[2505.13275] Hamiltonian Neural PDE Solvers through Functional Approximation

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.

View PDF HTML (experimental) 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…

saved by

• Gilbert Chang

related reading

• proceedings.mlr.press/v202/ma23a/ma23a.pdfproceedings.mlr.press
• Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningarxiv.org
• Learning Neural Constitutive Laws from Motion Observations for Generalizable PDE Dynamicsproceedings.mlr.press
• [1907.03452] Deep splitting method for parabolic PDEsarxiv.org
• Beyond Message Passing: a Physics-Inspired Paradigm for Graph Neural Networksthegradient.pub
• [2604.21691] There Will Be a Scientific Theory of Deep Learningarxiv.org
• [2404.07200] Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivearxiv.org
• [2205.09653] Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networksarxiv.org
• Neural Manifold Ordinary Differential Equationsarxiv.org
• Physics-constrained machine learning for scientific computing - Amazon Scienceamazon.science
• Zongyi Li | Fourier Neural Operatorzongyi-li.github.io
• Statistical Mechanics of Deep Learningganguli-gang.stanford.edu