalability of the two-scale optimization scheme.

Figure 1: Illustration of our two-scale optimization scheme, which groups the
design variables into a coarse-scale mesh based on their physical locations in
the design space. For the computational morphogenesis problem, we incorporate
both global and local information from the coarse-scale and fine-scale mesh to
provide input for training the deep learning (DL) model to generate synthetic
gradients. As illustrated in the bottom left corner, the block size $N_{B}$
determines the ratio between the side of an element of the fine-scale mesh to
the side of an element of the coarse-scale mesh.

###  III-D ONSG for Computational Morphogenesis

In this section, we showcase the application of ONSG on a challenging and
representative class of PDE-constrained optimization problems: the optimal
structure design [40, 20]. The goal of the optimal structural design is to
find the best structure that can sustain a given load. The corresponding PDE-
constrained optimization formulation is stated as [40]:

$\displaystyle\min_{\bm{z}}$
$\displaystyle\mathcal{J}(\bm{z},\bm{u}(\bm{z}))=\bm{f}^{\top}\bm{u}(\bm{z})$
(4) subject to
$\displaystyle\mathcal{V}(\bm{z})=\bm{v}^{\top}\mathbf{P}\bm{z}-V_{\text{max}}\leq
0$ $\displaystyle 0\leq z_{i}\leq 1\quad\forall i\in\\{1,\ldots,N_{d}\\},$

where $\bm{v}$ is a constant vector collecting element volumes, $\mathbf{P}$
is the density filter matrix [40] and $V_{\text{max}}$ is the maximum volume
imposed on the design. In the about NAND formulation, the implicit function
$\bm{u}(\bm{z})$ is defined through the discretized PDE constraints
$\bm{r}(\bm{z},\bm{u})=\mathbf{K}(\bm{z})\bm{u}-\bm{f}=\bm{0}$, where
$\mathbf{K}$ and $\bm{f}$ are the stiffness matrix and the global force
vector, respectively. According to Eq. (III-A), the gradients of the objective
function $\mathcal{J}(\cdot)$ at iteration $k$ is computed as

$g_{i}^{(k)}=-\big{(}\bm{u}^{(k)}\big{)}^{\top}~{}\frac{\partial\mathbf{K}}{\partial
z_{i}}\big{(}\bm{z}^{(k)}\big{)}~{}\bm{u}^{(k)},$ (5)

where $g_{i}^{(k)}$ is the $i$-th component of $\bm{g}^{(k)}$. We emphasize
that, to compute the gradient $\bm{g}^{(k)}$, one needs to first solve
$\bm{u}^{(k)}$ through $\mathbf{K}(\bm{z}^{(k)})\bm{u}-\bm{f}=\bm{0}$, which
is very computationally expensive for large-scale optimization problems.
Iterative solvers such as preconditioned conjugate gradient (PCG) and multi-
grid methods are typically used. On the other hand, the gradient of the volume
constraint function $\mathcal{V}$ is simply given by
$\partial\mathcal{V}/\partial\bm{z}=\textbf{P}^{\top}\bm{v}$, which does not
require expensive computation. Once the gradients are obtained, we use the
optimality criteria (OC) method [40], which the standard optimizer for (4), to
update design variables $\bm{z}$.

To apply the two-scale optimization scheme to computational morphogenesis, we
segment the design variables $\bm{z}$ into local subsets (3D blocks) of
uniform dimensions $N_{B}\times N_{B}\times N_{B}$, where $N_{B}$ is the block
size, as illustrated in Figure 1. The resulting coarse-scale discretized PDE
constraints become
$\bm{r}_{C}(\bm{z}_{C},\bm{u}_{C})=\mathbf{K}_{C}(\bm{z}_{C})\bm{u}_{C}-\bm{f}_{C}=\bm{0}$,
where $\mathbf{K}_{C}$ and $\bm{f}_{C}$ are the coarse-scale stiffness matrix
and the force vector, respectively.

###  III-E Model Architecture

We explored several different architectures for the deep learning model,
including fully-connected neural networks, convolutional neural networks and
its variants residual networks [41]. We finally choose fully-connected
DenseNet [42] since it performs the best on various design domains. Our model
connects all layers in a fully-connected neural network and preserve the feed-
forward nature. To be more specific, each layer obtains additional inputs from
all preceding layers and passes on its own output to all subsequent layers. We
concatenate the incoming features to serve as input of each layer. Our model
is different from DenseNet: we use fully-connected layers while DenseNet uses
convolutional layers. The hyperparameters of our model, such as the number of
layers, learning rate and hidden layer size, are provided in the experiment
section.

##  IV Experiments

In this section, we conduct extensive experiments to demonstrate the
effectiveness of ONSG in speeding up large-scale PDE-constrained optimization
problems. We first show the scalability of the proposed ONSG framework by
increasing the scale of the optimization problem. Then, we study the impact of
an important ONSG hyperparameter, the block size $N_{B}$, which can further
speedup the optimization process. We evaluate ONSG by tracking the objective
achieved along with the wall-clock time. Also, we plot the final solution for
each benchmark problem. For all the experiments, we fairly compare ONSG with
the standard optimizer, running with exactly the same computational resources.

Benchmark problems. We aim to construct a challenging benchmark, and choose
two large-scale PDE-constrained optimization problems in optimal structure
design: 1) a cantilever beam design problem, whose dimensions and boundary
conditions are illustrated in Figure 2(a). The design domain is fixed on its
face $x=0$ and subjected to a distributed load $\tau=1$ in the negative $z$
direction at the lower edge of the face $x=2$. We construct three fine-scale
meshes (Mesh 1–3) from lowest to highest level of refinement. The volume
fraction is $V_{\text{max}}=12\%$, and the density filter radius is $R=0.08$;
2) a Messerschmitt–Bölkow–Blohm (MBB) beam design problem, whose dimensions
and boundary conditions are illustrated in Figure 2(b). A downward force $F=1$
is applied on the center of the top surface of the design domain. We construct
a highly refined mesh (Mesh 4) for this problem. The volume fraction and the
density filter radius are set to $V_{\text{max}}=12\%$ and $R=0.08$,
respectively. Both are large-scale optimal design problems in 3D, and their
refined meshes have more than one million design variables. One can refer to
Table I for the exact number of design variables (# DVs) and the dimension of
the stiffness matrix $\mathbf{K}$ of each mesh.

(a) Cantilever

(b) MBB

Figure 2: Illustration of the design domains of two benchmark problems. TABLE I: Statistics of the benchmark problems. | Cantilever | MBB
---|---|---
Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4
$N_{B}=5$ | $N_{B}=2$ | $N_{B}=4$ | $N_{B}=8$
# DVs | 86K | 250K | 1.5M | 1.4M | 1.4M | 1.4M
K dim | 276K | 788K | 4.5M | 4.1M | 4.1M | 4.1M
$\textbf{K}_{C}$ dim | 3K | 8K | 40K | 71K | 10K | 3K

Competitor method. We compare our ONSG method with the standard optimizer in
the literature of large-scale structural design [20, 26]. Note that the
standard optimizer also employs GPUs to solve the discretized PDE constraints,
which is the most computationally intensive part in the optimization process.
The standard optimizer uses the PCG method with the Jacobi preconditioner
[43], which is shown to be the state of the art in terms of efficiency [44].
We use the official GPU parallel computing implementation of the PCG method in
Matlab.

