Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908750111047680 |
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| author | Hammad, M. M. |
| author_facet | Hammad, M. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02492 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations Hammad, M. M. Numerical Analysis Machine Learning 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. |
| title | Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2601.02492 |