Volume 63, pp. 468-495, 2025.

An approach to discrete operator learning based on sparse high-dimensional approximation

Daniel Potts and Fabian Taubert

Abstract

We present a dimension-incremental method for function approximation in bounded orthonormal product bases to learn the solutions of various differential equations. Therefore, we decompose the source function of the differential equation using parameters like Fourier or spline coefficients and treat the solution of the differential equation as a high-dimensional function with respect to the spatial variables, to these parameters, and also to further possible parameters from the differential equation itself. Finally, we learn this function in the sense of sparse approximation in a suitable function space by detecting coefficients of the basis expansion with the largest absolute values. Investigating the corresponding indices of the basis coefficients yields further insights into the structure of the solution as well as its dependency on the parameters and their interactions. This allows a reasonable generalization to even higher dimensions and therefore better resolutions of the decomposed source function.

Full Text (PDF) [592 KB], BibTeX , DOI: 10.1553/etna_vol63s468

Key words

sparse approximation, nonlinear approximation, high-dimensional approximation, dimension-incremental algorithm, partial differential equations, operator learning

AMS subject classifications

35C09, 35C11, 41A50, 42B05, 65D15, 65D30, 65D32, 65D40, 65T40