用时空均流思想提升物理PDE求解精度与效率
Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
- 将均流思想扩展至时空域，实现物理状态连续演化建模
- 相比基线方法，精度更高且推理速度更快，支持任意积分长度
- 对分布外初始条件和不同分辨率具有强泛化能力，适合复杂物理模拟
深度学习方法如PINNs和神经算子虽在求解偏微分方程（PDE）方面取得进展，但常难以捕捉物理系统的连续积分特性，要么依赖忽略积分视角的点态残差，要么依赖预离散的时间网格。受近期用于高效求解生成型常微分方程的连续时间积分器MeanFlow启发，本文提出时空均流（Spatio-Temporal MeanFlow），作为新型PDE求解器，学习物理状态在有限区间内的演化过程。通过将生成速度场替换为物理PDE算子，将多步数值积分转化为可自由控制积分长度的高效预测。关键在于，将原版均流约束从时间域扩展至时空域，耦合时间演化与空间一致性，形成统一框架，自然适配时变与稳态PDE。大量基准实验表明，该方法在精度和推理效率上优于代表性基线。此外，所提出的积分约束使模型在分布外初始条件和不同空间分辨率下表现出优异泛化性能。
原文摘要 · Abstract (English)
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.
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