Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow

TL;DR

This paper introduces Spatio-Temporal MeanFlow, a novel physics-informed neural PDE solver that captures the integral nature of physical systems, improving accuracy, efficiency, and generalization over existing methods.

Contribution

It extends MeanFlow to spatio-temporal domains, transforming PDE solving into efficient prediction and coupling time evolution with spatial consistency.

Findings

Achieves superior accuracy and inference efficiency on benchmark PDEs.

Generalizes well to out-of-distribution initial conditions.

Handles varying spatial resolutions effectively.

Abstract

Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of 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 freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the 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 benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out-of-distribution initial conditions and varying spatial resolutions.