pde

This paper develops a PAC-Bayesian framework for physics-informed machine learning, providing high-probability generalization guarantees for unbounded losses. It proposes a multi-task perspective that jointly handles data fidelity, PDE residuals, and boundary conditions, and introduces a self-bounding learning algorithm.

This paper identifies neural network training as a search through Hamilton-Jacobi initial-value problems, showing that residual networks, transformers, and RNNs discretize the same class of viscous Hamilton-Jacobi equations. It derives quantitative consequences including minimax optimal generalization rates, adversarial robustness bounds, and a closed-form influence function.