A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

Introduction

In this work, we analyze the suitability of deep neural networks (DNNs) for the numerical solution of parametric problems. Such problems connect a parameter space with a solution state space via a so-called parametric map, . One special case of such a parametric problem arises when the parametric map results from solving a partial differential equation (PDE) and the parameters describe physical or geometrical constraints of the PDE such as, for example, the shape of the physical domain, boundary conditions, or a source term. Applications that lead to these problems include modeling unsteady and steady heat and mass transfer, acoustics, fluid mechanics, or electromagnetics, .

Solving a parametric PDE for every point in the parameter space of interest individually, typically leads to two types of problems. First, if the number of parameters of interest is excessive—a scenario coined many-query application—then the associated computational complexity could be unreasonably high. Second, if the computation time is severely limited, such as in real-time applications, then solving even a single PDE might be too costly.

A core assumption to overcome the two issues outlined above is that the solution manifold, i.e., the set of all admissible solutions associated with the parameter space, is inherently low-dimensional. This assumption forms the foundation for the so-called reduced basis method (RBM). A reduced basis discretization is then a (Galerkin) projection on a low-dimensional approximation space that is built from snapshots of the parametrically induced manifold, .

Constructing the low-dimensional approximation spaces is typically computationally expensive because it involves solving the PDEs for multiple instances of parameters. These computations take place in a so-called offline phase—a step of pre-computation, where one assumes to have access to sufficiently powerful computational resources. Once a suitable low-dimensional space is found, the cost of solving the associated PDEs for a new parameter value is significantly reduced and can be performed quickly and online, i.e., with limited resources, . We will give a more thorough introduction to RBMs in Section 2. An extensive survey of works on RBMs, which can be traced back to the seventies and eighties of the last century (see for instance ), is beyond the scope of this paper. We refer, for example, to [33, Chapter 1.1], and [12, Chapter 1.9] for (historical) studies of this topic.

In this work, we show that the low-dimensionality of the solution manifold also enables an efficient approximation of the parametric map by DNNs. In this context, the RBM will be, first and foremost, a tool to model this low-dimensionality by acting as a blueprint for the construction of the DNNs.

over $H$. We will call optimizing the empirical loss the learning procedure.

In view of PDEs, the approach proposed above can be rephrased in the following way. We are aiming to produce a function from a parameter set to a state space based on a few snapshots only. This function should satisfy the involved PDEs as precisely as possible, and the evaluation of this function should be very efficient even though the construction of it can potentially be computationally expensive.

In the above-described sense, a parametric PDE problem almost perfectly matches the definition of a statistical learning problem. Indeed, the PDEs and the metric on the state space correspond to a (deterministic) distribution $\rho$ and a loss function. Moreover, the snapshots are construed as the training data, and the offline phase mirrors the learning procedure. Finally, the parametric map is the prediction rule.

One of the most efficient learning methods nowadays is deep learning. This method describes a range of learning procedures to solve statistical learning problems where the hypothesis class $H$ is taken to be a set of DNNs, . These methods outperform virtually all classical machine learning techniques in sufficiently complicated tasks from speech recognition to image classification. Strikingly, training DNNs is a computationally very demanding task that is usually performed on highly parallelized machines. Once a DNN is fully trained, however, its application to a given input is orders of magnitudes faster than the training process. This observation again reflects the offline-online phase distinction that is common in RBM approaches.

Based on the overwhelming success of these techniques and the apparent similarities of learning problems and parametric problems it appears natural to apply methods from deep learning to statistical learning problems in the sense of (partly) replacing the parameter-dependent map by a DNN. Very successful advances in this direction have been reported in .

2 Our Contribution

In the applications mentioned above, the combination of DNNs and parametric problems seems to be remarkably efficient. In this paper, we present a theoretical justification of this approach. We address the question to what extent the hypothesis class of DNNs is sufficiently broad to approximately and efficiently represent the associated parametric maps. Concretely, we aim at understanding the necessary number of parameters of DNNs required to allow a sufficiently accurate approximation. We will demonstrate that depending on the target accuracy the required number of parameters of DNNs essentially only scales with the intrinsic dimension of the solution manifold, in particular, according to its Kolmogorov $N$-widths. We outline our results in Subsection 1.2.1. Then, we present a simplified exposition of our argument leading to the main results in Subsection 1.2.2.

