# A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

## TL;DR

This paper provides a theoretical framework for understanding how deep neural networks can efficiently approximate solutions to parametric PDEs by leveraging low-dimensional solution manifolds, resulting in superior approximation rates.

## Contribution

It introduces a novel theoretical analysis showing neural networks can exploit low-dimensional structures in parametric PDE solutions, improving approximation efficiency.

## Key findings

- Neural networks can approximate parametric PDE solutions with sizes depending mainly on reduced basis dimensions.
- The approach yields significantly better approximation rates than classical neural network results.
- The method applies broadly to various parametric PDEs using the existence of small reduced bases.

## Abstract

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.