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Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains
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Proposes Poisson-Dirac Neural Networks (PoDiNNs), which use a Dirac structure to unify port-Hamiltonian and Poisson formulations so a single model can learn coupled multi-domain ODE systems with interpretable bivector couplings.
Problem
Data-driven modeling of dynamical systems is largely limited to mechanical systems and treats systems as monolithic, failing to capture couplings across electrical, rotational, hydraulic, and mechanical domains (multiphysics) and degenerate, dissipative, or externally forced dynamics.
Approach
Represent a system's components as ports in a Dirac structure whose bivector explicitly encodes coupling patterns between components, with neural networks modeling only component-wise energy functions and resistive characteristics; this separates coupling (bivector) from component behavior and handles degeneracy, dissipation, and external inputs in one framework.
Key results
- PoDiNNs achieved the best MSE and VPT on all seven datasets versus Neural ODEs, Dissipative SymODENs, and PNNs, with VPT reported as 0.856 on relative-coordinate system (b) versus 0.153 for Neural ODEs.
- On Chua's circuit (e), learned bivector coupling reproduced Kirchhoff's current and voltage laws and the true circuit topology across all 10 trials.
- On the hydraulic tank system (g), PoDiNNs recovered cross-sectional area coefficients 1.0 and -0.3 accurate to five significant figures in the bivector.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Neural ODEs with PoDiNNs | Mass-spring-damper (a), relative coordinates with external force · Mass-spring-damper (a) relative, external force | VPT: 0.128 -> 0.622 | Helped |
| Replace Neural ODEs with PoDiNNs | Mass-spring-damper (a'), absolute coordinates with external force · Mass-spring-damper (a') absolute, external force | VPT: 0.097 -> 0.437 | Helped |
| Replace Neural ODEs with PoDiNNs | Mass-spring-damper (b), relative coordinates with moving boundary · Mass-spring-damper (b) relative, moving boundary | VPT: 0.153 -> 0.856 | Helped |
| Replace Neural ODEs with PoDiNNs | Mass-spring-damper (b'), absolute coordinates with moving boundary · Mass-spring-damper (b') absolute, moving boundary | VPT: 0.135 -> 0.581 | Helped |
| Replace Neural ODEs with PoDiNNs | Mass-spring-damper (c) with redundant observations · Mass-spring-damper (c) relative, redundant observations | VPT: 0.099 -> 0.863 | Helped |
| Replace Neural ODEs with PoDiNNs | FitzHugh-Nagumo circuit · FitzHugh-Nagumo | VPT: 0.322 -> 0.649 | Helped |
| Replace Neural ODEs with PoDiNNs | Chua's circuit · Chua's circuit | VPT: 0.287 -> 0.469 | Helped |
| Replace Neural ODEs with PoDiNNs | DC motor multiphysics system · DC motor | VPT: 0.276 -> 0.923 | Helped |
| Replace Neural ODEs with PoDiNNs | Hydraulic tank multiphysics system · Hydraulic tank | VPT: 0.045 -> 0.918 | Helped |
| Assume fewer than the correct 3 dampers (ablation) | PoDiNNs on system (b), relative coordinates · Mass-spring-damper (b) relative, moving boundary | performance | Hurt |
| Assume more than the correct 3 dampers (ablation) | PoDiNNs on system (b), relative coordinates · Mass-spring-damper (b) relative, moving boundary | performance | No clear effect |
Surprising findings
- Neural ODEs diverged on the redundant mass-spring system (c), with MSE around 2490 in relative coordinates.
- Redundant dampers were typically ignored by learning zero damping or zero coupling, and this pattern was visible in training-set performance.
Limitations
- Cannot represent DAE systems such as ideal capacitors in parallel with voltage sources, and PDE systems are left to future work.
- Requires assuming the number and type of energy-dissipating components and the flow type of resistive elements, which are usually unknown.
- Coupling patterns in complex circuits may be non-unique due to Norton and Thevenin equivalences; the learned bivector need not satisfy the Jacobi identity.
Experimental setup
Datasets: seven simulated dynamical systems: mass-spring-damper systems (a), (b), (c), FitzHugh-Nagumo circuit (d), Chua's circuit
Benchmarks: seven simulated multi-domain ODE datasets (a)-(g)
Metrics: MSE valid prediction time (VPT)
Baselines: Neural ODEs Dissipative SymODENs PNNs HNNs
Builds on: Hamiltonian Neural Networks Poisson neural networks Dissipative SymODENs Neural ODEs Constrained HNNs
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