chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data

TL;DR

This paper introduces chebgreen, a data-driven, mesh-independent library that learns and interpolates Empirical Green's Functions for 1D systems with unknown PDEs, using neural networks and Chebyshev basis representations.

Contribution

It develops a novel method to learn and interpolate Green's functions from data for systems with unknown PDEs, combining neural networks and Chebyshev basis techniques.

Findings

Successfully models Green's functions for unseen parameters

Enables interpolation of singular functions on a manifold of Quasimatrices

Provides a mesh-independent approach for PDE system analysis

Abstract

In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns an Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network from which we subsequently construct a bivariate representation in a Chebyshev basis. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials.