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.
翻译：求解高维偏微分方程是一个关键挑战，现代数据驱动的求解器往往缺乏可靠性及严格的误差保证。我们提出了仿真校准的科学机器学习框架，该框架能够在无需重新训练的情况下，在推理阶段系统性地改进预训练的PDE求解器。我们的核心思想是采用缺陷校正方法，推导出一个称为“结构保持的缺陷定律”的新PDE，该方程精确描述了给定代理模型的误差。由于该方程保留了原始问题的结构，我们可以利用传统的随机模拟器高效求解，从而修正初始的机器学习解。我们证明了SCaSML框架能够实现更快的收敛速度，其最终误差受限于代理模型误差与仿真误差的乘积。在维度高达160的挑战性PDE问题上，SCaSML将包括物理信息神经网络和高斯过程在内的多种代理模型的误差降低了20-80%。SCaSML的代码已发布于https://github.com/Francis-Fan-create/SCaSML。
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