Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks

2024-12-15

arXiv.org

AI Summary

This work addresses the modeling challenge of coupled dynamical systems with algebraic constraints—such as electrical networks—by introducing the first composable, physics-structure-preserving learning framework. Methodologically, it proposes (1) neural port-Hamiltonian differential-algebraic equations (N-PHDAEs), where unknown terms in port-Hamiltonian DAEs are parameterized by neural networks while explicitly embedding port-Hamiltonian structure and energy conservation; (2) an automatic-differentiation-based index-reduction algorithm that equivalently transforms high-index neural DAEs into differentiable neural ODEs for end-to-end training; and (3) component-wise independent training and plug-and-play compositional modeling. Experiments demonstrate that, on nonlinear circuit long-term forecasting, the method improves both prediction accuracy and algebraic constraint satisfaction by an order of magnitude over neural ODE baselines. Furthermore, modular modeling capability is validated on real-world DC microgrid measurement data.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We develop compositional learning algorithms for coupled dynamical systems. While deep learning has proven effective at modeling complex relationships from data, compositional couplings between system components typically introduce algebraic constraints on state variables, posing challenges to many existing data-driven approaches to modeling dynamical systems. Towards developing deep learning models for constrained dynamical systems, we introduce neural port-Hamiltonian differential algebraic equations (N-PHDAEs), which use neural networks to parametrize unknown terms in both the differential and algebraic components of a port-Hamiltonian DAE. To train these models, we propose an algorithm that uses automatic differentiation to perform index reduction, automatically transforming the neural DAE into an equivalent system of neural ordinary differential equations (N-ODEs), for which established model inference and backpropagation methods exist. The proposed compositional modeling framework and learning algorithms may be applied broadly to learn control-oriented models of dynamical systems in a variety of application areas, however, in this work, we focus on their application to the modeling of electrical networks. Experiments simulating the dynamics of nonlinear circuits exemplify the benefits of our approach: the proposed N-PHDAE model achieves an order of magnitude improvement in prediction accuracy and constraint satisfaction when compared to a baseline N-ODE over long prediction time horizons. We also validate the compositional capabilities of our approach through experiments on a simulated D.C. microgrid: we train individual N-PHDAE models for separate grid components, before coupling them to accurately predict the behavior of larger-scale networks.

Problem

• Develop algorithms for learning coupled dynamical systems
• Model constrained dynamical systems using neural networks
• Improve prediction accuracy in electrical network simulations

Innovation

• Neural networks parametrize port-Hamiltonian DAE terms
• Automatic differentiation enables index reduction
• Compositional learning for coupled electrical networks