Experimental settings. All the experiments are conducted on a GPU-enabled
(Nvidia Titan Xp) Linux machine powered by Intel Xeon Silver 4116 processors
at 2.10GHz with 256GB RAM. The GPU memory is 12GB. To ensure fair comparison,
we allocate exactly the same computational resources (CPU, GPU and memory) for
all the experiments. For ONSG experiments, we fix the block size $N_{B}=5$ on
three Cantilever meshes, and have different block sizes $N_{B}=2,4,8$ on the
MBB mesh. For the deep learning model in ONSG, we use a fully-connected neural
network with 4 hidden layers. The size of each hidden layer is set to 1000 for
the Cantilever experiments, and set to $200,500,1000$ for the MBB experiments
with $N_{B}=2,4,8$ respectively. The learning rate is set to 0.0005 initially,
and decays by half every 500 iterations. We train the neural network for 2000
iterations for the Cantilever experiments, and 3000 iterations for the MBB
experiments. We use $N_{I}=10$ initial iterations with exact gradients. We fix
the online update interval $N_{F}=10$ for the Cantilever problems, and set
$N_{F}=50,45,10$ for the MBB experiments with $N_{B}=2,4,8$ respectively. We
set the total number of iterations $K=200$ for all the experiments.

Evaluation metrics. We use the optimization objective to evaluate the model
accuracy. Since we are minimizing the compliance, the objective is the lower
the better. To evaluate the model efficiency, we time each experiment and
report the wall-clock time in seconds. We also track the intermediate
objective during the optimization process, and plot the objective curve along
with the wall-clock time. With the plot, we compare the objective achieved by
different methods given certain amount of time. Also, we compare the time cost
of different methods to achieve a certain objective value.

###  IV-A Model Accuracy

As stated in Eq. (4), the goal of the design problems is to minimize the
compliance of the structure, which is the objective of the optimization
process. Thus, the objective is the lower the better. Our method uses the
online neural synthetic gradients to replace the standard gradients, which
will guide the optimization to a different path. So we first check the
accuracy of our method, i.e., how close it is to the standard optimizer in
terms of the final objective achieved. We report the experimental results in
Table II. For all the cases, ONSG achieves comparable or even better
objective, compared to the standard optimizer. Since the optimization problem
is highly non-convex, ONSG is able to drive the optimization to a better local
minima and beat the standard optimizer to get negative difference in many
cases.

In Figure 3 and 4, we visualize the final designs obtained by the standard
optimizer and our method. For the Cantilever design problem, there is no
visual difference observed between our design and the standard one, and both
get smoother and higher-quality designs on finer meshes. For the MBB design
problem, there is minor difference in our designs compared to the standard
one. The standard design is more complex, which has several small members.
According to the final objectives in Table II, our method actually finds
better designs with lower compliance values.

The experimental results above demonstrate that ONSG has comparable or even
better performance compared to the standard optimizer.

Figure 3: Final Cantilever designs obtained by the standard optimizer and our method. Figure 4: Final MBB designs obtained by the standard optimizer and our method. TABLE II: Final objective achieved by the standard optimizer and our method on two benchmark problems. The objective is the lower the better. | Cantilever | MBB
---|---|---
Mesh 1 | Mesh 2 | Mesh 3 | Mesh 4
$N_{B}=5$ | $N_{B}=2$ | $N_{B}=4$ | $N_{B}=8$
Standard | 370.11 | 379.54 | 383.51 | 221.30 | 221.30 | 221.30
ONSG | 368.61 | 379.37 | 383.75 | 220.28 | 220.52 | 220.92
Difference | $-0.40\%$ | $-0.05\%$ | $0.06\%$ | $-0.46\%$ | $-0.35\%$ | $-0.17\%$

###  IV-B Model Scalability

ONSG employs the two-scale optimization scheme to decomposes the global
structure into local ones, which is important for solving large-scale optimal
design problems. To evaluate the scalability of ONSG, and quantitatively show
the impact of the two-scale scheme, we record the number of model parameters
and the corresponding GPU memory cost of the Cantilever design problem in
Table III. For the single-scale model, as we increase the scale of mesh from
Mesh 1 to Mesh 3, the number of parameters in the deep learning model
increases drastically and so does the GPU memory required for training the
model. Actually the single-scale model runs out of GPU memory on Mesh 2 and
Mesh 3. On the contrary, ONSG with the two-scale scheme maintains a constant
number of parameters on all meshes. The GPU memory cost is much less compared
to the single-scale model, and has only increased by a modest margin, which is
caused by more training data being cached on GPU.

The two-scale scheme requires to solve the coarse-scale PDE constraints with
additional computational cost. In fact, this extra cost is almost negligible
since the dimensionality of the coarse-scale stiffness matrix $\mathbf{K}_{C}$
is cubically smaller than the original stiffness matrix $\mathbf{K}$, as
listed in Table I. Therefore, with this small extra cost, the two-scale scheme
has much better scalability and can easily handle large-scale optimization
problems on a single GPU.

TABLE III: Number of model parameters and the corresponding GPU memory cost of the Cantilever design problem. A hyphen indicates running out of GPU memory. Scheme |  | Mesh 1 | Mesh 2 | Mesh 3
---|---|---|---|---
Single-scale | # params | 175M | 867M | 3B
GPU memory | 3.9GB | - | -
Two-scale | # params | 3.3M | 3.3M | 3.3M
GPU memory | 0.7GB | 0.7GB | 0.9GB

(a) Cantilever Mesh 1

(b) Cantilever Mesh 2

(c) Cantilever Mesh 3

(d) MBB Mesh 4

Figure 5: Objective curves on the Cantilever and MBB design problems. The
objective is the lower the better.

(a) Cantilever

(b) MBB

Figure 6: Wall-clock time of the standard optimizer and our method. The
corresponding speedup of our method is labeled in red above the bars. For
Cantilever, Mesh 1–3 are from the smallest to the largest scale. For MBB, we
experiment on a large-scale mesh (Mesh 4), and vary the block size $N_{B}$.

###  IV-C Model Efficiency

To evaluate the model efficiency, we time the optimization process for both
the standard optimizer and our method, and compare their time cost. We first
study the performance on the Cantilever design problem. For Mesh 1–3, the
wall-clock time is reported in Figure 6(a), and the objective curves along
with the wall-clock time are plotted in Figure 5(a)–5(c). From these
experimental results, we can draw several conclusions: 1) ONSG is
significantly faster than the standard optimizer in all cases; 2) ONSG
achieves higher speedup on larger scale meshes. On Mesh 3 with over one
million design variables, ONSG achieves 4.2x speedup, which is higher than the
2.4x speedup on Mesh 2 and the 1.5x speedup on Mesh 1. For the standard
optimizer, when the number of design variables increases, the dimensionality
of the stiffness matrix in the PDE constraints $\mathbf{K}$ grows drastically
(as listed in Table I), so that on large-scale mesh it is very time-consuming
to compute the gradients in every iteration. ONSG employs neural synthetic
gradients as a shortcut to skip the expensive computation in most iterations,
and it saves more time on larger scale meshes; 3) the objective curves in
Figure 5(a)–5(c) show that at the beginning ONSG maintains the objective close
to the standard one, and then overtakes the standard optimizer and quickly
goes down. This is reasonable since the deep learning model has few training
data at the beginning, and the synthetic gradients are not that accurate. As
the optimization proceeds, more training data is collected to update the model
online, so that the synthetic gradients become more accurate and guide the
optimization back on track.

