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. A central challenge is how to integrate data with PDE constraints in a principled way to improve generalization. In this paper, we characterize the statistical estimation and generalization properties of learning solutions to the heat equation with a noisy source function from finite, randomly sampled data. We first derive a novel variational formulation, which we characterize as the minimizer of loss function for an energy regularized and ellipticity debiased Sobolev regression. By formulating the learning task as a supervised problem over spa...
| Authors | Zixiang Zhou, Xiaohui Chen, Shanghua Teng |
|---|---|
| Primary category | math.ST |
| Published | 10 October 2026 |
| Last updated at source | 10 October 2026 |
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