Neural Differential-Algebraic Equations with Dirac Structures for Coupling Dynamical Systems

Deep learning can be used to model a dynamical system as a single vector field over its state space to predict its trajectories. A real physical system, however, is an interconnection of components, and a model that reflects this internal structure is interpretable and reusable beyond prediction. Existing methods either assume this structure to be known in advance or, to learn it, assume the system to be reducible to an ordinary differential equation (ODE), learn only the reduced ODE, and require all states to be observed. We instead propose a neural network model that represents the system as a differential-algebraic equation (DAE) whose interconnection is a Dirac structure in kernel representation, and identify the internal structure by learning the constitutive relations and the interconnection from data. This implicit representation retains the algebraic constraints and composes trained subsystems without retraining. Training on the DAE residual further allows identification from boundary observations alone. Experiments show that the method accurately identifies the internal structure and predicts the trajectories of systems beyond the reach of existing methods.