Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective

Tanya Marwah, Zachary Chase Lipton, Jianfeng Lu, Andrej Risteski
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:24139-24172, 2023.

Abstract

A burgeoning line of research has developed deep neural networks capable of approximating the solutions to high dimensional PDEs, opening related lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most theoretical analyses thus far have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional $\mathcal{E}(u) = \int_\Omega L(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $\epsilon$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ \epsilon), p^{\log(1/\epsilon)}\}}\right)$. By a classical result due to Barron (1993), this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, \epsilon, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-marwah23a, title = {Neural Network Approximations of {PDE}s Beyond Linearity: A Representational Perspective}, author = {Marwah, Tanya and Lipton, Zachary Chase and Lu, Jianfeng and Risteski, Andrej}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {24139--24172}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/marwah23a/marwah23a.pdf}, url = {https://proceedings.mlr.press/v202/marwah23a.html}, abstract = {A burgeoning line of research has developed deep neural networks capable of approximating the solutions to high dimensional PDEs, opening related lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most theoretical analyses thus far have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional $\mathcal{E}(u) = \int_\Omega L(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $\epsilon$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ \epsilon), p^{\log(1/\epsilon)}\}}\right)$. By a classical result due to Barron (1993), this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, \epsilon, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.} }
Endnote
%0 Conference Paper %T Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective %A Tanya Marwah %A Zachary Chase Lipton %A Jianfeng Lu %A Andrej Risteski %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-marwah23a %I PMLR %P 24139--24172 %U https://proceedings.mlr.press/v202/marwah23a.html %V 202 %X A burgeoning line of research has developed deep neural networks capable of approximating the solutions to high dimensional PDEs, opening related lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most theoretical analyses thus far have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional $\mathcal{E}(u) = \int_\Omega L(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $\epsilon$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ \epsilon), p^{\log(1/\epsilon)}\}}\right)$. By a classical result due to Barron (1993), this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, \epsilon, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.
APA
Marwah, T., Lipton, Z.C., Lu, J. & Risteski, A.. (2023). Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:24139-24172 Available from https://proceedings.mlr.press/v202/marwah23a.html.

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