###  IV-D Further Speedup

In previous experiments on the Cantilever design problem, we fix the block
size $N_{B}=5$ and our method ONSG achieves up to 4.2x speedup over the
standard method. Now we discuss how the block size $N_{B}$ can have impact on
speedup, and explore whether our method can achieve further speedup by varying
$N_{B}$.

In the extreme case, if we set $N_{B}=1$, there is no difference between the
coarse-scale and the original fine-scale PDE constraints, i.e.,
$\mathbf{K}=\mathbf{K}_{C}$, which roughly reduces ONSG to the standard
optimizer and there should be no speedup anymore. In the other extreme case,
if we set $N_{B}$ to be large enough to cover the entire mesh (suppose we have
enough GPU memory), there is only one block to generate the training data. In
this case, the learning becomes more difficult due to the high-dimensional
input and the small number of training samples. Thus we have to increase the
frequency of supervision for the deep learning model, and use larger deep
neural networks for model training, which will definitely diminish the speedup
of ONSG. Based on our analysis of the two extremes, there should be a trade-
off of the block size $N_{B}$ to achieve the best speedup.

We conduct experiment on the MBB design problem by setting $N_{B}=2,4,8$
respectively. As reported in Table II, the final objectives achieved are even
better than the standard optimizer, showing that ONSG can maintain perfect
accuracy with larger block size. The wall-clock time is reported in Figure
6(b). With $N_{B}=4$, ONSG achieves 7.5x speedup over the standard optimizer,
which almost doubles the speedup of other two cases. The corresponding
objective curves are plotted in Figure 5(d). We can see that the $N_{B}=2$
curve takes long time to drop, since the coarse-scale PDE constraints are more
refined and takes longer to solve. The $N_{B}=4$ curve is at the bottom, which
is the fastest among all the curves. These experimental results demonstrate
that a good trade-off of the block size $N_{B}$ can lead to further speedup of
ONSG.

##  V Conclusions and Future Work

We propose a framework named ONSG to speed up computational morphogenesis.
ONSG employs the optimization history to train a deep learning model, and
obtains online synthetic gradients from the deep learning model to replace
exact gradients as a shortcut to save computational time. To enable the
framework to handle large-scale optimization problems, we propose a two-scale
optimization scheme, which decomposes the design variables into local subsets
but still incorporates global information. With the two-scale scheme, ONSG can
efficiently deal with computational morphogenesis problems with millions of
design variables. In the future, we plan to formulate other PDE-constrained
optimization applications into our framework, such as optimal control and
inverse estimation problems.

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# Report of the Topical Group on Solid State Detectors and Tracking for
Snowmass 2021

Anthony Affolder  Artur Apresyan  Steven Worm
for the Snowmass Instrumentation Frontier Solid State Detector and Tracking
community:
M. Albrow  D. Ally  D. Ambrose  E. Anderssen  N. Apadula  P. Asenov  W.
Armstrong  M. Artuso  A. Barbier  P. Barletta  L. Bauerdick  D. Berry  M.
Bomben  M. Boscardin  J. Brau  W. Brooks  M. Breidenbach  J. Buckley  V. Cairo
R. Caputo  L. Carpenter  M. Centis-Vignali  M. Cerullo  A. Collu  F. Chlebana
G.-F. Dalla-Betta  M. Demarteau  G. Deptuch  K. Di Petrillo  G. D’Amen  A.
Dragone  N.T. Fourches  M. Garcia-Sciveres  G. Giacomini  C. Gingu  N. Graf
C. Grace  S. Griso  L. Greiner  C. Haber  G. Haller  K. Harris  T. Heim  U.
Heinz  R. Heller  M.T. Hedges  R. Herbst  M.R. Hoeferkamp  T. Holmes  S.E.
Holland  S.-C. Hsu  R. Islam  M. Jadhav  S. Jindariani  S. Joosten  A. Jung
S. Karmarkar  C. Kenney  C. Kierans  J. Kim  S. Kim  S. Klein  A. Koshy  K.
Krizka  A. Lai  L. Lee  L. Linssen  R. Lipton  T. Liu  C. Madrid  T. Mahajan
T. Markiewicz  B. Markovic  S. Mazza  M. Mazziotta  Y. Mei  P. Merkel  J.
Metcalfe  Z.-E. Meziani  A. Minns  F. Moscatelli  P. Murat  J. Muth  B.
Nachman  S. Nahn  M. Narain  E.A. Narayanan  T. Nelson  J. Nielsen  S.
Oktyabrsky  J. Ott  F.R. Palomo  D. Passeri  R. Patti  T. Peltola  C. Peña  C.
Peng  C. Renard  P. Reimer  C. Rogan  L. Rota  H. Sadrozinski  J. Segal  A.
Schwartzman  B. Schumm  M. Scott  S. Seidel  A. Seiden  B. Sekely  X. Shi  E.
Sichtermann  N. Sinev  J. Sonneveld  L. Spiegel  A. Steinhebel  D. Strom
D.M.S. Sultan  A. Sumant  V. Tokranov  A. Tricoli  W. Trischuk  A. Tumasyan
L. Uplegger  C. Vernieri  H. Wang  P. Wagenknecht  H. Weber  S. Xie  M.
Yakimov  Z. Ye  C. Young  M. Żurek

##  1 Abstract

Tracking detectors are of vital importance for collider-based high energy
physics (HEP) experiments. The primary purpose of tracking detectors is the
precise reconstruction of charged particle trajectories and the reconstruction
of secondary vertices. The performance requirements from the community posed
by the future collider experiments require an evolution of tracking systems,
necessitating the development of new techniques, materials and technologies in
order to fully exploit their physics potential. In this article we summarize
the discussions and conclusions of the 2022 Snowmass Instrumentation Frontier
subgroup on Solid State and Tracking Detectors (Snowmass IF03).

##  2 Executive Summary

Tracking detectors are of vital importance for collider-based high energy
physics (HEP) experiments. The primary purpose of tracking detectors is the
precise reconstruction of charged particle trajectories and the reconstruction
of secondary vertices. The performance requirements from the community posed
by the future collider experiments require an evolution of tracking systems,
necessitating the development of new techniques, materials and technologies in
order to fully exploit their physics potential.

