Reproduce “Hamiltonian Neural PDE Solvers through Functional Approximation” on OpenResearch

About this paper

Anthony Zhou, Amir Barati Farimani

Hamiltonian Neural Solver (HNS) represents a PDE’s Hamiltonian as a learnable scalar functional, differentiates it with respect to the input field, and uses the resulting variational derivative to advance the dynamics. Across advection, KdV, and shallow-water systems, the approach improves long-horizon stability and out-of-distribution behavior, although it is slower than conventional neural solvers and is intended primarily for Hamiltonian or conservative dynamics.

Key insights

• Accurate scalar functional regression does not guarantee accurate derivatives: on the linear test, HNS reaches roughly 1.15×10⁻³ derivative MSE, versus 0.083 for an MLP and 0.066 for an FNO.
• Encoding the Hamiltonian structure improves temporal extrapolation and preserves energy-like quantities more effectively than directly predicting states; global field conditioning is particularly important for nonlocal or derivative-dependent Hamiltonians such as KdV.
• The conservation-based inductive bias is most valuable under distribution shift: HNS is not always best in-distribution, but it substantially outperforms competing operators on unseen shallow-water Gaussian-pulse initial conditions.

Summary by alphaXiv.