A Supervised Learning Scheme for Hamilton-Jacobi Equation via Density Coupling

Learn about a supervised learning approach for solving first-order Hamilton-Jacobi partial differential equations in high dimensions through this 40-minute conference presentation. Discover how geometric structures of Wasserstein Hamiltonian flows are leveraged via a density coupling strategy to create an effective computational scheme. Explore how this method can be reformulated as a regression problem using Bregman divergence as the loss function, with data generated through particle formulations of Wasserstein Hamiltonian flow. Examine the theoretical foundation including posterior estimates on L1 residual based on coupling density support, and review numerical examples demonstrating the approach's effectiveness across different Hamiltonian systems. Gain insights into this collaborative research bridging machine learning techniques with classical PDE theory, presented by Haomin Zhou from Georgia Institute of Technology at IPAM's Scientific Machine Learning Workshop.