VSL- Variational Spectral Learning in TensorFlow: Reference Implementation and Experiments

This is the first release of the Variational Spectral Learning code (reference implementation) used to generate the figures/analyses in the manuscript titled "Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations" by M. M. Hammad. Variational spectral learning (VSL) is a coefficient-space machine learning framework for solving partial differential equations (PDEs) that unifies variational formulations, spectral discretization, and modern gradient-based optimization. The key idea is to represent the unknown solution as a finite spectral expansion in tensor-product Chebyshev bases over space and time, using Dirichlet-satisfying spatial modes to enforce homogeneous boundary conditions analytically. Rather than training a high-dimensional field surrogate, VSL learns a compact coefficient vector by minimizing differentiable space–time functionals constructed from either strong-form least-squares residuals or weak (Galerkin) projected residuals.