Minimax Rates for imperfect Physics-Informed Regression
Abstract
Physics-informed machine learning incorporates physical knowledge into data-driven models, often through a partial differential equation (PDE) that the target is assumed to satisfy. However, how much this knowledge improves statistical performance remains unclear, especially when the physical model is imperfect. Since satisfying a PDE implies being differentiable, a natural benchmark for physics-informed regression is the best achievable performance when smoothness is the only prior information about the target function. We show that there is a characteristic threshold for the modeling error above which the physical prior cannot improve the convergence rate. The form of this threshold is common to every linear PDE and certain nonlinear PDEs. Below it, the gain depends on the equation type. For classes of elliptic and hyperbolic PDEs with constant coefficients, we establish convergence rates that improve continuously as the modeling error decreases, and prove their minimax optimality up to logarithmic factors. With exact physics, the rates correspond to learning in dimension d-1 rather than d, reflecting the freedom left by unknown boundary or initial data. We also give conditions for computing the estimator with near-linear complexity and show how to tune its hyperparameters without knowing the modeling error.
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BibTeX
@misc{doumeche2026minimax,
title = {{Minimax Rates for imperfect Physics-Informed Regression}},
author = {Nathan Doumèche},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29532},
eprint = {2610.29532},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05820420v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimax Rates for imperfect Physics-Informed Regression
AU - Nathan Doumèche
PY - 2026
DA - 2026/10/19
PB - HAL
DO - 10.48550/arxiv.2610.29532
UR - https://hal.science/hal-05820420v1
AB - Physics-informed machine learning incorporates physical knowledge into data-driven models, often through a partial differential equation (PDE) that the target is assumed to satisfy. However, how much this knowledge improves statistical performance remains unclear, especially when the physical model is imperfect. Since satisfying a PDE implies being differentiable, a natural benchmark for physics-informed regression is the best achievable performance when smoothness is the only prior information about the target function. We show that there is a characteristic threshold for the modeling error above which the physical prior cannot improve the convergence rate. The form of this threshold is common to every linear PDE and certain nonlinear PDEs. Below it, the gain depends on the equation type. For classes of elliptic and hyperbolic PDEs with constant coefficients, we establish convergence rates that improve continuously as the modeling error decreases, and prove their minimax optimality up to logarithmic factors. With exact physics, the rates correspond to learning in dimension d-1 rather than d, reflecting the freedom left by unknown boundary or initial data. We also give conditions for computing the estimator with near-linear complexity and show how to tune its hyperparameters without knowing the modeling error.
ER -