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

TL;DR

This paper introduces a dimension-incremental sparse approximation method for solving differential equations by learning high-dimensional functions with basis coefficients, enabling better resolution and understanding of parameter dependencies.

Contribution

It proposes a novel sparse high-dimensional approximation approach using basis coefficient detection for differential equation solutions, improving scalability and interpretability.

Findings

Effective in approximating solutions of differential equations

Provides insights into parameter dependencies and interactions

Enhances resolution in high-dimensional function approximation

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 into parameters like Fourier or Spline coefficients and treat the solution of the differential equation as a high-dimensional function w.r.t. the spatial variables, these parameters and also 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 on the structure of the solution as well as its dependency on the parameters and their interactions and allows for a reasonable generalization to even higher dimensions and therefore better resolutions of the decomposed source function.