发表机构
Ocean University of China; University of Glasgow（中国海洋大学; 格拉斯哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对自主PDE长时间外推的误差问题，提出误差界感知与先验引导的神经残差框架，经多基准案例验证，其外推误差低于数值先验及十种对比方法，可提升无真实轨迹监督的PDE长时间模拟精度。
AI 中文摘要
精确模拟由偏微分方程（PDE）描述的系统的长时间演化是科学计算的核心任务。在现有基于深度学习的PDE求解方法中，神经算子通常依赖大量轨迹数据，而物理知情方法在长时间外推过程中往往稳定性有限。对于适定的自主PDE，可通过重复组合固定步长的演化算子生成长时间轨迹，因此长时间外推取决于对该算子的近似误差及该误差在递归组合下的传播的控制。据此，我们提出一种无需真实轨迹监督训练的数值先验引导、物理约束的方法：低成本数值先验降低了近似单步演化算子的难度，而弱形式PDE残差为误差传播界中的单步误差项提供了可计算代理。我们在涵盖四类PDE的五个基准案例上验证了该方法，并在排除训练和模型选择中使用真实轨迹的统一协议下，与十种物理知情学习方法进行比较。结果表明，在所有五个案例中，所提方法相较于数值先验降低了长时间外推误差，且在每个案例中均优于最优的竞争基线，从而在无需真实轨迹监督的情况下提升了不同PDE的长时间模拟精度。本文开发的源代码将在稿件录用后公开。
英文摘要
Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among existing deep learning?based approaches for solving PDEs, neural operators typically rely on extensive trajectory data, whereas physics-informed meth?ods often exhibit limited stability during long-time extrapolation. For a well-posed autonomous PDE, long-time trajectories can be generated by repeated composition of a fixed-step evolution operator; hence, long-time extrapolation depends on controlling the approximation error of this operator and the propagation of that error under recursive composition. Accordingly, we propose a numerical-prior-guided, physics-constrained method trained without ground-truth trajectory supervision: a low-cost numerical prior reduces the difficulty of approximating the one?step evolution operator, while a weak-form PDE residual provides a computable proxy for the one-step error term in the error?propagation bound. We validate the method on five benchmark cases spanning four PDE classes and compare it with ten physics?informed learning methods under a unified protocol that excludes ground-truth trajectories from training and model selection. The results indicate that, in all five cases, the proposed method reduces long-time extrapolation error relative to the numerical prior and outperforms the best competing baseline in each case, thereby improving long-time simulation accuracy across different PDEs without ground-truth trajectory supervision. The source code developed for this paper will be made publicly available upon acceptance of the manuscript.
