Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

2026-05-29

AI Summary

This work addresses the high memory overhead and inadequate physical structure modeling in existing sequential models for solving time-dependent partial differential equations (PDEs). It introduces, for the first time, an oscillatory state-space model with explicit physical meaning as an inductive bias into physics-informed neural networks (PINNs). By integrating linear oscillator-driven temporal evolution with a PDE-aware spectral basis representation, the proposed method enables closed-form spatial differentiation, strictly enforces boundary conditions, and scales effectively to high-dimensional problems. Evaluated on both forward and inverse tasks—including PDEs up to 100 dimensions—the approach significantly outperforms current sequence-based PINNs, achieving higher accuracy while substantially reducing memory consumption.

Abstract

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

Problem

• time-dependent PDEs
• physics-informed neural networks
• temporal evolution
• structured dynamics
• memory efficiency

Innovation

• oscillatory state-space models
• physics-informed neural networks
• spectral basis
• temporal evolution
• high-dimensional PDEs