The main contributions of this work is given by an approximation result with DNNs based on ReLU activation functions. Here, we aim to learn a variation of the parametric map

We assume that there exists a basis of a high-fidelity discretization of $\mathcal{H}$ which may potentially be quite large. Let $\mathbf{u}_{y}$ be the coefficient vector of $u_{y}$ with respect to the high-fidelity discretization. Moreover, we assume that there exists a RB approximating $u_{y}$ sufficiently accurately for every $y\in\mathcal{Y}$.

Theorem 4.3 then states that, under some technical assumptions, there exists a DNN that approximates the discretized solution map

up to a uniform error of $\epsilon>0$, while having a size that is polylogarithmical in $\epsilon$, cubic in the size of the reduced basis, and at most linear in the size of the high-fidelity basis.

This result highlights the common observation that, if a low-dimensional structure is present in a problem, then DNNs are able to identify it and use it advantageously. Concretely, our results show that a DNN is sufficiently flexible to benefit from the existence of a reduced basis in the sense that its size in the complex task of solving a parametric PDE does not or only weakly depend on the high-fidelity discretization and mainly on the size of the reduced basis.

The main result is based on four pillars that are described in detail in Subsection 1.2.2: First, we show that DNNs can efficiently solve linear systems, in the sense that, if supplied with a matrix and a right-hand side, a moderately-sized network outputs the solution of the inverse problem. Second, the reduced-basis approach allows reformulating the parametric problem, as a relatively small and parametrized linear system. Third, in many cases, the map that takes the parameters to the stiffness matrices with respect to the reduced basis and right-hand side can be very efficiently represented by DNNs. Finally, the fact that neural networks are naturally compositional allows combining the efficient representation of linear problems with the NN implementing operator inversion.

In practice, the approximating DNNs that we show to exist need to be found using a learning algorithm. In this work, we will not analyze the feasibility of learning these DNNs. The typical approach here is to apply methods based on stochastic gradient descent. Empirical studies of this procedure in the context of learning deep neural networks were carried out in . In particular, we mention the recent study in , which analyzes precisely the set-up described in this work and finds a strong impact of the approximation-theoretical behavior of DNNs on their practical performance.

2.2 Simplified Presentation of the Argument

In this section, we present a simplified outline of the arguments leading to the approximation result described in Subsection 1.2.1. In this simplified setup, we think of a ReLU neural network (ReLU NN) as a function

It is intuitively clear from the exponential convergence in (1.3) and proved in [69, Proposition 3] that the size of a NN approximating the scalar multiplication on $^{2}$ up to an error of $\epsilon>0$ is $\mathcal{O}(\log_{2}(1/\epsilon))$.

As a next step, we use the approximate scalar multiplication to approximate a multiplication operator for matrices by ReLU NNs. A matrix multiplication of two matrices of size $d\times d$ can be performed using $d^{3}$ scalar multiplications. Of course, as famously shown in , a more efficient matrix multiplication can also be carried out with less than $d^{3}$ multiplications. However, for simplicity, we focus here on the most basic implementation of matrix multiplication. Hence, the approximate multiplication of two matrices with entries bounded by $1$ can be performed by NN of size $\mathcal{O}(d^{3}\log_{2}(1/\epsilon))$ with accuracy $\epsilon>0$. We make this precise in Proposition 3.7. Along the same lines, we can demonstrate how to construct a NN emulating matrix-vector multiplications.

The map from the parameters to the associated stiffness matrices of the Galerkin discretization of the parametric PDE with respect to a reduced basis can be well approximated by NNs.

The map from the parameters to the right-hand side of the parametric PDEs discretized according to the reduced basis can be well approximated by NNs.

3 Potential Impact and Extensions

We believe that the results of this article have the potential to significantly impact the research on NNs and parametric problems in the following ways:

Theoretical foundation: We offer a theoretical underpinning for the empirical success of NNs for parametric problems which was observed in, e.g., . Indeed, our result, Theorem 4.3, indicates that properly trained NNs are as efficient in solving parametric PDEs as RBMs if the complexity of NNs is measured in terms of free parameters. On a broader level, linking deep learning techniques for parametric PDE problems with approximation theory opens the field up to a new direction of thorough mathematical analysis.

