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Structure-informed operator learning for parabolic Partial Differential Equations

Fred Espen Benth, Nils Detering, Luca Galimberti

TL;DR

This paper presents a framework for learning the solution map of a backward parabolic Cauchy problem, using Fr\'echet space neural networks to address this operator learning problem, and provides an alternative to Deep Operator Networks.

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 Fréchet 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.

Structure-informed operator learning for parabolic Partial Differential Equations

TL;DR

This paper presents a framework for learning the solution map of a backward parabolic Cauchy problem, using Fr\'echet space neural networks to address this operator learning problem, and provides an alternative to Deep Operator Networks.

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 Fréchet 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.

Paper Structure

This paper contains 8 sections, 7 theorems, 82 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A brief introduction to neural networks in infinite dimensions
  3. A continuity analysis of the nonlinear operator maps
  4. The parabolic Cauchy problem
  5. Solution maps and continuity
  6. A numerical case study
  7. Basis functions
  8. Numerical example

Key Result

Theorem 2.2

Assume $\sigma:\mathfrak{X}\rightarrow\mathfrak{X}$ is a continuous, bounded and separating activation function and suppose that $F\in C(\mathfrak{X};\mathfrak{Y})$. Then, for given compact set $\mathcal{K}\subset\mathfrak{X}$ and $\epsilon>0$ there exist $d\in\mathbb N$, $d$ independent unit vector where $\mathcal{N}_d$ is defined in nn-def-inf.

Figures (3)

  • Figure 1: First 5 basis functions $e_1,\ldots,e_5$ (left) and 5 random samples of $c$ from $\mathcal{K}$ (right)
  • Figure 2: Box plots of the error distributions. The left figure shows the result from the Fréchet neural network, and the right figure the results with DeepONet.
  • Figure 3: Comparison of error distributions for various values of $x$. The first row shows the result from the Fréchet neural network, and the second row shows the results with DeepONet.

Theorems & Definitions (15)

  • Definition 2.1: Separation property
  • Theorem 2.2
  • Remark 3.2
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Corollary 3.7
  • proof
  • ...and 5 more