Decoupling Universal Laws and Environmental Heterogeneity: A Physics-Inspired Framework for Robust Spatio-Temporal Forecasting

Most spatio-temporal forecasting models assume in-distribution data and can degrade sharply under non-stationary environments. Existing methods for handling distribution shift largely rely on discrete graph inference, making it difficult to disentangle universal dynamics from environment-specific changes and to respect the continuous physical nature of spatio-temporal fields. To this end, we propose STPDE, a general framework that reformulates spatio-temporal dynamics as the evolution of inhomogeneous partial differential equations. STPDE explicitly decomposes dynamics into an Invariant Diffusion Operator that captures universal mechanisms and an Environment Basis Manifold that parameterizes local heterogeneous media. We show that the Green's function of the Laplacian can be effectively approximated by linear attention, enabling global diffusion at scale. Combined with stochastic environment perturbations, STPDE improves robustness under heterogeneous and shifting environments. Extensive experiments on in-distribution forecasting, out-of-distribution generalization, few-shot cross-city transfer, and continual learning demonstrate consistent improvements over state-of-the-art baselines with competitive computational efficiency.