Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations

TL;DR

This paper presents variational spectral learning (VSL), a machine learning approach that solves PDEs by optimizing spectral coefficients within a variational framework, combining spectral discretization with modern deep learning tools.

Contribution

VSL introduces a novel spectral learning framework that recasts PDE solutions into differentiable energies, enabling efficient optimization with machine learning techniques.

Findings

Achieves accuracy comparable to classical spectral methods.

Effectively solves benchmark elliptic and parabolic PDEs.

Utilizes TensorFlow for robust optimization of spectral coefficients.

Abstract

We introduce variational spectral learning (VSL), a machine learning framework for solving partial differential equations (PDEs) that operates directly in the coefficient space of spectral expansions. VSL offers a principled bridge between variational PDE theory, spectral discretization, and contemporary machine learning practice. The core idea is to recast a given PDE \[ \mathcal{L}u = f \quad \text{in} \quad Q=\Omega\times(0,T), \] together with boundary and initial conditions, into differentiable space-time energies built from strong-form least-squares residuals and weak (Galerkin) formulations. The solution is represented as a finite spectral expansion \[ u_N(x,t)=\sum_{n=1}^{N} c_n\,\phi_n(x,t), \] where $\phi_n$ are tensor-product Chebyshev bases in space and time, with Dirichlet-satisfying spatial modes enforcing homogeneous boundary conditions analytically. This yields a compact linear parameterization in the coefficient vector $\mathbf{c}$, while all PDE complexity is absorbed into the variational energy. We show how to construct strong-form and weak-form space-time functionals, augment them with initial-condition and Tikhonov regularization terms, and minimize the resulting objective with gradient-based optimization. In practice, VSL is implemented in TensorFlow using automatic differentiation and Keras cosine-decay-with-restarts learning-rate schedules, enabling robust optimization of moderately sized coefficient vectors. Numerical experiments on benchmark elliptic and parabolic problems, including one- and two-dimensional Poisson, diffusion, and Burgers-type equations, demonstrate that VSL attains accuracy comparable to classical spectral collocation with Crank-Nicolson time stepping, while providing a differentiable objective suitable for modern optimization tooling.