Differentiable Modeling for Low-Inertia Grids: Benchmarking PINNs, NODEs, and DP for Identification and Control of SMIB System

The transition toward low-inertia power systems demands modeling frameworks that provide not only accurate state predictions but also physically consistent sensitivities for control. While scientific machine learning offers powerful nonlinear modeling tools, the control-oriented implications of different differentiable paradigms remain insufficiently understood. This paper presents a comparative study of Physics-Informed Neural Networks (PINNs), Neural Ordinary Differential Equations (NODEs), and Differentiable Programming (DP) for modeling, identification, and control of power system dynamics. Using the Single Machine Infinite Bus (SMIB) system as a benchmark, we evaluate their performance in trajectory extrapolation, parameter estimation, and Linear Quadratic Regulator (LQR) synthesis. Our results highlight a fundamental trade-off between data-driven flexibility and physical structure. NODE exhibits superior extrapolation by capturing the underlying vector field, whereas PINN shows limited generalization due to its reliance on a time-dependent solution map. In the inverse problem of parameter identification, while both DP and PINN successfully recover the unknown parameters, DP achieves significantly faster convergence by enforcing governing equations as hard constraints. Most importantly, for control synthesis, the DP framework yields closed-loop stability comparable to the theoretical optimum. Furthermore, we demonstrate that NODE serves as a viable data-driven surrogate when governing equations are unavailable.

Learning Subgrid-Scale Models in Discontinuous Galerkin Methods with Neural Ordinary Differential Equations for Compressible Navier--Stokes Equations

The growing computing power over the years has enabled simulations to become more complex and accurate. However, high-fidelity simulations, while immensely valuable for scientific discovery and problem solving, come with significant computational demands. As a result, it is common to run a low-fidelity model with a subgrid-scale model to reduce the computational cost, but selecting the appropriate subgrid-scale models and tuning them are challenging. We propose a novel method for learning the subgrid-scale model effects when simulating partial differential equations using neural ordinary differential equations in the context of discontinuous Galerkin (DG) spatial discretization. Our approach learns the missing scales of the low-order DG solver at a continuous level and hence improves the accuracy of the low-order DG approximations as well as accelerates the filtered high-order DG simulations with a certain degree of precision. We demonstrate the performance of our approach through multidimensional Taylor--Green vortex examples at different Reynolds numbers and times, which cover laminar, transitional, and turbulent regimes. The proposed method not only reconstructs the subgrid-scale from the low-order (1st-order) approximation but also speeds up the filtered high-order DG (6th-order) simulation by two orders of magnitude.

Learning Subgrid-scale Models with Neural Ordinary Differential Equations

We propose a new approach to learning the subgrid-scale model effects when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results using the two-scale Lorenz 96 ODE and the convection-diffusion PDE are used to illustrate this approach.