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Stable Port-Hamiltonian Neural Networks
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Proposes stable port-Hamiltonian neural networks (sPHNN), which parameterize a convex, normalized Hamiltonian (FICNN) plus neural structure, dissipation and input matrices to guarantee global asymptotic stability of learned nonlinear dynamics without projection.
Problem
Data-driven identification of nonlinear dynamical systems often yields unstable or physically implausible forecasts that extrapolate poorly, especially from sparse data.
Approach
Constrain the port-Hamiltonian Hamiltonian to be a convex, positive definite, radially unbounded function (FICNN normalized to a strict minimum at a known or learned equilibrium), so the Hamiltonian serves as a global Lyapunov function; dissipation matrix R is Cholesky-parameterized, giving global asymptotic stability without projection.
Key results
- On the thermal food processing surrogate with 3 augmented dimensions, sPHNN predictions remain stable near thermal equilibrium while NODE and PHNN become unstable; under 25% noise sPHNN test RMSE is 3.177 vs 5.231 for bPHNN and 24.224 for PHNN.
- On the additive manufacturing surrogate, sPHNN total test RMSE peaks below 30 K, with errors nearly an order of magnitude lower than NODE and PHNN which increase rapidly due to instabilities.
- On cascaded tanks, sPHNN, sPHNN-LM and bPHNN achieve comparable accuracy while NODE and PHNN extrapolate poorly; sPHNN-LM identified equilibrium between -0.196 and 0.432 V, and the sPHNN-LM equilibrium error on the spinning body was 0.010 IQM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the convex, stability-constrained sPHNN instead of bPHNN | Thermal food processing surrogate · Noise-free test set, 25% training noise | RMSE: 5.231 -> 3.177 | Helped |
| Use the convex, stability-constrained sPHNN instead of NODE | Thermal food processing surrogate · Noise-free test set, 25% training noise | RMSE: 4.362 -> 3.177 | Helped |
| Use the convex, stability-constrained sPHNN instead of PHNN | Thermal food processing surrogate · Noise-free test set, 5% training noise | RMSE: 28.233 -> 1.170 | Helped |
| Learn the Hamiltonian minimum instead of fixing it (ablation) | sPHNN on the thermal food processing surrogate · Noise-free test set, 5% training noise | RMSE: 1.170 -> 2.420 | Hurt |
| Learn the Hamiltonian minimum instead of fixing it (ablation) | sPHNN on the thermal food processing surrogate · Noise-free test set, 25% training noise | RMSE: 3.177 -> 3.677 | Hurt |
| Add augmented state dimensions to sPHNN (ablation) | sPHNN on the thermal food processing surrogate · Thermal food processing surrogate | RMSE | Helped |
| Add augmented state dimensions to NODE or PHNN | NODE and PHNN on the thermal food processing surrogate · Thermal food processing surrogate | RMSE | Hurt |
| Use sPHNN-LM instead of sPHNN when training data are scarce (ablation) | sPHNN on the thermal food processing surrogate · Scarce-data regime | RMSE | Hurt |
| Use sPHNN instead of bPHNN with three augmented dimensions | Thermal food processing surrogate · Different training dataset sizes | RMSE | Helped |
| Use sPHNN instead of bPHNN on additive-manufacturing field predictions | Additive manufacturing surrogate · Test trajectories | RMSE | Helped |
| Use a convex, globally stable Hamiltonian instead of an unconstrained Hamiltonian | sPHNN versus PHNN on the additive manufacturing surrogate · Test trajectories | RMSE | Helped |
Surprising findings
- Trajectory fitting of NODE and sNODE failed to converge, attributed to discontinuities from projection-based stability enforcement.
- Adding augmented dimensions degraded NODE and PHNN performance due to instabilities, whereas sPHNN benefited from them.
Limitations
- Global stability limits sPHNN to systems that appear globally stable in the region of interest; multistable systems are not covered.
- Convexity is sufficient but not necessary, and input warping was not needed in experiments, so expressiveness is constrained.
- Equilibrium-learning variant (sPHNN-LM) is less accurate in scarce-data regimes.
Experimental setup
Datasets: spinning rigid body simulation (Euler equations) cascaded tanks dataset conjugate heat transfer chicken breast simulation (Le et al.) additive manufacturing FE heat conduction simulation
Benchmarks: spinning rigid body energy prediction cascaded tanks RMSE and extrapolation thermal food processing surrogate RMSE additive manufacturing POD-latent surrogate RMSE
Metrics: RMSE interquartile mean energy median RMSE under noise
Baselines: NODE PHNN bPHNN sNODE sPHNN-LM
Builds on: port-Hamiltonian systems neural ODE input convex neural networks OnsagerNet stable neural ODE (projection-based)
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