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

• URL: http://arxiv.org/abs/2302.11061v1
• Date: Tue, 21 Feb 2023 23:33:29 GMT
• Title: Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition
• Authors: Luis Espath, Pouria Behnoudfar, and Raul Tempone
• Abstract summary: We further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients. Our PiSL computational framework enjoys spectral (exponential) convergence. We assess the capabilities of our method in various numerical examples including the Storm of the Century' with satellite data from 1993.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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