Understanding the role of the ambient dimension: It has been repeatedly observed that NNs seem to offer approximation rates of high-dimensional functions that do not deteriorate exponentially with increasing dimension, .

In this context, it is interesting to identify the key quantity determining the achievable approximation rates of DNNs. Possible explanation for approximation rates that are essentially independent from the ambient dimension have been identified if the functions to be approximated have special structures such as compositionality, , or invariances, . In this article, we identify the highly problem-specific notion of the dimension of the solution manifold as a key quantity determining the achievable approximation rates by NNs for parametric problems. We discuss the connection between the approximation rates that NNs achieve and the ambient dimension in detail in Section 5.

Identifying suitable architectures: One question in applications is how to choose the right NN architectures for the associated problem. Our results show that NNs of sufficient depth and size are able to produce very efficient approximations. Nonetheless, it needs to be mentioned that our results do not yield a lower bound on the number of layers and thus it is not clear whether deep NNs are indeed necessary.

This work is a step towards establishing a theory of deep learning-based solutions of parametric problems. However, given the complexity of this field, it is clear that many more steps need to follow. We outline a couple of natural further questions of interest below:

General parametric problems: Below we restrict ourselves to coercive, symmetric, and linear parametric problems with finitely many parameters. There exist many extensions to, e.g. noncoercive, nonsymmetric, or nonlinear problems, , or to infinite parameter spaces, see e.g. . It would be interesting to see if the methods proposed in this work can be generalized to these more challenging situations.

Bounding the number of snapshots: The interpretation of the parametric problem as a statistical learning problem has the convenient side-effect that various techniques have been established to bound the number of necessary samples $N$, such that the empirical loss (1.1) is very close to the expected loss. In other words, the generalization error of the minimizer of the learning procedure is small, meaning that the prediction rule performs well on unseen data. (Here, the error is measured in a norm induced by the loss function and the underlying probability distribution.). Using these techniques, it is possible to bound the number of snapshots required for the offline phase to achieve a certain fidelity in the online phase. Estimates of the generalization error in the context of high-dimensional PDEs have been deduced in, e.g., .

Special NN architectures: This article studies the feasibility of standard feed-forward NNs. In practice, one often uses special architectures that have proved efficient in applications. First and foremost, almost all NNs used in applications are convolutional neural networks (CNNs), . Hence a relevant question is to what extent the results of this work also hold for such architectures. It was demonstrated in that there is a direct correspondence between the approximation rates of CNNs and that of standard NNs. Thus we expect that the results of this work translate to CNNs.

Another successful architecture is that of residual neural networks (ResNets), . These neural networks also admit skip-connections, i.e., do not only connect neurons in adjacent layers. This architecture is by design at least as powerful as a standard NN and hence inherits all approximation properties of standard NNs.

Necessary properties of neural networks: In this work, we demonstrate the attainability of certain approximation rates by NNs. It is not clear if the presented results are optimal or if there are specific necessary assumptions on the architectures, such as a minimal depth, a minimal number of parameters, or a minimal number of neurons per layer. For approximation results of classical function spaces such lower bounds on specifications of NNs have been established for example in . It is conceivable that the techniques in these works can be transferred to the approximation tasks described in this work.

A special instance of such a function of interest is given by $f(\mathbf{A})\coloneqq e^{t\mathbf{A}},~{}t>0,$ which is analytic and plays an important role in the treatment of initial value problems.

Numerical studies: In a practical learning problem, the approximation-theoretical aspect only describes one part of the problem. Two further central factors are the data generation and the optimization process. It is conceivable that in comparison to these issues, approximation theoretical considerations only play a negligible role. To understand the extent to which the result of this paper is relevant for applications, comprehensive studies of the theoretical set-up of this work should be carried out. A first one was published recently in .

4 Related Work

In this section, we give an extensive overview of works related to this paper. In particular, for completeness, we start by giving a review of approximation theory of NNs without an explicit connection to PDEs. Afterward, we will see how NNs have been employed for the solution of PDEs.

The first and most fundamental results on the approximation capabilities of NNs were universality results. These results claim that NNs with at least one hidden layer can approximate any continuous function on a bounded domain to arbitrary accuracy if they have sufficiently many neurons, . However, these results do not quantify the required sizes of NNs to achieve these rates. One of the first results in this direction was given in . There, a bound on the sufficient size of NNs with sigmoidal activation functions approximating a function with finite Fourier moments is presented. Further results describe approximation rates for various smoothness classes by sigmoidal or even more general activation functions, .

