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

• URL: http://arxiv.org/abs/2501.18715v2
• Date: Wed, 12 Feb 2025 17:41:57 GMT
• Title: chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data
• Authors: Harshwardhan Praveen, Jacob Brown, Christopher Earls,
• Abstract summary: We present a mesh-independent, data-driven library, chebgreen, to model one-dimensional systems. We learn an Empirical Green's Function for the associated, but hidden, boundary value problem. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library.
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• 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.

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