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

• URL: http://arxiv.org/abs/2601.02492v1
• Date: Mon, 05 Jan 2026 19:03:58 GMT
• Title: Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations
• Authors: M. M. Hammad,
• Abstract summary: We introduce variational spectral learning (VSL), a machine learning framework for solving partial differential equations (PDEs)<n>VSL offers a principled bridge between variational PDE theory, spectral discretization, and contemporary machine learning practice.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 \[ \mathcal{L}u = f \quad \text{in} \quad Q=Ω\times(0,T), \] 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 \[ u_N(x,t)=\sum_{n=1}^{N} c_n\,φ_n(x,t), \] where $φ_n$ 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.

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