AI and the World's Hard Problems · Theoretical computer science · arXiv · 2026-10-22
Minimax Rates for imperfect Physics-Informed Regression
Physics-informed machine learning incorporates physical knowledge, often a PDE that the target is assumed to satisfy, but how much this improves statistical performance when the physical model is imperfect has been unclear. The study shows there is a characteristic threshold for the modeling error above which the physical prior cannot improve the convergence rate, while below it the gain depends on the equation type. For classes of elliptic and hyperbolic PDEs with constant coefficients, the established convergence rates improve continuously as the modeling error decreases and are minimax optimal up to logarithmic factors; with exact physics, the rates correspond to learning in dimension d-1 rather than d.