Operator Compression with Deep Neural Networks

TL;DR

This paper introduces a neural network-based method for compressing parameterized partial differential operators, enabling efficient surrogate modeling and fast online computations for complex multiscale problems.

Contribution

It proposes directly approximating the coefficient-to-surrogate map with neural networks, improving compression ratios and computational efficiency over traditional numerical methods.

Findings

Achieves large compression ratios for multiscale operators

Enables fast online surrogate evaluations via neural network forward passes

Demonstrates effectiveness on elliptic heterogeneous diffusion operators

Abstract

This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example.