For the non-differentiable activation function ReLU, first rates of approximation were identified in for classes of smooth functions, in for piecewise smooth functions, and in for oscillatory functions. Moreover, NNs mirror the approximation rates of various dictionaries such as wavelets, , general affine systems, , linear finite elements, , and higher-order finite elements, .

4.2 Neural Networks and PDEs

A well-established line of research is that of solving high-dimensional PDEs by NNs assuming that the NN is the solution of the underlying PDE, e.g., . In this regime, it is often possible to bound the size of the involved NNs in a way that does not scale exponentially with the underlying dimension. In that way, these results are quite related to our approaches. Our results do not seek to represent the solution of a PDE as a NN, but a parametric map. Moreover, we analyze the complexity of the solution manifold in terms of Kolmogorov $N$-widths. Finally, the underlying spatial dimension of the involved PDEs in our case would usually be moderate. However, the dimension of the parameter space could be immense.

One of the first approaches analyzing NN approximation rates for solutions of parametric PDEs was carried out in . In that work, the analyticity of the solution map $y\mapsto u_{y}$ and polynomial chaos expansions with respect to the parametric variable are used to approximate the map $y\mapsto u_{y}$ by ReLU NNs of moderate size. Moreover, we mention the works which apply NNs in one way or another to parametric problems. These approaches study the topic of learning a parametric problem but do not offer a theoretical analysis of the required sizes of the involved NNs. These results form our motivation to study the constructions of this paper.

Finally, we mention that the setup of the recent numerical study is closely related to this work.

5 Outline

In Section 2, we describe the type of parametric PDEs that we consider in this paper, and we recall the theory of RBs. Section 3 introduces a NN calculus which is the basis for all constructions in this work. There we will also construct the NN that maps a matrix to its approximate inverse in Theorem 3.8. In Section 4, we construct NNs approximating parametric maps. First, in Theorem 4.1, we approximate the parametric maps after a high-fidelity discretization. Afterward, in Subsection 4.2, we list two broad examples where all assumptions which we imposed are satisfied.

We conclude this paper in Section 5 with a discussion of our results in light of the dependence of the underlying NN complexities in terms of the governing quantities.

To not interrupt the flow of reading, we have deferred all auxiliary results and proofs to the appendices.

6 Notation

Parametric PDEs and Reduced Basis Methods

In this section, we introduce the type of parametric problems that we study in this paper. A parametric problem in its most general form is based on a map $\mathcal{P}\colon\mathcal{Y}\to\mathcal{Z}$, where $\mathcal{Y}$ is the parameter space and $\mathcal{Z}$ is called solution state space $\mathcal{Z}$. In the case of parametric PDEs, $\mathcal{Y}$ describes certain parameters of a partial differential equation, $\mathcal{Z}$ is a function space or a discretization thereof, and $\mathcal{P}(y)\in\mathcal{Z}$ is found by solving a PDE with parameter $y$.

We will place several assumptions on the PDEs underlying $\mathcal{P}$ and the parameter spaces $\mathcal{Y}$ in Section 2.1. Afterward, we give an abstract overview of Galerkin methods in Section 2.2 before recapitulating some basic facts about RBs in Section 2.3.

In the following, we will consider parameter-dependent equations given in the variational form

$\mathcal{Y}$ is the parameter set specified in Assumption 2.1,

$f_{y}\in\mathcal{H}^{*}$ is the parameter-dependent right-hand side of (2.1),

$u_{y}\in\mathcal{H}$ is the solution of (2.1).

Throughout this paper, we impose the following assumptions on Equation (2.1).

In [12, Section 1.2], it has been demonstrated that if $\mathcal{Y}$ is a compact subset of some Banach space $V$, then one can describe every element in $\mathcal{Y}$ by a sequence of real numbers in an affine way. To be more precise, there exist $(v_{i})_{i=0}^{\infty}\subset V$ such that for every $y\in\mathcal{Y}$ and some coefficient sequence $\mathbf{c}_{y}$ whose elements can be bounded in absolute value by 1 there holds $y=v_{0}+\sum_{i=1}^{\infty}(\mathbf{c}_{y})_{i}v_{i},$ implying that $\mathcal{Y}$ can be completely described by the collection of sequences $\mathbf{c}_{y}$. In this paper, we assume these sequences $\mathbf{c}_{y}$ to be finite with a fixed, but possibly large support size.

