2023/02/21 by Luis Espath, Espath, Luis, Pouria Behnoudfar +3
Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Gaussian Processes and Bayesian Inference #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2302.11061
openalex publication_date 2023/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \citeEsp21 based on a discrete L2 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.