Spatiotemporal decoupled physics-informed Stone-Weierstrass neural operator for long-time prediction of time-dependent parametric PDEs

TL;DR

This paper introduces PI-SWNO, a novel neural operator architecture that decouples space and time to improve long-term predictions of time-dependent PDEs, addressing accuracy and stability issues.

Contribution

It proposes a spatiotemporally decoupled, physics-informed neural operator based on Stone-Weierstrass theorem, enhancing stability and reducing memory use for long-time PDE predictions.

Findings

Mitigates error accumulation over long time horizons.

Reduces training costs and memory consumption.

Ensures continuity and convergence in full-time solutions.

Abstract

Driven by rapid advances in artificial intelligence and modern GPU computing capabilities, deep learning methods based on the optimization paradigm have provided new pathways to solve spatiotemporal physical problems, whose mathematical core lies in solving partial differential equations (PDEs). As an emerging class of function-space learning methods, neural operators (NOs) have exhibited great potential in efficient PDE solving. However, existing mainstream neural operator frameworks suffer from critical bottlenecks when modeling time-dependent PDEs over long time horizons, including accuracy degradation, insufficient stability, high training costs, and excessive memory consumption, which severely limit their practical deployment. To address these challenges in long-time prediction with neural operators, we propose a novel spatiotemporally decoupled physics-informed neural operator architecture, termed the physics-informed Stone-Weierstrass neural operator (PI-SWNO). The design is theoretically grounded in the decoupling paradigm combining time-invariant spatial basis functions with time-varying evolution coefficients, as well as the Stone-Weierstrass approximation theorem. By encoding spatial and temporal information via two separate subnetworks, the framework structurally mitigates the accumulation of errors over extended time intervals. Furthermore, we introduce a time-marching batch-wise sampling strategy to resolve the memory bottleneck of full-range modeling over extended time spans, ensuring continuity and convergence of full-time-domain solutions.