Bengining overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Iductive Bias

Recent advances in machine learning theory showed that interpolation to noisy samples using over-parameterized machine learning algorithms always leads to inconsistency. However, this work surprisingly discovers that interpolated machine learning can exhibit benign overfitting and consistency when using physics-informed learning for supervised tasks governed by partial differential equations (PDEs) describing laws of physics. An analysis provides an asymptotic Sobolev norm learning curve for kernel ridge(less) regression addressing linear inverse problems involving elliptic PDEs. The results reveal that the PDE operators can stabilize variance and lead to benign overfitting for fixed-dimensional problems, contrasting standard regression settings. The impact of various inductive biases introduced by minimizing different Sobolev norms as implicit regularization is also examined. Notably, the convergence rate is independent of the specific (smooth) inductive bias for both ridge and ridgeless regression. For regularized least squares estimators, all (smooth enough) inductive biases can achieve optimal convergence rates when the regularization parameter is properly chosen. The smoothness requirement recovers a condition previously found in the Bayesian setting and extends conclusions to minimum norm interpolation estimators.