Symmetry, uniform continuity, and coercivity of the bilinear forms: We assume that for all $y\in\mathcal{Y}$ the bilinear forms $b_{y}$ are symmetric, i.e.

Hence, by the Lax-Milgram lemma (see [57, Lemma 2.1]), Equation (2.1) is well-posed, i.e. for every $y\in\mathcal{Y}$ and every $f_{y}\in\mathcal{H}^{*}$ there exists exactly one $u_{y}\in\mathcal{H}$ such that (2.1) is satisfied and $u_{y}$ depends continuously on $f_{y}$.

Compactness of the solution manifold: We assume that the solution manifold

2 High-Fidelity Approximations

In practice, one cannot hope to solve (2.1) exactly for every $y\in\mathcal{Y}$. Instead, if we assume for the moment that $y$ is fixed, a common approach towards the calculation of an approximate solution of (2.1) is given by the Galerkin method, which we will describe shortly following [33, Appendix A] and [57, Chapter 2.4]. In this framework, instead of solving (2.1), one solves a discrete scheme of the form

3 Theory of Reduced Bases

The starting point of this theory lies in the concept of the Kolmogorov $N$-width which is defined as follows.

The next statement gives a (generally sharp) upper bound which relates the possibility of constructing small snapshot RBs directly to the Kolmogorov $N$-width.

For the underlying discretization matrix, one can demonstrate (see for instance [57, Section 3.4.1]) that

is the coefficient vector of the RB solution if expanded with respect to the high-fidelity basis $\left(\varphi_{i}\right)_{i=1}^{D}.$ Finally, as in Equation 2.3, we obtain

Neural Network Calculus

We start by introducing a formal definition of a NN. Afterward, we introduce several operations, such as parallelization and concatenation that can be used to assemble complex NNs out of simpler ones. Unless stated otherwise we follow the notion of where most of this formal framework was introduced. First, we introduce a terminology for NNs that allows us to differentiate between a NN as a family of weights and the function implemented by the NN. This implemented function will be called the realization of the NN.

where $\mathbf{x}_{L}$ results from the following scheme:

First of all, we note that it is possible to concatenate two NNs in the following way.

We call $\Phi^{1}{\raisebox{2.0pt}{\tiny\newmoon}\,}\Phi^{2}$ the concatenation of $\Phi^{1},$ $\Phi^{2}$.

We now introduce the sparse concatenation of two NNs.

We will see later in Lemma 3.6 the properties of the sparse concatenation of NNs. We proceed with the second operation that we can perform with NNs. This operation is called parallelization.

The following lemma was established in [21, Lemma 5.4] and examines the properties of the sparse concatenation as well as of the parallelization of NNs.

If the input dimension of $\Phi^{1}$, which shall be denoted by $n_{1}$, equals the output dimension of $\Phi^{2},$ and $n_{2}$ is the input dimension of $\Phi^{2},$ then

$L\left(\Phi^{1}\odot\Phi^{2}\right)\leq L\left(\Phi^{1}\right)+L\left(\Phi^{2}\right),$

$M\left(\Phi^{1}\odot\Phi^{2}\right)\leq M\left(\Phi^{1}\right)+M\left(\Phi^{2}\right)+M_{1}\left(\Phi^{1}\right)+M_{L\left(\Phi^{2}\right)}\left(\Phi^{2}\right)\leq 2M\left(\Phi^{1}\right)+2M\left(\Phi^{2}\right),$

$M_{1}\left(\Phi^{1}\odot\Phi^{2}\right)=M_{1}\left(\Phi^{2}\right),$

$M_{L\left(\Phi^{1}\odot\Phi^{2}\right)}\left(\Phi^{1}\odot\Phi^{2}\right)=M_{L\left(\Phi^{1}\right)}\left(\Phi^{1}\right).$

2 A Neural Network Based Approach Towards Matrix Inversion

up to an $\|\cdot\|_{2}$- error of $\epsilon$.

