chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data
This work provides a data-driven method for modeling physical systems where the underlying equations are unknown, which is incremental as it builds on existing interpolation and neural network techniques for specific boundary value problems.
The authors tackled the problem of modeling one-dimensional systems with unknown governing PDEs by learning Empirical Green's Functions from data, resulting in a mesh-independent library called chebgreen that can interpolate these functions at unseen control parameters using Rational Neural Networks and Chebyshev basis representations.
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.