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

Abstract:

Addressing time-dependent partial differential equations (PDEs) constitutes a critical challenge within computational science and engineering. While Physics-Informed Neural Networks (PINNs) derive solutions to PDEs directly from their governing equations, accurately modeling temporal evolution continues to pose significant difficulties. Recent methodologies relying on general-purpose sequence models parameterize time evolution by capturing temporal dependencies; however, these approaches fail to explicitly encode the structured dynamics inherent to PDE solutions. Furthermore, their memory demands often scale poorly with respect to sequence length and resolution, thereby restricting their utility in high-dimensional or large-scale applications.

This study presents a novel PINN framework that integrates oscillatory state-space dynamics to effectively represent the modal structure of PDE solutions. The proposed architecture utilizes a linear-oscillator-based mechanism for temporal evolution, complemented by a PDE-aware spectral basis for spatial representation. This specific design facilitates closed-form spatial differentiation and ensures the consistent enforcement of boundary conditions. The methodology is rigorously tested across forward, inverse, and high-dimensional PDE problems, extending to scenarios involving up to 100 spatial dimensions. Empirical results demonstrate that this approach yields superior accuracy and lower memory consumption when compared to contemporary sequence-model-based PINN techniques. Ultimately, this research underscores the advantages of embedding structured dynamical priors into the temporal evolution of neural PDE solvers, advocating for the development of PINN architectures that are both computationally efficient and aligned with physical principles.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC