Nonlinear integro-differential operator regression with neural networks

TL;DR

This paper presents a neural network-based regression method to identify nonlinear integro-differential operators directly from data, avoiding the need for predefined operator libraries.

Contribution

It introduces a novel approach combining neural networks and Fourier transforms to learn a broad class of nonlinear operators from data.

Findings

Successfully recovered operators in fractional heat and Kuramoto-Sivashinsky equations

Demonstrated the method's ability to fit complex nonlinear operators

Validated the approach with numerical solutions

Abstract

This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We verify that this method can recover the spatial operators in the fractional heat equation and the Kuramoto-Sivashinsky equation from numerical solutions of the equations.