用缺陷校正提升高维偏微分方程求解精度，无需重训练。
Physics-Informed Inference Time Scaling for Solving High-Dimensional PDE via Defect Correction
- 引入缺陷校正机制，构建结构保持的误差模型。
- 160维问题上将各类代理模型误差降低20%至80%。
- 适合需高可靠性求解的科学计算场景，如物理模拟。
求解高维偏微分方程是关键挑战，现代数据驱动求解器常缺乏可靠性与严格误差保证。本文提出仿真校准科学机器学习（SCaSML）框架，在推理阶段无需重训练即可系统提升预训练的PDE求解器性能。核心思想是采用缺陷校正方法，推导出一个新的偏微分方程——缺陷保持律，精确描述给定代理模型的误差。由于该方程保留了原问题的结构，可高效使用传统随机模拟器求解，并修正初始机器学习解。理论证明SCaSML具有更快收敛率，最终误差由代理模型与仿真误差的乘积决定。在高达160维的复杂PDE问题上，该方法使PINNs、高斯过程等各类代理模型的误差降低20%至80%。代码已开源：https://github.com/Francis-Fan-create/SCaSML。
原文摘要 · Abstract (English)
Solving high-dimensional partial differential equations (PDEs) is a critical challenge where modern data-driven solvers often lack reliability and rigorous error guarantees. We introduce Simulation-Calibrated Scientific Machine Learning (SCaSML), a framework that systematically improves pre-trained PDE solvers at inference time without any retraining. Our core idea is to use defect correction method that derive a new PDE, termed Structural-preserving Law of Defect, that precisely describes the error of a given surrogate model. Since it retains the structure of the original problem, we can solve it efficiently with traditional stochastic simulators and correct the initial machine-learned solution. We prove that SCaSML achieves a faster convergence rate, with a final error bounded by the product of the surrogate and simulation errors. On challenging PDEs up to 160 dimensions, SCaSML reduces the error of various surrogate models, including PINNs and Gaussian Processes, by 20-80%. Code of SCaSML is available at https://github.com/Francis-Fan-create/SCaSML.
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