Towards Numerical Method-Informed Neural Networks for PDE Learning

Pasquale De Luca () and Livia Marcellino ()
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Pasquale De Luca: Department of Science and Technology, Parthenope University of Naples, Centro Direzionale Isola C4, 80143 Naples, Italy
Livia Marcellino: Department of Science and Technology, Parthenope University of Naples, Centro Direzionale Isola C4, 80143 Naples, Italy

Mathematics, 2025, vol. 13, issue 15, 1-24

Abstract: Solving stiff partial differential equations with neural networks remains challenging due to the presence of multiple time scales and numerical instabilities that arise during training. This paper addresses these limitations by embedding the mathematical structure of implicit–explicit time integration schemes directly into neural network architectures. The proposed approach preserves the operator splitting decomposition that separates stiff linear terms from non-stiff nonlinear terms, inheriting the stability properties established for these numerical methods. We evaluate the methodology on Allen–Cahn equation dynamics, where interface evolution exhibits the multi-scale behavior characteristic of stiff systems. The structure-preserving architecture achieves improvements in solution accuracy and long-term stability compared to conventional physics-informed approaches, while maintaining proper energy dissipation throughout the evolution.

Keywords: structure-preserving neural networks; implicit–explicit methods; stiff partial differential equations; Allen–Cahn equation; operator splitting; physics-informed learning (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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