Cole J. Errico, Alberto Padovan, Daniel J. Bodony
Non-intrusive reduced-order modeling techniques are necessary for systems that are simulated using black-box solvers or known only from data. For systems exhibiting large transients and operating far away from equilibria, current non-intrusive models often exhibit poor forecasting accuracy and can even be unstable in infinite or finite time. Recent developments have addressed the stability issue by seeking structure-preserving latent-space architectures when reducing Hamiltonian or Lagrangian full-order dynamics, or by enforcing global stability via Lyapunov-informed parameterizations in the latent space. However, such developments do not necessarily improve the forecasting accuracy of the resulting models, since these formulations achieve dimensionality reduction using orthogonal projections that accidentally truncate dynamically-important states. In this paper, we address both issues by introducing a non-intrusive framework designed to simultaneously identify globally-asymptotically-stable latent-space dynamics, and oblique projection operators capable of capturing the sensitivity mechanisms of the system. In particular, given a Lyapunov-based parameterization of the latent-space tensors, and a matrix-manifold parameterization of the oblique projection operators, we fit a model against high-fidelity training trajectories. Furthermore, we show that the gradient of the objective function can be written in closed form using adjoint-based backpropagation in the latent space, eliminating the need for automatic differentiation. We compare our formulation with state-of-the-art methods on a three-dimensional system of ordinary differential equations, and a two-dimensional lid-driven cavity flow at Reynolds number Re=8300. We demonstrate that our models are not only globally asymptotically stable (as expected by construction), but they are also significantly more accurate.