Hamiltonian Neural PDE Solvers through Functional Approximation

TL;DR

This paper introduces Hamiltonian Neural PDE Solvers that leverage functional approximation to respect conservation laws, improve stability, and generalize better in solving PDEs with neural networks.

Contribution

It extends Hamiltonian neural networks to PDEs by representing Hamiltonian functionals with neural kernels, enabling energy conservation and better generalization.

Findings

HNS effectively conserves energy-like quantities.

HNS shows improved stability over traditional neural PDE solvers.

HNS generalizes well to longer time horizons and unseen initial conditions.

Abstract

Designing neural networks within a Hamiltonian framework offers a principled way to ensure that conservation laws are respected in physical systems. While promising, these capabilities have been largely limited to discrete, analytically solvable systems. In contrast, many physical phenomena are governed by PDEs, which govern infinite-dimensional fields through Hamiltonian functionals and their functional derivatives. Building on prior work, we represent the Hamiltonian functional as a kernel integral parameterized by a neural field, enabling learnable function-to-scalar mappings and the use of automatic differentiation to calculate functional derivatives. This allows for an extension of Hamiltonian mechanics to neural PDE solvers by predicting a functional and learning in the gradient domain. We show that the resulting Hamiltonian Neural Solver (HNS) can be an effective surrogate model through improved stability and conserving energy-like quantities across 1D and 2D PDEs. This ability to respect conservation laws also allows HNS models to better generalize to longer time horizons or unseen initial conditions.