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Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains

Razmik Arman Khosrovian, Takaharu Yaguchi, Hiroaki Yoshimura, Takashi Matsubara

TL;DR

This work proposes Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics and demonstrates that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Abstract

Deep learning has achieved great success in modeling dynamical systems, providing data-driven simulators to predict complex phenomena, even without known governing equations. However, existing models have two major limitations: their narrow focus on mechanical systems and their tendency to treat systems as monolithic. These limitations reduce their applicability to dynamical systems in other domains, such as electrical and hydraulic systems, and to coupled systems. To address these limitations, we propose Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics. This framework enables a unified representation of various dynamical systems across multiple domains as well as their interactions and degeneracies arising from couplings. Our experiments demonstrate that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains

TL;DR

This work proposes Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics and demonstrates that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Abstract

Deep learning has achieved great success in modeling dynamical systems, providing data-driven simulators to predict complex phenomena, even without known governing equations. However, existing models have two major limitations: their narrow focus on mechanical systems and their tendency to treat systems as monolithic. These limitations reduce their applicability to dynamical systems in other domains, such as electrical and hydraulic systems, and to coupled systems. To address these limitations, we propose Poisson-Dirac Neural Networks (PoDiNNs), a novel framework based on the Dirac structure that unifies the port-Hamiltonian and Poisson formulations from geometric mechanics. This framework enables a unified representation of various dynamical systems across multiple domains as well as their interactions and degeneracies arising from couplings. Our experiments demonstrate that PoDiNNs offer improved accuracy and interpretability in modeling unknown coupled dynamical systems from data.

Paper Structure

This paper contains 33 sections, 3 theorems, 25 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background Theory and Related Work
  3. Hamiltonian Systems
  4. Degeneracy, Dissipation, and External Inputs
  5. Poisson-Dirac Neural Networks
  6. Dirac Structure
  7. Flows and Efforts for Components
  8. Bivector for Representing Coupled Systems
  9. Discussions for Comparisons and Limitations
  10. Experiments and Results
  11. Experimental Settings
  12. Results
  13. Conclusion
  14. Ethics Statement
  15. Reproducibility Statement
  16. ...and 18 more sections

Key Result

Theorem 1

Let $V$ be a vector space and $\Delta$ a vector subspace. Define the annihilator $\Delta^\circ$ of $\Delta$ as $\Delta^\circ = \{ {\bm{\alpha}}\in V^* \mid \langle{\bm{\alpha}},{\bm{v}}\rangle = 0 \text{ for all } {\bm{v}} \in \Delta \}\subset V^*.$ Then, $D_V = \Delta \oplus \Delta^\circ\subset V\o

Figures (7)

  • Figure 1: Conceptual diagram of PoDiNNs.
  • Figure 2: Diagrams of systems that provide datasets. Detailed definitions are found in Appendix \ref{['appendix:datasets']}.
  • Figure 3: Visualizations of example results. Each top panel shows ground truth trajectories, while the other panels show the absolute errors of all 10 trials in semi-transparent color. See also Fig. \ref{['fig:results2']}.
  • Figure 4: The identified characteristics of $R_2$.
  • Figure 4: Impact of # Components and VPT.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: Courant1990Yoshimura2006
  • Theorem 1: Courant1990Yoshimura2006
  • Definition 2: Courant1990Yoshimura2006VanderSchaft2014
  • Theorem 2
  • Definition 3: Poisson-Dirac Neural Network
  • Remark 1: Coupling as Non-zero Elements of Bivector
  • Remark 2: Degeneracy of Dynamics as Degeneracy of Bundle Map
  • Remark 3: Coordinate Transformation by Bivector
  • Remark 4: Multiphysics
  • proof : Proof of Theorem \ref{['thm:construction_Dirac']}Yoshimura2006
  • ...and 5 more