Shinhoo Kang

Shinhoo Kang, Hai V. Nguyen, Tan Bui-Thanh

Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus match values and tangents at the supervised states yet curve differently from the truth, remaining locally accurate while drifting toward spurious attractors and distorting long-time statistics. We show that enforcing second-order consistency mitigates these failures, but forming the full Hessian is prohibitive in high dimensions. We propose model-constrained randomized Jacobian matching, which compares the Jacobians of the true and learned vector fields at randomly perturbed inputs. A Taylor expansion shows that the expected randomized Jacobian loss decomposes into the nominal Jacobian mismatch plus a Hessian mismatch scaled by the noise variance, implicitly enforcing second-order consistency at $\mathcal{O}(d^2)$ cost without forming the $\mathcal{O}(d^3)$ Hessian tensor. Using only Jacobian evaluations, the method scales to high dimensions where explicit Hessian matching does not. Numerical experiments confirm that second-order methods are robust. For Lorenz~63, first-order methods produce catastrophic Lyapunov-exponent outliers under minimal temporal supervision, which second-order methods eliminate while recovering the correct attractor. For coupled Lorenz~96, an out-of-distribution forcing sweep separates the methods: all agree up to $F=16$, but beyond $F=18$ only second-order methods preserve the invariant measure and Lyapunov spectrum. On both systems, randomized Jacobian matching performs comparably to explicit Hessian matching at much lower cost.

Shinhoo Kang, Francis X. Giraldo, Tan Bui-Thanh

We propose IMEX HDG-DG schemes for planar and spherical shallow water systems. Of interest is subcritical flow, where the speed of the gravity wave is faster than that of nonlinear advection. In order to simulate these flows efficiently, we split the governing system into a stiff part describing the gravity wave and a non-stiff part associated with nonlinear advection. The former is discretized implicitly with the HDG method while an explicit Runge-Kutta DG discretization is employed for the latter. The proposed IMEX HDG-DG framework: 1) facilitates high-order solutions both in time and space; 2) avoids overly small time-step sizes; 3) requires only one linear system solve per time stage; 4) relative to DG generates smaller and sparser linear systems while promoting further parallelism. Numerical results of various test cases demonstrate that our methods are comparable to explicit Runge-Kutta DG schemes in terms of accuracy while allowing for much larger time step sizes.

Shinhoo Kang, Tan Bui-Thanh, Todd Arbogast

We develop a high-order hybridized discontinuous Galerkin (HDG) method for a linear degenerate elliptic equation arising from a two-phase mixture of mantle convection or glacier dynamics. We show that the proposed HDG method is well-posed by using an energy approach. We derive ${\it a priori}$ error estimates for the proposed HDG method on simplicial meshes in both two- and three-dimensions. The error analysis shows that the convergence rates are optimal for both the scaled pressure and the scaled velocity for non-degenerate problems and are sub-optimal by half order for degenerate ones. Several numerical results are presented to confirm the theoretical estimates. We also enhance the HDG solutions by post-processing. The superconvergence rates of $(k+2)$ and $(k+\frac{3}{2})$ are observed for both a non-degenerate case and a degenerate case away from the degeneracy. Degenerate problems with low regularity solutions are also studied, and numerical results show that high-order methods are beneficial in terms of accuracy.

Shinhoo Kang, Emil M. Constantinescu

We propose a new approach to learning the subgrid-scale model 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 with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.

Shinhoo Kang, Emil M. Constantinescu

Computational advances have fundamentally transformed the landscape of numerical simulations, enabling unprecedented levels of complexity and precision in modeling physical phenomena. While these high-fidelity simulations offer invaluable insights for scientific discovery and problem solving, they impose substantial computational requirements. Consequently, low-fidelity models augmented with subgrid-scale parameterizations are employed to achieve computational feasibility. We introduce an end-to-end differentiable framework for solving the compressible Navier--Stokes equations. This integrated approach combines a differentiable discontinuous Galerkin (DG) solver with a neural network source term. Through the implementation of neural ordinary differential equations (NODEs) for network parameter optimization, our methodology ensures continuous interaction with the governing equations throughout the training process. We refer to this approach as NODE-DG. This hybrid approach combines the accuracy of numerical methods with the efficiency of machine learning, offering the following key advantages: (1) improved accuracy of low-order DG approximations by capturing subgrid-scale dynamics; (2) robustness against nonuniform or missing temporal data; (3) elimination of operator-splitting errors; (3) total mass conservation; and (4) a continuous-in-time operator that enables variable time step predictions, which accelerate projected high-order DG simulations. We demonstrate the performance of the proposed framework through two examples: two-dimensional Kelvin--Helmholtz instability and three-dimensional Taylor--Green vortex examples.

Shinhoo Kang, Sangwook Kim, Sehyun Yun

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.