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

TL;DR

PoDiNNs are a new neural network framework based on geometric mechanics that can model coupled dynamical systems across various domains with improved accuracy and interpretability.

Contribution

The paper introduces PoDiNNs, a novel framework unifying port-Hamiltonian and Poisson formulations to model diverse and coupled dynamical systems from data.

Findings

Enhanced modeling accuracy for coupled systems

Improved interpretability of dynamical models

Applicable across multiple physical domains

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.