The basic ingredient for the construction of NNs emulating a matrix inversion is the following result about NNs emulating the multiplication of two matrices.

for any $(\mathbf{vec}(\mathbf{A}),\mathbf{vec}(\mathbf{B}))\in K^{Z}_{d,n,l}$ we have

for any $\mathbf{vec}(\mathbf{A})\in K^{1-\delta}_{d}$ we have

In the proof of Theorem 3.8, we approximate the function mapping a matrix to its inverse via the Neumann series and then emulate this construction by NNs. There certainly exist alternative approaches to approximating this inversion function, such as, for example, via Chebyshev matrix polynomials (for an introduction of Chebyshev polynomials, see for instance [65, Chapter 8.2]). In fact, approximation by Chebyshev matrix polynomials is more efficient in terms of the degree of the polynomials required to reach a certain approximation accuracy. However, emulation of Chebyshev matrix polynomials by NNs either requires larger networks than that of monomials or, if they are represented in a monomial basis, coefficients that grow exponentially with the polynomial degree. In the end, the advantage of a smaller degree in the approximation through Chebyshev matrix polynomials does not seem to set off the drawbacks described before.

Neural Networks and Solutions of PDEs Using Reduced Bases

$C^{\mathbf{u}}_{L}$ depends affine linearly on

$C^{\mathbf{u}}_{M}$ depends affine linearly on

Theorem 4.3 guarantees the existence of two moderately sized NNs the realizations of which approximate the discretized solution maps:

suggests that $(y,x)\mapsto u_{y}(x)$ can be approximated with essentially the cost of approximating the respective function in (4.1). Many basis elements that are commonly used for the high-fidelity representation can indeed be approximated very efficiently by realizations of NNs, such as, e.g., polynomials, finite elements, or wavelets .

2 Examples of Neural Network Approximation of Parametric Maps

We will state the following examples already in their variational formulation and note that they fulfill the requirements of Assumption 2.1. We also remark that the presented examples represent only a small portion of problems to which our theory is applicable.

2.2 Example II: Linear Elasticity Equation

$f_{y}(v)\coloneqq y_{3}\int_{\Gamma_{1}}\mathbf{n}\cdot v\,d\mathbf{x}+y_{4}\int_{\Gamma_{2}}\mathbf{n}\cdot v\,d\mathbf{x}+y_{5}\int_{\Gamma_{3}}\mathbf{n}\cdot v\,d\mathbf{x},$ and where $\mathbf{n}$ denotes the outward unit normal on $\partial\Omega.$

For a more thorough discussion of this example (a special case of the linear elasticity equation which describes the displacement of some elastic structure under physical stress on its boundaries), we refer to [57, Chapter 2.1.2, Chapter 2.3.2, Chapter 8.6].

Discussion: Dependence of Approximation Rates on Involved Dimensions

Note that, while having comparable complexity, the NN-based approach is more versatile than using a Galerkin scheme and can be applied even when the underlying PPDE is fully unknown as long as sufficiently many snapshots are available.

Acknowledgments

M.R. would like to thank Ingo Gühring for fruitful discussions on the topic.

G. K. acknowledges partial support by the Bundesministerium für Bildung und Forschung (BMBF) through the Berlin Institute for the Foundations of Learning and Data (BIFOLD), Project AP4, RTG DAEDALUS (RTG 2433), Projects P1, P3, and P8, RTG BIOQIC (RTG 2260), Projects P4 and P9, and by the Berlin Mathematics Research Center MATH+, Projects EF1-1 and EF1-4. P.P. was supported by a DFG Research Fellowship “Shearlet-based energy functionals for anisotropic phase-field methods”. R.S. acknowledges partial support by the DFG through grant RTG DAEDALUS (RTG 2433), Project P14.

References

Appendix A Proofs of the Results from Section 3

In this subsection, we will prove Proposition 3.7. As a preparation, we first prove the following special instance under which $M(\Phi^{1}{\raisebox{2.0pt}{\tiny\newmoon}\,}\Phi^{2})$ can be estimated by $\max\left\{M(\Phi^{1}),M(\Phi^{2})\right\}$.

