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Hamiltonian Neural PDE Solvers through Functional Approximation
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Proposes Hamiltonian Neural Solvers (HNS), which parameterize a Hamiltonian functional as a learnable integral kernel functional (neural field kernel) and predict functional derivatives via autograd to evolve PDEs with energy conservation.
Problem
Neural PDE surrogates lack conservation structure; Hamiltonian neural networks have been limited to discrete particle systems, and learning infinite-dimensional Hamiltonian functionals (function-to-scalar maps with well-defined functional derivatives) is not addressed.
Approach
Represent the Hamiltonian functional as a kernel integral with a learnable, conditioned neural field (FiLM-conditioned SIREN) so that autograd yields functional derivatives; train in the gradient domain and evolve the PDE via J (finite differences) and an Adams-Bashforth integrator.
Key results
- On 1D KdV, HNS achieves correlation time 151.75 vs 125.25 for the best Unet baseline (Base) and is stable beyond training horizon.
- On 2D SWE Pulse (OOD), HNS Hamiltonian relative L2 error is 0.0003 versus 0.0158 for Unet and 0.1326 for FNO.
- HNS uses roughly half the parameters of baselines (e.g., 32K vs 65K for Adv), but inference is slower than FNO and Unet in 1D and 2D.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an integral-kernel functional instead of the strongest baseline | Functional approximation models · Linear and nonlinear toy functionals, Base/OOD/Disc. | functional and functional derivative MSE: 2.47e-5 -> 3.00e-16; 0.046 -> 2.13e-7; 49.3 -> 0.737; 0.066 -> 1.15e-3; 0.122 -> 1.15e-3; 2.70 -> 0.077; 0.016 -> 2.05e- | Helped |
| Add nonlinear kernel to the base HNS architecture (ablation) | HNS · Advection | rollout error: 1.248 -> 0.028 | Helped |
| Add FiLM conditioning after the nonlinear kernel (ablation) | HNS · Advection | rollout error: 0.028 -> 0.016 | Helped |
| Add SIREN kernel architecture after FiLM conditioning (ablation) | HNS · Advection | rollout error: 0.016 -> 0.003 | Helped |
| Add global conditioning after SIREN (ablation) | HNS · Advection | rollout error: 0.003 -> 0.057 | Hurt |
| Add nonlinear kernel to the base HNS architecture (ablation) | HNS · KdV | correlation time: 51.75 -> 73.00 | Helped |
| Add FiLM conditioning after the nonlinear kernel (ablation) | HNS · KdV | correlation time: 73.00 -> 65.25 | Hurt |
| Add SIREN kernel architecture after FiLM conditioning (ablation) | HNS · KdV | correlation time: 65.25 -> 73.25 | Helped |
| Add global conditioning after SIREN (ablation) | HNS · KdV | correlation time: 73.25 -> 151.75 | Helped |
| Add nonlinear kernel to the base HNS architecture (ablation) | HNS · SWE-Sines | rollout error: 0.113 -> 0.034 | Helped |
| Add FiLM conditioning after the nonlinear kernel (ablation) | HNS · SWE-Sines | rollout error: 0.034 -> 0.030 | Helped |
| Add SIREN kernel architecture after FiLM conditioning (ablation) | HNS · SWE-Sines | rollout error: 0.030 -> 0.026 | Helped |
Surprising findings
- Adding a loss on the Hamiltonian error harms HNS performance, hypothesized to be because learning the identity mapping is easiest.
- Gradient-domain learning with conventional Unet architectures disproportionately harms energy conservation, implying functional-derivative meaningfulness of the architecture matters.
Limitations
- Theory for nonlinear functionals is underdeveloped; only linear functional approximation is proven.
- Inference overhead from autograd, operator J evaluation, and numerical quadrature scales poorly with grid size; slower than FNO/Unet.
- Only applicable to conservative Hamiltonian systems; dissipative systems excluded.
Experimental setup
Datasets: 1D Advection simulations 1D KdV simulations 2D shallow water equations simulations (PyClaw, sinusoidal and Gaussian pulse ICs)
Benchmarks: 1D Advection 1D KdV 2D Shallow Water Equations (Sines and Pulse) toy polynomial functionals
Metrics: rollout error correlation time relative L2 Hamiltonian error inference time validation MSE for functionals and derivatives
Baselines: FNO Unet PINO Transolver MLP
Builds on: Hamiltonian Neural Networks neural operators (Fourier Neural Operator) FiLM SIREN
Closest papers by meaning
- Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guarantees (NeurIPS 2024 · similarity 0.89)
- Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks (ICML 2025 · similarity 0.87)
- Stable Port-Hamiltonian Neural Networks (NeurIPS 2025 · similarity 0.86)
- Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling (NeurIPS 2024 · similarity 0.86)
- Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws (ICML 2024 · similarity 0.86)
- Learned Lagrangian Models of PDEs via Euler–Lagrange Residual Minimization (NeurIPS 2026 · similarity 0.86)
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