Relative to the currently operating systems and their upgrades, the technical
requirements for tracking detectors (trackers) in the next 20-40 years are
significantly more stringent, such as tolerances to fluences 1-2 orders of
magnitude higher, larger areas at lower costs, segmentation and position
resolution 2-4 times finer and precision timing resolution, radiation length
per layer from 0.1-1% $\rm{X_{0}}$ and integration of novel radiation-hard
materials.

Technological developments currently underway aim to address these issues, and
the successful completion of the programs outlined below requires focused
efforts from the community on the steady development and refinement of
existing technologies, and the pursuit of novel “blue sky” technologies to
enable transformative progress. The HEP community gathered at Seattle Snowmass
Summer Meeting in 2022 identified the following key directions for the near-
term priorities of the solid-state tracking:

  * •

IF03-1: Develop high spatial resolution pixel detectors with precise per-pixel
time resolution to resolve individual interactions in high-collision-density
environments

  * •

IF03-2: Adapt new materials and fabrication/integration techniques for
particle tracking in harsh environments, including sensors, support structures
and cooling

  * •

IF03-3: Realize scalable, irreducible-mass trackers in extreme conditions

  * •

IF03-4: Progress advanced modeling for simulation tools, developing required
extensions for new devices, to drive device design.

  * •

IF03-5: Provide training and workforce maintenance to enable future tracking
systems to be designed, developed, constructed and simulated.

  * •

IF03-6: Nurture collaborative networks, provide technology benchmarks and
roadmaps and funding in order to develop required technologies on necessary
time scales, costs and scope.

Eight white papers on solid state trackers were submitted during the 2022
Snowmass [1, 2, 3, 4, 5, 6, 7, 8], focusing on a broad set of R&D topics that
the US community is actively pursuing. While these papers don’t completely
capture the full breadth and depth of the R&D carried out in the US, they
highlight the key challenges that future trackers will face and summarize the
main research thrusts actively pursued by the community.

Throughout the tracker community, many new, novel tracking technologies are
being investigated. Silicon pixel detectors based on Monolithic Active Pixels
(MAPS) are, for example, being developed for a wide range of applications.
Recent devices show significant improvements in pixel granularity, readout
speed and radiation tolerance. Also addressing detector speed and radiation
performance are so-called 3D sensors, where the collection electrodes are
orthogonal to the plane of the detector. Developed in both silicon and diamond
subtrates, 3D sensors serve as a first step into wide-bandgap semiconductor
materials. Next, thin film detectors offer a radical new way to manufacture
large-area and low-cost tracking detectors. Finally quantum-enabled detection
is being investigated for trackers in novel architectures and materials, with
the promise of achieving extremely fast response and high granularity with
only a minimal detector thickness.

Advanced 4-dimensional trackers with ultra fast timing (10-30 ps) and extreme
spatial resolution (O(few $\mu$m)) represent a new avenue in the development
of silicon trackers, enabling a variety of physics studies which would remain
out of experimental reach with the existing technologies. Several technology
solutions are being currently pursued by the community to address the
challenges posed by various experiments, both for the sensors and the
associated electronics. Bringing in technological innovation and fully
exploiting the potential of future detectors through the usage of ultra fast
timing is a unique and exciting opportunity for the particle physics
community. In order to reach this goal, it is of paramount importance in the
coming years to undertake thoroughly the R&D studies to investigate how to
best combine timing with spatial information.

The packaging and integration of the sensors and readout electronics will
become more critical for future experiments, as device segmentation decreases
to mitigate the increased track density. Bump bonding technologies have nearly
reached their limits; more advanced 3D packing technologies including wafer-
to-wafer and die-to-wafer hybrid bonding and through silicon vias (TSV) have
the potential to meet the goals of future particle physics experiments.
Collaboration between research groups and industrial partners with this
expertise is crucial to introduce this technology to the HEP community. This
research will take consider time and efforts to have confidence to use these
technique in future large-scale tracking systems, and as such need to being
urgently.

Detector mechanics will also play a significant role in future detector’s
performance. Material necessary for cooling and structural stability will be
the lower bound on the radiation length for future tracking systems. Increased
segmentation will lead naturally to larger power densities; in order to
minimize material, solutions with integrated services and cooling are
necessary. A holistic approach to design, simulation and manufacturing will be
required. Novel materials, new cooling and composite manufacturing technique
will need to be developed in order to reach the targeted performance.

To develop these new technologies, simulations of the properties of silicon
and novel sensor materials throughout the lifetime of the experiments will be
critical. These studies can drive device design included implant locations,
size and strength to the most promising directions of development. To reach
their full potential, further developments are needed to improve accuracy,
precision and new devices. With this research, the performance of future
experiments can be better predicted for its full life cycle prior to
construction.

##  3 Requirements

Tracking detectors are key to address science challenges and their development
is based on our understanding of fundamental laws of physics. Therefore, there
is a “virtuous cycle” between fundamental laws of nature and enabling detector
concepts and techniques. Such virtual cycle must remain strong and unbroken
and in turn leads to a greater physics discoveries.

At the Energy Frontier, there are two major drivers for detector R$\&$D:
detectors for future hadron and muon colliders and development of advanced
technologies for the future ${e^{+}e^{-}}$ machines. The challenge of the most
intense tracking regions at future hadron colliders will be addressed by
hybrid pixel detectors, and the breakthrough technology is expected to come
from CMOS sensor developments, which are currently being pushed to
unprecedented levels of radiation hardness and rate capabilities. For future
applications in particle physics, alternative technologies, by employing
beyond state-of-the-art interconnection technologies, such as 3D vertical
integration, through-silicon-vias (TSV), or micro bump-bonding, which, while
retaining the advantages of separate and optimized fabrication processes for
sensor and electronics, would allow fine pitch interconnect of multiple chips.
The Basic Research Needs for High Energy Physics Detector Research &
Development document [9] compiles a list of requirements for trackers at the
next generation of energy frontier experiments focused on precision Higgs and
SM physics, and searches for BSM phenomena.

Emerging novel vertex and tracking detector technologies are the vital
backbone for the success at a future electron-positron machines. These will
operate in an environment with high (continuous or bunched) beam currents, a
minimum distance from beam axis of about 20 mm, a requirement of $<5~{}\mu m$
single point resolution, high granularity ($<30\times 30~{}\mu m^{2}$), power
dissipation ($<50~{}mW/cm^{2}$), low mass ($\sim 0.1~{}\%$ of $X_{0}$, or 100
$\mu m$ Si-equivalent per layer). The clear challenge is unprecedented spatial
resolution, to be achieved with ultra-small pixels, and thus extremely low
material budget. Very thin detector assemblies are mandatory, while providing
high stiffness and stability. The tracking resolutions should enable high-
precision reconstruction of the recoil mass in the $e^{+}e^{-}\rightarrow Zh$
process, as shown in Table 1, and allow very efficient $b$ and $c$ tagging and
$\tau$-lepton identification through the reconstruction of secondary vertices.
The detector solenoid magnetic field must be limited to 2 T for the Tera-Z
operation at future circular electron-positron colliders, to avoid a blow up
of the vertical beam emittance and a resulting loss of luminosity. The 2 T
magnetic field limit is not a significant problem since the momentum scale of
the produced partons is typically distributed around 50 GeV and does not
exceed 182.5 GeV. In general, the future demands at electron-positron Higgs
Factories for high resolution (granularity) and low material budget on one
hand, and low power consumption on the other hand, exceed significantly what
is the state-of-the-art today.