Let \Phi=\big{(}(\mathbf{A}_{1},\mathbf{b}_{1}),\dots,(\mathbf{A}_{L},\mathbf{b}_{L})\big{)}, and $\mathbf{a},\mathbf{D}$ as in the statement of the lemma. Then the result follows if

It is clear that $\|\mathbf{a}\mathbf{A}_{L}\|_{0}$ is less than the number of nonzero columns of $\mathbf{A}_{L}$ which is certainly bounded by $\|\mathbf{A}_{L}\|_{0}$. The same argument shows that $\|\mathbf{a}\mathbf{b}_{L}\|_{0}\leq\|\mathbf{b}_{L}\|_{0}$. This yields (A.1).

where $I(\gamma)=0$ if $\gamma=0$ and $I(\gamma)=1$ otherwise. Also,

where, for a matrix $\mathbf{G}$, $\mathbf{G}_{l,-}$ denotes the $l$-th row of $\mathbf{G}$. Moreover, we have that for all $l\leq n$

Now we are ready to prove Proposition 3.7.

Without loss of generality, assume that $Z\geq 1$. By [21, Lemma 6.2], there exists a NN $\times^{Z}_{\epsilon}$ with input dimension $2$, output dimension $1$ such that for $\Phi_{\epsilon}\coloneqq\times^{Z}_{\epsilon}$

We have, for all $i\in\{1,\dots,d\},k\in\{1,\dots,n\},j\in\{1,\dots,l\}$, that $L\left(\Phi^{Z}_{i,k,j;\epsilon}\right)=L\left(\times^{Z}_{\epsilon}\right)$ and by Lemma A.1 that $\Phi^{Z}_{i,k,j;\epsilon}$ satisfies (A.2), (A.3), (A.4) with $\Phi_{\epsilon}\coloneqq\Phi^{Z}_{i,k,j;\epsilon}$. Moreover, we have by (A.5)

which by Lemma 3.6 is a NN with $n\cdot(d+l)$-dimensional input and $1$-dimensional output such that (A.2) holds with $\Phi_{\epsilon}\coloneqq\Phi^{Z}_{i,j;\epsilon}$. Moreover, by Lemmas A.1 and 3.6 and by (A.3) we have that

Additionally, by Lemmas 3.6 and A.1 and (A.4), we obtain

which yields (ii) of the result. Moreover, by Lemma 3.6 and (A.8) it follows that

Equation (A.9) establishes (iv) of the asserted result. Finally, we have for any $(\mathbf{vec}(\mathbf{A}),\mathbf{vec}(\mathbf{B}))\in K^{Z}_{d,n,l}$ that

This demonstrates that (v) holds and thereby finishes the proof. ∎

A.2 Proof of Theorem 3.8

for every $(\mathbf{vec}(\mathbf{A}),\mathbf{vec}(\mathbf{B}))\in K^{Z}_{d,d,d}$ we have

One consequence of the ability to emulate the multiplication of matrices is that we can also emulate the squaring of matrices. We make this precise in the following definition.

for all $\mathbf{vec}(\mathbf{A})\in K^{Z}_{d}$ we have

for every $\mathbf{vec}(\mathbf{A})\in K^{1/2}_{d}$ we have

and $\Phi^{1/2,d}_{2^{j};\epsilon}$ satisfies (i),(ii),(iii). Now we define

By the triangle inequality, we obtain for any $\mathbf{vec}(\mathbf{A})\in K_{d}^{1/2}$

By construction of $\Phi^{1/2,d}_{2^{j+1};\epsilon}$, we know that

Therefore, using the triangle inequality and the fact that $\|\cdot\|_{2}$ is a submultiplicative operator norm, we derive that

where the penultimate estimate follows by the induction hypothesis (A.10) and $\epsilon<1/4$. Hence, since $\|\cdot\|_{2}$ is a submultiplicative operator norm, we obtain

where we used $\left\|\mathbf{A}^{2^{j}}\right\|_{2}\leq 1/4$ and the induction hypothesis (A.10). Applying (A.13) and (A.12) to (A.11) yields

The estimates (A.14) and (A.15) complete the proof of the assertions (iv) and (v) of the proposition statement. Now we estimate the size of $\Phi^{1/2,d}_{2^{j+1};\epsilon}$. By the induction hypothesis and Lemma 3.6(a)(i), we obtain

which implies (i). Moreover, by the induction hypothesis and Lemma 3.6(a)(ii), we conclude that

implying (ii). Finally, it follows from Lemma 3.6(a)(iii) in combination with the induction hypothesis as well Lemma 3.6(a)(iv) that

for any $\mathbf{vec}(\mathbf{A})\in K^{1/2}_{d}$ we have

By construction and Lemma 3.6(a)(iv), we first observe that

Moreover, we obtain by the induction hypothesis as well as Lemma 3.6(a)(iii) in combination with Lemma 3.6(b)(iv) that

