Abstract
High-dimensional partial differential equations (PDEs) pose significant computational challenges across fields ranging from quantum chemistry to economics and finance. Although scientific machine learning (SciML) techniques offer approximate solutions, they often suffer from bias and neglect crucial physical insights. Inspired by inference-time scaling strategies in language models, we propose Simulation-Calibrated Scientific Machine Learning (SCaSML), a physics-informed framework that dynamically refines and debiases the SCiML predictions during inference by enforcing the physical laws. SCaSML leverages derived new physical laws that quantifies systematic errors and employs Monte Carlo solvers based on the Feynman-Kac and Elworthy-Bismut-Li formulas to dynamically correct the prediction. Both numerical and theoretical analysis confirms enhanced convergence rates via compute-optimal inference methods. Our numerical experiments demonstrate that SCaSML reduces errors by 20-50% compared to the base surrogate model, establishing it as the first algorithm to refine approximated solutions to high-dimensional PDE during inference. Code of SCaSML is available at this https URL.
Abstract (translated)
高维偏微分方程（PDEs）在从量子化学到经济学和金融学等多个领域中带来了重大的计算挑战。虽然科学机器学习（SciML）技术提供了解决问题的近似方法，但这些方法往往存在偏差，并且忽略了重要的物理见解。受语言模型推理时扩展策略的启发，我们提出了一个基于物理学信息的框架——模拟校准的科学机器学习（SCaSML），该框架在推理过程中动态地通过强制执行物理定律来修正和减少SciML预测中的偏差。SCaSML利用新推导出的物理法则量化系统误差，并采用基于费曼-卡茨公式和伊柳斯比斯穆特李公式的蒙特卡洛求解器，以动态方式校正预测结果。 无论是数值分析还是理论分析都证实了通过计算优化的推理方法可以提高收敛速度。我们的数值实验表明，与基础代理模型相比，SCaSML减少了20-50%的误差，这标志着它成为第一个在推理过程中修正高维PDE近似解的算法。SCaSML的代码可以在[此链接](https://example.com)获取。
URL
https://arxiv.org/abs/2504.16172
