Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow

Deep learning paradigms, such as Physics-Informed Neural Networks (PINNs) and neural operators, have significantly advanced the solving of partial differential equations (PDEs). However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre-discretized temporal grids. Drawing inspiration from MeanFlow, a continuous-time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi-step numerical integration into an efficient prediction with a controllable integration length. Crucially, we extend the original MeanFlow loss from the temporal to spatio-temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time-dependent and stationary PDEs. Comprehensive experiments on canonical benchmarks demonstrate that our approach achieves superior accuracy over representative baselines. Furthermore, the proposed integral constraint enables excellent zero-shot generalization to out-of-distribution initial conditions and varying spatial resolutions.