This shows (iii). To show (iv), we perform a similar estimate as the one following (A.11). By the triangle inequality,

By the construction of $\Phi^{1/2,d}_{k;\epsilon}$ and the Proposition 3.7, we conclude that

where we have used that $\left(\frac{1}{2}\right)^{t}\leq\frac{1}{4}$ for $t\geq 2$. This shows (iv). In addition, by an application of the triangle inequality, we have that

This shows (v). Now we analyze the size of $\Phi^{1/2,d}_{k;\epsilon}$. We have by Lemma 3.6(a)(i) in combination with Lemma 3.6(b)(i) and by the induction hypothesis that

which implies (i). Finally, we address the number of non-zero weights of the resulting NN. We first observe that, by Lemma 3.6(a)(ii),

where we have used the definition of the parallelization for the first two equalities, Lemma 3.6(b)(iii) for the third equality, Lemma 3.6(a)(ii) for the fourth inequality as well as the properties of $\Phi^{\mathbf{Id}}_{d^{2},L}$ in combination with Proposition A.3(iii) for the last inequality. Moreover, by the definition of the parallelization of two NNs with different numbers of layers, we conclude that

Proposition A.4 only provides a construction of a NN the ReLU-realization of which emulates a power of a matrix $\mathbf{A}$, under the assumption that $\|\mathbf{A}\|_{2}\leq 1/2$. We remove this restriction in the following corollary by presenting a construction of a NN $\Phi^{Z,d}_{k;\epsilon}$ the ReLU-realization of which approximates the map $\mathbf{A}\mapsto\mathbf{A}^{k},$ on the set of all matrices $\mathbf{A}$ the norms of which are bounded by an arbitrary $Z>0$.

for any $\mathbf{vec}(\mathbf{A})\in K^{Z}_{d}$ we have

Let $\left((\mathbf{A}_{1},\mathbf{b}_{1}),...,(\mathbf{A}_{L},\mathbf{b}_{L})\right)\coloneqq\Phi^{1/2,d}_{k;\frac{\epsilon}{2\max\left\{1,Z^{k}\right\}}}$ according to Proposition A.4. Then the NN

fulfills all of the desired properties. ∎

We have seen how to construct a NN that takes a matrix as an input and computes a power of this matrix. With this tool at hand, we are now ready to prove Theorem 3.8.

We have for any $\mathbf{vec}(\mathbf{A})\in K^{1-\delta}_{d}$

This completes the proof of (iii). Moreover, (iv) is a direct consequence of (iii). Now we analyze the size of the resulting NN. First of all, we have by Lemma 3.6(b)(i) and Corollary A.5 that

which implies (i). Moreover, by Lemma 3.6(b)(ii), Corollary A.5 and the monotonicity of the logarithm, we obtain

Appendix B Proof of Theorem 4.3

Due to the fact that for two invertible matrices $\mathbf{M},\mathbf{N},$

where we have used Assumption 4.1, Equation (2.9) and Equation (B.2). Now we turn our attention to estimating II. First, observe that for every $y\in\mathcal{Y}$ by the triangle inequality and Remark B.2, that

Finally, by construction we can conclude that

This implies (iii) of the assertion. Now, by Equation (2.7) we obtain

completing the proof of (iv). Finally, for all $y\in\mathcal{Y}$ we estimate

and, by Lemma 3.6(a)(ii) in combination with Theorem 3.8(ii), we obtain

We proceed with estimating II. It is not hard to see from Assumption 4.2 that

where we applied Proposition B.3(i) and chose a suitable constant

We now note that if we establish (iii), then (iv) follows immediately by Lemma 3.6(a)(ii). Thus, we proceed with proving (iii). First of all, by Lemma 3.6(a)(ii) in combination with Proposition 3.7 we have

Next, by Lemma 3.6(b)(ii) in combination with Proposition B.3 as well as Assumption 4.1 and Assumption 4.2 we have that