Structure-informed operator learning for parabolic Partial Differential Equations

TL;DR

This paper introduces a novel neural network framework that leverages basis functions and structural information to learn solution maps of backward parabolic PDEs, offering advantages over existing DeepONet methods.

Contribution

The paper proposes a structure-informed operator learning approach using basis functions, providing an alternative to DeepONets for infinite-dimensional PDE solution maps.

Findings

Effective learning of solution maps demonstrated through numerical experiments

Advantages over DeepONets highlighted in proof-of-concept results

Leverages basis functions to encode structural information in operator learning

Abstract

In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fr\'echet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets.