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Preprint · Preprints.org · September 24, 2026

A Spectral–Physics-Informed Neural Network Hybrid with Energy-Constrained Training for the Allen–Cahn Equation: Forward and Inverse Problems

Zhuldyz Issa, Nauryzbay Adil, Zh. A. Abdiramanov, Yessimkhan Zhumazhanov, Dastan Urnek

Zhanars Abdiramanov
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Abstract

Physics-informed neural networks (PINNs) offer a mesh-free alternative to classical discretizations of the Allen–Cahn equation, but suffer spectral bias near sharp interfaces and give no structural guarantee that the Ginzburg–Landau energy decreases, a property tied to the double-well potential’s u↦−u symmetry. We propose a Spectral-PINN hybrid whose network outputs a time-dependent vector of Chebyshev (1D) or Fourier (2D) spectral coefficients rather than the field value directly, so spatial derivatives are exact and, in 2D, reflection symmetry is preserved to machine precision by construction. On a benchmark problem the Spectral-PINN cuts the L2 error by a factor of ∼49 relative to a vanilla PINN while training ∼5.4× faster, with zero violations under an added energy-constrained loss. A stress test up to K=256 modes, however, shows accuracy degrading, not improving, even under proportionally larger training budgets, exposing an optimization bottleneck absent in classical spectral methods. A systematic sweep of the inverse problem (recovering the interface width ε) rejects our own initial hypothesis that its ∼38% recovery bias is a truncation artifact, since the bias stays flat across a 6× range of K. In two dimensions, the Fourier-Spectral-PINN’s symmetry violation remains at machine precision across all tested scales, whereas a vanilla PINN’s violation grows measurably with time; at longer horizons (T=10), the energy-constrained loss keeps enforcing zero energy violations even once solution accuracy has fully collapsed, dissociating this structural guarantee from correctness. Finally, we formalize in Lean 4 with Mathlib both the discrete algebraic lemmas and the continuous energy-dissipation theorem itself, as a general Hilbert-space gradient-flow result resting only on Lean’s standard axioms.

Model Reduction and Neural NetworksNeural Networks and ApplicationsNumerical methods for differential equationsAllen-Cahn equationphysics-informed neural networksspectral coefficients

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