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Efficiently Parameterized Neural Metriplectic Systems

Anthony Gruber, Kookjin Lee, Haksoo Lim, Noseong Park, Nathaniel Trask

TL;DR

This work develops Neural Metriplectic Systems (NMS) to learn metriplectic (GENERIC) dynamics from data while strictly preserving energy and entropy properties. By leveraging exterior algebra, NMS achieves a compact, nonredundant parameterization of the reversible and irreversible brackets that scales quadratically with the state dimension and the rank of dissipation. The authors prove universal approximation and time-generalization error bounds and demonstrate superior accuracy and scalability over previous methods, even when entropic variables are unobserved. Practical experiments on two gas containers and a thermoelastic double pendulum show NMS outperforms baselines and preserves thermodynamic laws, highlighting its potential for data-driven, physics-consistent modeling and future model-reduction applications.

Abstract

Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic data. Besides being provably energy conserving and entropy stable, the proposed approach comes with approximation results demonstrating its ability to accurately learn metriplectic dynamics from data as well as an error estimate indicating its potential for generalization to unseen timescales when approximation error is low. Examples are provided which illustrate performance in the presence of both full state information as well as when entropic variables are unknown, confirming that the proposed approach exhibits superior accuracy and scalability without compromising on model expressivity.

Efficiently Parameterized Neural Metriplectic Systems

TL;DR

This work develops Neural Metriplectic Systems (NMS) to learn metriplectic (GENERIC) dynamics from data while strictly preserving energy and entropy properties. By leveraging exterior algebra, NMS achieves a compact, nonredundant parameterization of the reversible and irreversible brackets that scales quadratically with the state dimension and the rank of dissipation. The authors prove universal approximation and time-generalization error bounds and demonstrate superior accuracy and scalability over previous methods, even when entropic variables are unobserved. Practical experiments on two gas containers and a thermoelastic double pendulum show NMS outperforms baselines and preserves thermodynamic laws, highlighting its potential for data-driven, physics-consistent modeling and future model-reduction applications.

Abstract

Metriplectic systems are learned from data in a way that scales quadratically in both the size of the state and the rank of the metriplectic data. Besides being provably energy conserving and entropy stable, the proposed approach comes with approximation results demonstrating its ability to accurately learn metriplectic dynamics from data as well as an error estimate indicating its potential for generalization to unseen timescales when approximation error is low. Examples are provided which illustrate performance in the presence of both full state information as well as when entropic variables are unknown, confirming that the proposed approach exhibits superior accuracy and scalability without compromising on model expressivity.
Paper Structure (17 sections, 4 theorems, 11 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 4 theorems, 11 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Previous and Related Work
  4. Soft constrained methods.
  5. Hard constrained methods.
  6. Model reduction.
  7. Formulation and Analysis
  8. Exterior algebra
  9. Learnable metriplectic operators
  10. Specific parameterizations
  11. Approximation and error
  12. Algorithm
  13. Examples
  14. Two gas containers
  15. Thermoelastic double pendulum
  16. ...and 2 more sections

Key Result

Lemma 3.2

Let $K\subset\mathbb{R}^n$. For all $~x\in K$, the operator $~L:K\to\mathbb{R}^{n\times n}$ satisfies $~L^\intercal = -~L$ and $~L\nabla S = ~0$ for some $S:K\to\mathbb{R}$, $\nabla S \neq ~0$, provided there exists a non-unique bivector field $\mathsf{A}:U\to{\bigwedge}^2\mathbb{R}^n$ and equivalen Similarly, for all $~x\in K$ a positive semi-definite operator $~M:K\to\mathbb{R}^{n\times n}$ sati

Figures (2)

  • Figure 1: A visual depiction of the NMS architecture.
  • Figure 2: The ground-truth and predicted position, momentum, instantaneous entropy, and energies for the two gas containers example in the training (white), validation (yellow), and testing (red) regimes.

Theorems & Definitions (12)

  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.7
  • Remark 3.8
  • Theorem 3.9
  • Remark 3.10
  • ...and 2 more