PGMNO: A physics-Guided mamba neural operator framework for partial differential equations.

Accurately modeling the long-term evolution of complex physical systems governed by partial differential equations (PDEs) remains a central challenge in operator learning. In this work, we introduce the Physics-Guided Mamba Neural Operator (PGMNO), a framework specifically designed to capture long-range spatiotemporal dependencies. PGMNO overcomes the limitations of conventional single-step and Transformer-based operators by unifying linear multistep numerical methods with structured state space models (SSMs). To ensure temporal stability, the framework employs multistep temporal modeling in the forward pass, complemented by an implicit backward differentiation formula (BDF)-based scheme during training. Simultaneously, PGMNO leverages the efficient kernel integration properties of SSMs to achieve scalable and stable operator learning. We validate PGMNO on diverse PDE benchmarks. Experimental results demonstrate that PGMNO consistently outperforms state-of-the-art models in prediction accuracy, computational efficiency, and long-term stability. Additionally, the framework exhibits strong resolution-invariant extrapolation and generalization across varying spatial discretizations. By prioritizing clear methodological articulation, this work underscores the potential of unifying numerical integration with state space modeling to build robust surrogate solvers for PDE-governed dynamical systems.