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HHD-GP: Incorporating Helmholtz-Hodge Decomposition into Gaussian Processes for Learning Dynamical Systems
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Builds a Gaussian process model that learns a dynamical system's vector field as a sum of curl-free and div-free GP components via Helmholtz-Hodge decomposition, and adds symmetry-preserving kernels (SPHHD-GP) to make the decomposition identifiable.
Problem
Learning dynamical systems from sparse, noisy observations with physics-agnostic models that lack interpretability and data efficiency; div-free-only models cannot represent dissipative dynamics, and HHD decompositions are non-identifiable.
Approach
Place separate GP priors on curl-free (gradient of scalar potential) and div-free (curl of vector potential) components, sum them into an additive HHD-form GP, and impose Euclidean symmetry (via Haar-integration and GIM kernels) on each component to remove the harmonic non-identifiability.
Key results
- On the damped mass-spring system with 20 training points, SPHHD-GP reaches RMSE 2.02e-2 and VPT 8.52 versus 4.83e-2 and 2.92 for HHD-GP and 2.80e-2 and 5.28 for GIM-GP.
- On the Chua circuit, SPHHD-GP achieves derivative RMSE 4.1 and MNLL -2.72, outperforming GIM-GP (10.2, -1.92) and HHD-GP (12.0, -1.76).
- On the ocean current field, SPHHD-GP yields a continuous current prediction and recovers a non-constant divergence field, whereas Helmholtz-GP predicts an almost constant curl-free field.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Mass Spring | RMSE (×10−2): 4.83 ± 1.60 -> 2.02 ± 1.75 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Mass Spring | VPT: 2.92 ± 0.90 -> 8.52 ± 5.11 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Mass Spring | MNLL: -2.20 ± 0.29 -> -3.16 ± 0.85 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Pendulum | RMSE (×10−2): 34.64 ± 15.68 -> 13.34 ± 6.89 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Pendulum | VPT: 1.83 ± 0.42 -> 3.18 ± 1.37 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Damped Pendulum | MNLL: -0.79 ± 0.26 -> -1.56 ± 0.21 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Chua circuit | RMSEder: 12.0 ± 11.1 -> 4.1 ± 1.8 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Chua circuit | VPT: 5.1 ± 1.4 -> 13.0 ± 5.6 | Helped |
| add symmetry-preserving curl-free and divergence-free GP priors (ablation) | HHD-GP · Chua circuit | RMSEen: 7.4 ± 3.6 -> 0.7 ± 1.5 | Helped |
| add a curl-free GP component to a divergence-free GP | HHD-GP · Damped Mass Spring | RMSE (×10−2): 21.48 ± 38.11 -> 4.83 ± 1.60 | Helped |
Surprising findings
- Unconstrained HHD-GP and D-HNN fail to recover physically plausible energy trajectories and their energy error does not decrease with more data, indicating non-identifiability.
- Plain HHD-GP performs similarly to the independent-kernel Ind-GP baseline in most cases.
Limitations
- Experiments are on low-dimensional systems (2D and 3D), with symmetry groups chosen by hand.
- Ocean current results are assessed qualitatively against expert knowledge since no ground truth is available.
- Symmetry-preserving kernels require the system to have known symmetries, and the Haar integral can be computationally expensive.
Experimental setup
Datasets: simulated damped mass-spring data simulated damped pendulum data Chua circuit simulated data ocean current buoy velocity dataset from Helmholtz-GP work
Benchmarks: damped mass-spring damped pendulum Chua circuit ocean current field
Metrics: RMSE valid prediction time (VPT) mean negative log likelihood (MNLL) energy RMSE
Baselines: D-HNN Div-GP Ind-GP GIM-GP Helmholtz-GP
Builds on: Helmholtz-Hodge decomposition Gaussian processes with linear operators Group Integration Matrix kernel Haar-integration kernel Div-free GP for Hamiltonian dynamics
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