Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition

TL;DR

This paper advances Physics-informed Spectral Learning (PiSL) for discrete Helmholtz--Hodge decomposition, achieving spectral convergence by adaptively building sparse Fourier bases with regularization, combining supervised and unsupervised learning on sparse data.

Contribution

The work introduces an adaptive, sparse Fourier basis construction within PiSL for Helmholtz--Hodge decomposition, with spectral convergence and combined data and physics constraints.

Findings

Spectral (exponential) convergence demonstrated.

Effective decomposition from sparse satellite data.

Regularization improves stability and accuracy.

Abstract

In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. Moreover, our PiSL computational framework enjoys spectral (exponential) convergence. We regularize the minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the divergence- and curl-free constraints become a finite set of linear algebraic equations. The proposed computational framework combines supervised and unsupervised learning techniques in that we use data concomitantly with the projection onto divergence- and curl-free spaces. We assess the capabilities of our method in various numerical examples including the `Storm of the Century' with satellite data from 1993.