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 https://github.com/Francis-Fan-create/SCaSML.

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