Physics-Informed Inference Time Scaling for Solving High-Dimensional PDE via Defect Correction

Abstract

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.