NN-OpInf: an operator inference approach using structure-preserving composable neural networks

We propose neural network operator inference (NN-OpInf): a structure-preserving, composable, and minimally restrictive operator inference framework for the non-intrusive reduced-order modeling of dynamical systems. The approach learns latent dynamics from snapshot data, enforcing local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation, while also reflecting complex dynamics by supporting additive compositions of heterogeneous operators. We present practical training strategies and analyze computational costs relative to linear and quadratic polynomial OpInf (P-OpInf). Numerical experiments across several nonlinear and parametric problems demonstrate improved accuracy, stability, and robustness over P-OpInf and prior NN-ROM formulations, particularly when the dynamics are not well represented by polynomial models. These results suggest that NN-OpInf can serve as an effective drop-in replacement for P-OpInf when the dynamics to be modeled contain non-polynomial nonlinearities, offering potential gains in accuracy and out-of-distribution performance at the expense of higher training computational costs and a more difficult, non-convex learning problem.

Hyper-Reduced Autoencoders for Efficient and Accurate Nonlinear Model Reductions

Projection-based model order reduction on nonlinear manifolds has been recently proposed for problems with slowly decaying Kolmogorov n-width such as advection-dominated ones. These methods often use neural networks for manifold learning and showcase improved accuracy over traditional linear subspace-reduced order models. A disadvantage of the previously proposed methods is the potential high computational costs of training the networks on high-fidelity solution snapshots. In this work, we propose and analyze a novel method that overcomes this disadvantage by training a neural network only on subsampled versions of the high-fidelity solution snapshots. This method coupled with collocation-based hyper-reduction and Gappy-POD allows for efficient and accurate surrogate models. We demonstrate the validity of our approach on a 2d Burgers problem.