Energy-Guided Learning for Heat Equation with Noisy Source: Optimal Estimation and Generalization Bounds

Data and the governing partial differential equations (PDEs) of continuous-time dynamical systems provide rich structure for statistical learning, yet how to integrate data with PDE constraints in a principled manner remains a central challenge. We characterize the statistical estimation and generalization properties of learning weak and nonsmooth solutions to parabolic PDEs from finite, randomly sampled data. We introduce a variational formulation for a prototypical class of time-dependent heat equations with unknown forcing terms, in which the solution is characterized as the minimizer of an energy-regularized, ellipticity-debiased Sobolev regression objective. Moving beyond standard supervised learning and regression approaches, this formulation yields a learning framework that leverages auxiliary data—samples of the source function—to construct a physics-informed energy regularization. Our analysis reveals that learning in parabolic PDEs exhibits intrinsic anisotropy between spatial and temporal regularity, in sharp contrast to static elliptic problems, leading to fundamentally different statistical behaviors. We establish non-asymptotically rate-optimal estimation and generalization error bounds that explicitly capture the interplay among temporal sampling, spatial resolution, and solution smoothness. Our numerical experiments demonstrate substantial orders of magnitude of reduction in testing errors against various physics-informed neural network methods on various parabolic and hyperbolic PDEs.