Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition
This work provides an incremental improvement for researchers in computational physics and data science by combining supervised and unsupervised learning with physics constraints to handle sparse datasets.
The paper tackles the problem of solving the discrete Helmholtz-Hodge decomposition from sparse data by developing a Physics-informed Spectral Learning (PiSL) framework, achieving spectral (exponential) convergence in numerical examples such as the 'Storm of the Century' with 1993 satellite data.
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.