Table 1: Physics goals and detector requirements [9, 10]. Initial state | Physics goal | Detector | Requirement
---|---|---|---
$e^{+}e^{-}$ |  $h$ZZ sub-% | Tracker |  $\sigma_{p_{T}}/p_{T}$=0.2% for $p_{T}<$ 100 GeV
|  |  |  $\sigma_{p_{T}}/p_{T}^{2}=2\cdot 10^{-5}/$ GeV for $p_{T}>$ 100 GeV
| $hb\bar{b}/hc\bar{c}$ | Tracker |  $\sigma_{r\phi}=5\oplus 15(p\sin\theta^{\frac{3}{2}})^{-1}\mu$m
|  |  | 5$~{}\mu$m single hit resolution
|  |  | per track timing resolution of 10 ps
pp-100 TeV | Higgs & BSM | Tracker |  $\sigma_{p_{T}}/p_{T}$=0.5% for $p_{T}<$ 100 GeV
|  |  |  $\sigma_{p_{T}}/p_{T}^{2}=2\cdot 10^{-5}/$ GeV for $p_{T}>$ 100 GeV
|  |  |  $300$ MGy and $\approx$ 1018 neq/cm2
|  |  | per track timing resolution of 5 ps
$\mu^{+}\mu^{-}$ | Higgs & BSM | Tracker |  $\sigma_{p_{T}}/p_{T}$=0.2–0.3% for $p_{T}<$ 100 GeV
|  |  |  $\sigma_{p_{T}}/p_{T}^{2}=2\cdot 10^{-5}/$ GeV for $p_{T}>$ 100 GeV
|  |  | 100 kGy and $\approx$ 1015 neq/cm2
|  |  | 0.01 rad angular resolution
|  |  | 5$~{}\mu$m single hit resolution
|  |  | 30 ps timing resolution

Picosecond timing information will be the vital backbone for the success of
future hadron colliders, driven by the general push towards higher
luminosities at the next generation of colliders. The addition of the timing
information will allow one to implement 4D reconstruction of the primary
collision vertex - joint pattern recognition, based on timing and tracking
detectors. In particular, the required accuracy in the ATLAS and CMS at HL-LHC
will be dictated by the pile-up induced backgrounds and the nominal
interaction distribution within a single bunch crossing ($\sigma_{z}\sim$ 5
cm, $\sigma_{t}\sim$ 170 ps rms). Furthermore, a factor-of-five larger pileup
at a 100 TeV hadron collider than at the HL-LHC will pose stringent
requirements on the detector design [11]: future hadron colliders require
timing information of the order of 5-10 ps per track to suppress pileup and
correctly assign tracks to vertices. At a multi-TeV muon collider to reject a
significant fraction of beam induced background, accurate timing information
with a resolution of 30 ps is assumed to be available in the vertex detectors
[10]. Aiming for an excellent position and timing resolution ($\sim$ 10 ps and
$\sim$ 10 $\mu m$) with GHz counting capabilities to perform 4D tracking, LGAD
sensors represent a very attractive option for PID and TOF applications.

The 4D-tracking systems will play a crucial role in LLP exploration allowing
to efficiently reconstruct position-time vertice, including those with large
displacement, and reduce backgrounds including beam-induced contributions.
4D-trackers will also enable complementary TOF and dE/dx measurement in the
same sensor critical for PID and heavy stable charged particle (HSCP)
searches. In the latter case, the TOF measurement yields complementary
information of potential evidence of new HSCP’s, as recently reported by ATLAS
[12], and signal characterization in case of discovery. An ultimate concept is
to develop 4D real-time tracking system for a fast trigger decision and to
exploit 5D imaging reconstruction approach, if space-point, picosecond-time
and energy information are available at each point along the track.

At future hadron colliders, detector capabilities to reconstruct highly
boosted objects are fairly challenging (for instance, the average $Z$ boson
from $ZZ$ production would shower mostly within a single LHC calorimeter
cell). This challenge is accentuated by so-called “hyper-boosted” jets, whose
decay products are collimated into areas the size of single calorimeter cells.
Holistic detector designs that integrate tracking, timing, and energy
measurements are needed to mitigate for these conditions [13, 14, 15, 16, 17].

Future hadron and muon collider experiments at the Energy Frontier have an
additional requirement of radiation hardness of tracking systems. Despite
their quite complicate fabrication process, that can be both double- and
single-sided, 3D detectors are the most radiation-hard technology to-date
demonstrating good performance even up to a fluence of $3\times 10^{16}$
$neq/cm^{2}$ and time resolution of 30 ps. New developments are being pursued
to address the challenges of radiation hardness, small pitch with narrower
columns, thinner devices, better breakdown behaviour, improved on-wafer
selection and lower costs.

The Rare Processes and Precision Measurements (RPF) frontier encompasses
several different experimental programs who share the common aim of searching
for new physics through its manifestation in rare and forbidden decays in the
SM, as well as in precise measurements sensitive to subtle deviations from SM
expectations. Four main categories of experiments were considered: heavy
flavor, lepton flavor violation, electric dipole moment, and the dark sector.
This represents a data-driven approach to search for signatures that may lead
to some understanding of the key feature of the new physics, namely its mass
scale and coupling to SM particles. This approach has been very fruitful in
the past.

The continual exploration of new physics in heavy flavor is pursued by LHCb at
the LHC hadron collider[18], with a brand new detector being commissioned
right now, and an ambitious upgrade planned for the HL-LHC. In this phase, the
detector operates at a peak luminosity pf $1.7\times 10^{34}{\rm
cm^{-2}s^{-1}}$ and a pile-up of $\sim 40$. Belle II[19] is also consider
upgrades. The first goal is to reach an integrated luminosity of 50 ab-1 by
2035. Detector upgrades are considered, in the second long shutdown
possibilities include a CMOS vertex detector. Longer terms upgrades of the
luminosity are considered. Lighter quark flavor physics include rare $K$
decays, $\eta-\eta^{\prime}$ factories[20], and rare $\pi$ decays.

Charged lepton flavor violation is another pathway to uncover indirect
evidence for new physics. Example experiments that will study rare $\mu$
decays in the next few decades are MEG II, COMET and Mu3e (off-shore), and
Mu2e and its upgrade Mu2eII, taking data at Fermilab. If we consider
experiments sensitive to $\mu$ conversion in the field of a nucleus, the
signature of a signal event is an electron with an energy near the rest mass
of the muon, characteristic of neutrinoless $\mu\to\ e$ conversion in the
field of a nucleus. Separating the signal from the background from cosmic ray
processes requires low mass tracking with excellent momentum resolution. This
is currently accomplished with an extremely low mass gaseous detector. The
Mu3e experiment requires excellent momentum, position and tracking information
to achieve the desired 10-16 branching ratio sensitivity to $\mu^{+}\to
e^{*}e^{-}e^{+}$[21]. To accomplish this, a low-mass HV-CMOS tracking detector
is an integral component.

