Nonlinear integro-differential operator regression with neural networks
This addresses the challenge of operator regression in computational mathematics, but it appears incremental as it builds on existing neural network and Fourier transform techniques without broad validation.
The paper tackled the problem of learning nonlinear integro-differential operators from data by introducing a regression method that uses neural networks and Fourier transforms, and verified it by recovering operators from numerical solutions of the fractional heat and Kuramoto-Sivashinsky equations.
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.