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Efficiently Parameterized Neural Metriplectic Systems
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Proposes neural metriplectic systems (NMS), a hard-constrained parameterization of metriplectic (GENERIC) dynamics using exterior-algebra projections that scales quadratically in state dimension and rank, with approximation and error-bound results.
Problem
Learning metriplectic (GENERIC) dynamics from data that exactly conserve energy and produce entropy, with efficient parameterization that scales well with state dimension and rank of the irreversible operator, including when entropic variables are unobserved.
Approach
Parameterize the reversible and irreversible brackets L and M via orthogonal projections onto the complements of grad E and grad S (L = P_S A P_S with A skew, M = P_E B D B^T P_E with D PSD), which enforces symmetry and degeneracy by construction using only quadratically many learnable functions, and prove universal approximation and a time-error bound.
Key results
- On the two gas containers example, NMS achieves MSE .01 ± .02 and MAE .08 ± .06 versus GFINN at .07 and .13 and NODE at .12 and .25.
- On the thermoelastic double pendulum, NMS achieves MSE .38 ± .03 and MAE .42 ± .07, the best among compared methods (GFINN .40 / .43).
- NMS parameter count scales as (1/2)((n+r)^2-(n-r))+2 learnable scalar functions versus cubic for GNODE and GFINN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NMS instead of GFINN | Metriplectic learning on two gas containers, with unobserved entropy variables · Two gas containers test interval [0,t_test] | MSE: .07 $\pm$ .03 -> .01 $\pm$ .02 | Helped |
| Use NMS instead of GFINN | Metriplectic learning on two gas containers, with unobserved entropy variables · Two gas containers test interval [0,t_test] | MAE: .13 $\pm$.03 -> .08 $\pm$ .06 | Helped |
| Use NMS instead of GFINN | Metriplectic learning on the thermoelastic double pendulum · Thermoelastic double pendulum test set | MSE: .40 $\pm$ .03 -> .38 $\pm$ .03 | Helped |
| Use NMS instead of GFINN | Metriplectic learning on the thermoelastic double pendulum · Thermoelastic double pendulum test set | MAE: .43 $\pm$ .07 -> .42 $\pm$ .07 | Helped |
Surprising findings
- All baselines struggle to approximate the thermoelastic double pendulum dynamics over the entire training interval, with differences among methods small.
Limitations
- Scaling to realistic 3-D problems remains difficult, per the authors.
- Jacobi identity not enforced, so L may not generate a true Poisson structure; strong-form energy conservation traded against it.
- Requires nondegenerate systems (nonvanishing gradients of E and S).
Experimental setup
Datasets: two gas containers simulated trajectories thermoelastic double pendulum simulated trajectories
Benchmarks: two gas containers thermoelastic double pendulum
Metrics: MSE MAE
Baselines: NODE SPNN GNODE GFINN
Builds on: GNODE GFINN SPNN Neural ODE Adamax
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