Some of the experiments considered rely on state of the art solid state
detector systems to achieve their desired sensitivities. In the following we
will discuss the requirements of this broad experimental physics program with
respect to four main technological aspects: time stamp specifications on
single hits and tracks, spatial resolution and granularity, environmental
conditions such as radiation tolerance, and system aspects such as
scalability, material budget and cooling requirements.

A precision time stamp, of the order of tens of ps, makes it possible to
achieve a 4D tracking reconstruction algorithm, with the time dimension fully
integrated as a fourth dimension. The Vertex Locator (VELO) for the LHCb
Upgrade2 requires with 20 ps time resolution per track will maintain high
primary vertex finding efficiency at higher $pp$ collision density while also
providing a precision time stamp to each collision [18]. Similar time stamps
on other tracking subsystems, and particle identification devices
(calorimeter, hadron identification systems), enhances the speed of the all-
software trigger, and enables the use of time of flight for particle
identification. Dedicated timing layers in the vertex detector and in the mid-
section of the calorimeter, close to the point where the maximum shower is
developed, are also studied. Their hit resolution should be of the order of 25
ns or better. The VELO must do this while maintaining a high spatial
resolution at the 10 $\mu$m level. The requirements for a timing layer
integrated in the VELO telescope is a 25 ns time stamp at the hit level. An
option considered in this upgrade is a timing layer in the calorimeter region
of maximum shower development with similar specifications. An upgrade of the
NA62 kaon experiment also demonstrates the benefits of 4D tracking of the
incoming beam with sub-50 ps resolution, and then using time measurements in
final state detectors to match decay products to incoming particles at a rate
of 3 ${\rm GHz}$ [22]. Other experiments may require timing resolutions at the
sub-100 ps level, as well.

High spatial granularity while maintaining a low material budget is the most
desirable feature in several applications. For example, the MuPix pixel
sensors implemented for the M3e experiment are being studied as a possible
starting point for the design of the Upstream Tracker and Mighty Tracker for
the LHC upgrade [18]. Here a pixel size of $50\times 150~{}\mu m^{2}$ is
considered.

Experiments in RF also need development of tracking detectors with large areas
and low material budgets. Tracking in the proposed REDTOP experiment requires
radiation lengths per layer $\sim 0.1\%$ $\rm{X_{0}}$ if implemented using
silicon sensors [20]. Proposed future upgrades of the Belle 2 vertex detector
should have a radiation length per layer of 0.2-0.7% $\rm{X_{0}}$ [19].

The radiation environment is also a driver in the technology choices,
especially in applications involving hadron beams. For example, in LHCb
upgrade 2, the VELO detector needs to operate at fluences up to the
$10^{16}~{}n_{\rm eq}/{\rm cm}^{2}$ level, while the upstream tracker needs to
withstand fluences up to $\sim 3\times 10^{15}~{}n_{\rm eq}/{\rm cm}^{2}$.

##  4 Silicon Radiation Detectors for High Energy Physics Experiments

There are currently a variety of tools available for simulating the properties
of silicon sensors before and after irradiation. These tools include finite
element methods for device properties, dedicated annealing models, and
testbeam/full detector system models. No one model can describe all of the
necessary physics. Most of these models are either fully or partially
developed by HEP scientists and while there are many open-source tools, the
most precise device property simulations rely on expensive, proprietary
software.

The development of these simulations happens inside existing experimental
collaborations and within the RD50 Collaboration at CERN. RD50 is essential
for model research and development and provides an important forum for inter-
collaboration exchange.

While existing approaches are able to describe many aspects of signal
formation in silicon devices, even after irradiation and annealing, there is
significant research and development (R&D) required to improve the accuracy
and precision of these models and to be able to handle new devices (e.g. for
timing) and the extreme fluences of future colliders. The US particle physics
community can play a key role in this R&D program, but it will require
resources for training, software, testbeam, and personnel.

For example, there is a great need for (1) a unified microscopic model of
sensor charge collection, radiation damage, and annealing (no model can
currently do all three), (2) radiation damage models (for leakage current,
depletion voltage, charge collection) with uncertainties (and a database of
such models), and (3) a measurement program to determine damage factors and
uncertainties for particle types and energies relevant for current and future
colliders.

This program of work is critical for the future success of collider physics.
With these tools, the performance of future experiments would be better
predicted throughout their lifetimes. The simulations will also be critical in
directing the R& D for the next generation of complex sensors into the most
promising avenues to realize the necessary performance to reach our ultimate
physics goals.

##  5 Novel Sensors for Particle Tracking

The proposed future collider experiments pose unprecedented challenges for
event reconstruction, which require development of new approaches in detector
designs. Research in particle tracking detectors for high energy physics
application is underway with a goal of improving radiation hardness, achieving
improved position vertex and timing resolution, simplifying integration, and
optimizing power, cost, and material. Several technologies are currently being
investigated to develop technologies for the future, which approach these
goals in complementary ways.

Silicon sensors with “3D technology” have electrodes oriented perpendicular to
the plane of the sensor. These show promise for compensation of lost signal in
high radiation environments and for separation of pileup events by precision
timing. New 3D geometries involving p-type trench electrodes spanning the
entire thickness of the detector, separated by lines of segmented n-type
electrodes for readout, promise improved uniformity, timing resolution, and
radiation resistance relative to established devices operating effectively at
the LHC. The 3D technology is also being realized in diamond substrates, where
column-like electrodes may be placed inside the detector material by use of a
130 fs laser with wavelength 800 nm. Present research aims for operation with
adequate signal-to-noise ratio at fluences approaching $10^{18}~{}n_{\rm
eq}/{\rm cm}^{2}$, with timing resolution on the order of 10 ps.

Monolithic Active Pixel Sensors (MAPS), in which charge collection and readout
circuitry are combined in the same pixel, have been shown to be a promising
technology for high-granularity and low material budget detector systems. MAPS
have several advantages over traditional hybrid pixel detector technologies,
as they can be inexpensively fabricated in standard CMOS imaging processes,
back-thinned or back-illuminated, have demonstrated high radiation hardness,
and can be stitched or tiled for large, low-mass arrays. MAPS offer the
possibility of individual pixel readout at MHz speeds, with low power
consumption and powered from a low voltage supply.

Many MAPS pixel geometries have been explored, but the close connection of a
sensor and front-end amplifier, without the need for external
interconnections, holds the promise of reducing the input capacitance
significantly, and hence extremely low-noise designs are possible. The
reduction of the noise floor means that even small signals, for example from
thinned low-power sensors, can yield satisfactorily high S/N. The next
generation of MAPS will target improved performance in speed, resolution, and
radiation tolerance, and also incorporate developments in the systems
integration that are needed for large scale use at a reasonable cost.
Prototype detectors are currently being designed to addresses the integration
issues associated with making large arrays, by first developing an
intermediate scale array of tens of m2 which include full powering and readout
of large numbers of MAPS. Many different MAPS detector architectures are in
development, and are being adapted to a diverse set of applications and
environments; from hadron colliders to the EIC and ILC, to space-based tracker
applications in $\gamma$-ray satellites.

