PHySIM: Neural-Augmented Physical Simulation of Networked Systems

Network-based systems, including power grids, circuits, and gas pipelines, consist of interconnected subsystems governed by coupled equations. Simulating these systems requires solving algebraic and differential subsystem equations simultaneously while enforcing their shared physical boundary variables. Physics-based solvers exploit the subsystem structure but are slow to converge when subsystems have severe nonlinearities or behaviors driven by exogenous factors (weather, occupancy) that first-principles models cannot capture. Black-box approaches offer low-cost approximations that capture hidden behavior but cannot enforce conservation laws at inference time. These methods also need training data covering the network's full state space and struggle to generalize across topology changes. To address these limitations, we introduce PHySIM, a flexible framework for modeling network-based systems through subsystem partitioning, where each subsystem can be represented by either a physics-based model or a DNN coupled at inference time through shared state variables at subsystem boundaries. PHySIM implicitly captures interactions across coupled subsystems while combining the predictive capabilities of DNNs to capture hidden behaviors seamlessly with the explainability and accuracy of physics-based models. The coupled system is solved inside the Newton-Raphson solver loop that extracts the DNN's sensitivities via backpropagation and assembles them into the same Jacobian matrix as physics-based terms, thereby solving the coupled system simultaneously rather than sequentially. The inference-time coupling inherits the benefits of both physics-based and DNN models: exact conservation-law satisfaction; superlinear state-space scaling from DNN reuse; data requirements that scale with device complexity rather than network size; zero-shot topology generalization without retraining; and provable Newton stability linked directly to a Sobolev Jacobian training loss. We validate PHySIM across power systems, analog circuits, and gas pipeline networks, and demonstrate a capstone electrothermal grid simulation that correctly identifies voltage-collapse onset where physcs-based and black-box simulation baselines fail.