Model Reduction and Neural Networks for Parametric PDEs

TL;DR

This paper introduces a neural network framework for approximating input-output maps in parametric PDEs, combining model reduction and deep learning to achieve convergence and robustness in infinite-dimensional settings.

Contribution

It presents a novel data-driven approach that extends neural network approximation to infinite-dimensional spaces with proven convergence for certain classes of PDE-related maps.

Findings

Demonstrates convergence and robustness of the method

Shows effectiveness through numerical experiments

Compares favorably with existing algorithms

Abstract

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers' equation.