Thin film detectors have the potential to be fully integrated, while achieving
large area coverage and low power consumption with low dead material and low
cost. Thin flim transistor technology uses crystalline growth techniques to
layer materials, such that monolithic detectors may be fabricated by combining
layers of thin film detection material with layers of amplification
electronics using vertical integration.

The DoTPiX pixel architecture has been proposed on the principle of a single
n-channel MOS transistor, in which a buried quantum well gate performs two
functions: as a hole-collecting electrode and as a channel current modulation
gate. The quantum well gate is made with a germanium silicon substrate. The
active layers are of the order of 5$~{}\mu m$ below the surface, permitting
detection of minimum ionizing particles. This technology is intended to
achieve extremely small pitch size to enable trigger-free operation without
multiple hits in a future linear collider, as well as simplified
reconstruction of tracks with low transverse momentum near the interaction
point.

Lastly, a technology is under development in which a novel ultra-fast
scintillating material employs a semiconductor stopping medium with embedded
quantum dots. The candidate material, demonstrating very high light yield and
fast emission, is a GaAs matrix with InAs quantum dots. The first prototype
detectors have been produced, and pending research goals include demonstration
of detection performance with minimum ionizing particles, corresponding to
signals of about 4000 electron-hole pairs in a detector of 20$~{}\mu m$
thickness. A compatible electronics solution must also be developed. While the
radiation tolerance of the device is not yet known, generally quantum dot
media are among the most radiation hard semiconductor materials.

##  6 4-Dimensional trackers

Future collider experiments call for development of tracking detectops with
10-30 ps timing resolution, in addition to excellent position resolution, i.e.
4-Dimensional trackers. Time resolution of the future 4D tracker detector can
be factorized into contributions from the sensor itself, and those from the
readout electronics. The overall detector system should contain a sensor with
short drift time, high signal to noise, limited thickness in the path of a MIP
to reduce the Landau fluctuations, and small TDC bin size. Several
technologies to address these needs are being developed and are introduced in
this section.

One approach to meet these requirements utilizes Low Gain Avalanche Detectors
(LGADs), and adapts them to achieve the fine segmentation necessary for
tracking. The main limitation of the DC-coupled LGADs is the presence of JTE
to interrupt the gain layer, which introduces inactive regions on the sensors.
Several modifications to LGAD technology have been proposed and demonstrated
to make LGAD sensors suitable for tracking sensors with 100% fill-factor,
achieving excellent timing and position resolution. The most advanced high
granularity designs realized so far are the AC-coupled LGAD (AC-LGAD). Because
the gain layer is uninterrupted, the electrodes can be designed with smaller
pitch and size than standard LGADs. A key feature of AC-LGADs is the signal
sharing between electrodes, which can be used to obtain the spatial resolution
necessary for future colliders with a reduced number of readout channels,
achieving simultaneous 30 ps and 5 $\mu$m resolutions. Another design geared
towards 100% fill factor are the “Deep-Junction” (DJ-LGAD), which are formed
by abutting thin, highly-doped p$+$ and n$+$ layers. Trench-isolated (TI)
LGADs address the fill-factor limitation of LGADs by substituting the JTE with
narrow trenches filled with dielectric material. The width of the trenches is
about 1 $\mu$m, allowing for a substantial reduction of the distance between
gain layers of neighboring channels with respect to standard LGADs. Intense
R&D is ongoing to improve the radiation tolerance of LGADs beyond
$10^{16}~{}n_{\rm eq}/{\rm cm}^{2}$, by e.g. exploiting the observation that
deep and narrow gain layers are more radiation hard. Such “Buried Layer LGADs”
achieve the deep gain layer by implanting the boron layer at low energy and
then to burying it under a few microns of epitaxially grown silicon.

Another thrust to develop 4D-tracking sensors is through a usage of sensors
with 3D geometries, or closer integration with electronics via monolithic
pixel sensors, or adoption of new materials in sensor design. Since charge
carriers are collected perpendicularly to the sensor thickness in sensors with
3D-geometries, the time uncertainties due to nonuniform ionisation density,
delta rays and charge carrier diffusion are minimized. Prototypes have
demonstrated promising results with time resolution below 20 ps. New
materials, such as SiC detectors, can operate at very high temperatures, and
are known to exhibit significant radiation hardness and extremely low leakage
currents. Although the adoption of this technology in HEP has been slow,
recent advances in wafer size and quality are enabling compatibility with
processing tools developed for silicon devices, and allowing the price per
device to sharply decrease.

Going even beyond 4D-sensors are designs that aim to simultaneously measure
not only the position and time, but also the angle of passing tracks. These
kind of sensors would dramatically reduce the complexity of detector modules,
and enable unprecedented reconstruction capabilities on the front-end. Double
Sided LGADs (DS-LGADs) achieve this goals by adding a readout layer to the
p-side of the LGAD structure, which allows one to also measure signals from
the slower-drifting holes. The signal p-side collects two components, holes
from the primary ionization followed by the larger number of holes generated
at the gain layer. This provides a unique signature of the pattern of charge
deposit within the device that could enable measurement of the track angle as
well. Another approach to achieve the same goal is through detection of the
Shockley-Ramo current induced at the readout electrode from mobile charge
carriers within the sensor. This current has a very fast rising edge ($\sim$15
ps) and can be used to precisely timestamp track hits, and is a promising
direction for establishing 5D-sensor designs.

The timing ASICs under development for the ATLAS and CMS timing upgrades,
named ALTIROC and ETROC, represent revolutionary steps forward as the first
readout chips to bring O(10 ps) timing to collider experiments. However, they
are able to use significantly more space and power than high density ASICs
designed for trackers with fine pitch and limited material. Compared to RD53A,
the timing chips use several hundred times more power and area per channel.
The primary challenges to transform them into chips for 4D tracking will be to
minimize both the power consumption and the channel size. The high luminosity
of future hadron colliders will require trackers capable to survive in extreme
radiation environments (accumulating a dose of up to 30 GRad and $10^{18}$
neutrons/cm2). Currently there are several projects with the aim to make
advance in these areas. Efforts are geared towards a specific application,
however the goals are common: bandwidth optimization, low noise, low area,
time resolution and power dissipation. The CERN’s EP R&D WP5 chip aims to
reduce the total area of the circuitry by moving to 28 nm node with the design
of the low-power and compact TDC, while the Silicon-Germanium based ASIC
developed by Anadyne in TowerJazz 130 nm is expected to have 0.5 mW per
channel (front end and discriminator) with 10 ps of timing resolution for 5 fC
charge. The CFD-chip ASIC developed at FNAL in 65 nm node was demonstrated to
deliver $\sim$10 ps jitter for 15 fC of injected charge, using a constant-
fraction-discriminator implementation.

