Sliding physical invariant neural operator for long-term prediction of complex dynamics in physical systems

Operator learning aims to simulate the underlying physical system to solve an entire family of partial differential equations (PDEs), rather than focusing on a single instance of an equation. Current approaches encode initial conditions into physical invariants to guide the solution of multiple equations; however, these static physical invariants capture only short-term dynamics and fail to represent long-term evolution, thereby degrading both accuracy and stability over time. In this paper, we introduce the sliding physical invariant neural operator, which integrates physical invariants that slide dynamically with the evolution of the underlying PDEs. Compared with existing methods, our approach achieves substantial error reductions of 34.3%-79.8% in the training domain and 7.7%-76.5% in the future domain, underscoring its markedly enhanced generalization and long-term predictive performance across a wide range of PDE scenarios.