##  7 Integration

In the past years, HEP experiments have been mostly relying on bump bonding
for high density pixel sensor to ASIC connection. The bump bonding technology
was proven to be reliable; however, it is known to have several limitations:
it only works down to 20-50 $\mu m$ of pitch and has yield issues for finer
connections. Furthermore, the solder balls used for the connection increase
the input capacitance to the amplifier and hence the noise. With bump
bonding,the sensor or chip need side extensions to have external connections.
In terms of mechanical properties the resulting connection is subject to heat
stress since it involves different materials; the minimum thickness is also
limited since both chip and sensor need a thick enough support wafers in order
to meet alignment and bow requirements for the bump bonding process.

The introduction of more advanced packaging may solve many of these issues,
allowing for the improvement of performance, yield and processing. 3D
integration is a common widespread technology in industry, it allows tight
packaging of sensor and readout chip. Furthermore it allows to stacks multiple
chips in a single monolithic device. There are many technologies available for
3D integration, hybrid bonding is the most widely accepted: silicon covalent
oxide bonding combined with copper diffusion bonding to provide connectivity
between layers. The process can be done for wafer to wafer (w2w) or die to
wafer (d2w) assembly. Through Silicon Vias (TSVs) allow multiple planes to be
stacked and connected with external connections without the need of extensions
or silicon interposers. There are several advantages in using 3D integration
in sensor to chip connection:

  * •

Less space: 3D chips can be multi layer and do not need extension or
interposers for external connections. Reduction of single layer thickness,
after integration all supports can be removed

  * •

Very fine pitch bonding: down to a few micrometers

  * •

Better connections: faster and shorter than in circuit boards, with reduced
dissipated power

  * •

Better performance: reduced input capacitance and lower noise

  * •

Layered design: e.g. sensor + analog electronics + digital electronics, each
layer can be manufactured by different producers

  * •

Reduced thermal stress and increased heat dissipation since material is
homogeneous, also increased robustness

Advanced packaging and wafer to wafer bonding would facilitate several
applications both in HEP and outside of HEP: 4D tracking with high granularity
LGADs, 3D integrated SiPM with advanced signal processing, chip tile array
assembled at wafer level with no edges, small pixels connection (10 $\mu m$ or
less) that reduces the input capacitance of thin sensors (e.g. thin LGADs)
increasing both space and time precision, double sided connection for LGADs to
have readout on both sides, stacked 3D integrated chip, 3D network of
algorithm cells for advanced pattern recognition, zero mass trackers with very
thin and highly populated layers, sensor stacks for high energy X-ray
detection and decay detection. Finally it will advance the knowledge in
substrate engineering, useful, for example, in the fabrication of buried p-n
layers for LGADs, necessary for the production of, for example, DJ-LGAD and
buried junction LGAD.

Therefore electronics and sensor advanced packaging provide a variety of
technologies that can meet the needs of future particle physics experiments.
Combining these capabilities with silicon technologies developed for HEP, such
as LGADs and active edge sensors, will allow the design of sophisticated
detector systems that can meet the increasing challenges of next generation
experiments. The collaboration between research groups and industry with
established expertise in advanced packaging is crucial for the successful
introduction of this technology in the research community.

Current availability of such technologies is mixed. Leading edge foundries and
nodes now include 3D processes (Intel Forveros, TSMC 3DFabric, Samsung x-cube)
but are typically too expensive for HEP. Hybrid bonding is available from many
vendors that already work with HEP. The availability of TSVs is more limited.
Small pitch TSVs are available from some foundries at advanced nodes ($>$32
nm), in Silicon-on-Insulator (SOI) wafers, and from specialty foundries.
Larger pitch TSVs inserted in a completed wafer (via-last) are available from
a variety of packaging suppliers. Reliable 3D devices with these large pitch
($>$20 $\mu m$) external TSV connections can be accessed now with resources
available to HEP. It is important to build design and fabrication experience
with moderate scale companies now gearing up to address opportunities afforded
by the US semiconductor initiative.

It is urgent that this research is begun as it will take considerable effort
and time in order to have the confidence required to use these techniques in
future large-scale tracking systems.

##  8 Mechanics

Detector mechanics will play a significant role in future detectors’
performance; the necessary improvements will require simulations, novel ways
to reduce the total mass, as well as more integrated design concepts to save
on material budgets and optimize performance. The increased segmentation has
naturally lead to larger power densities requiring high performance material
support structures with integrated services. In many use-cases, the material
in these structures can be the limiting factor for the tracking performance of
the system. Particle detectors at future colliders rely on ever more precise
charged particle tracking devices, which are supported by structures
manufactured from composite materials. Various engineering techniques able to
solve challenges related to the design and manufacturing of future support
structures have been developed.

Future particle colliders, such as the high luminosity Large Hadron Collider
(HL-LHC), the Future Circular Collider (FCC-ee, FCC-hh), or the muon collider
(MuC) will collide particles at unprecedented rates and present a harsh
environment for future detectors. A holistic approach to the design and
manufacturing of detector support structures will be necessary to achieve
minimal weight systems and to potentially integrating electrical and cooling
services into structures. Novel techniques, materials and other design and
manufacturing solutions provide an avenue to solve challenges of increasingly
more complex and large tracking detectors. Complex stresses in composite
structures consisting of multiple parts are a consequence of different
manufacturing techniques utilized (compression molding, oven-cured lamination,
etc.). These stresses pose a severe challenge to current simulation tools.
Research and development efforts are underway to solve these challenges by
exploring multi-functional composite structures manufactured from highly
thermally conductive materials using different manufacturing techniques.
Machine learning based topology optimization have been employed to fulfill the
low-mass high-stiffness and thermal conductivity structure requirements.

Material savings due to novel approaches have the potential of reduction on
the order of 30-50% depending on more detailed R&D studies. This is an ideal
opportunity to explore the conjunction of latest techniques in composite
engineering involving machine learning based algorithms for heat transfer,
mechanical loading and micro-to-macro scale material response predictions for
the performance of the support structure mechanics. Current R&D efforts
provide a headway for future detector support structures relying on
transformational novel manufacturing techniques.

Different structures are current under study to achieve these goals which
include: carbon fiber support structures with integrated titanium pipes for
$\rm{CO_{2}}$ cooling, etched silicon and peak cooling micro-channels, Kapton-
based support structures and engineered air cooling.

Historically, mechanics has had limited support in the US and international
community. To reach the targeted performance of future tracking detector
systems, an increase of resources will be required to be successful.

##  9 Collaborative development programs

Collaborative efforts should be encouraged when possible to make efficient use
of resources and reduce the financial burden on individual groups, and to
enlarge the pool of expertise in the field. Especially critical is to
encourage and support young researchers to engage and pursue intensive
research programs to develop these new technologies, which will help to ensure
an expert workforce is sustained for the long term. Considering the
prohibitively high costs for productions of new technologies at commercial
foundries, collaborative submissions can provide the only viable solution for
research groups to access advanced production facilities. Such collaborations
promise to significantly expand the possibilities for small experiments or
small groups to join advanced detector R&D programs, work with others in the
field to build technology demonstrators, and help advance the solid-state
tracking technology.

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