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

TL;DR

It is suggested 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.

Abstract

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.

Figures (13)

• Figure 1: Operator evaluation cost ratios relative to linear and quadratic OpInf baselines as a function of reduced dimension $K=K$, assuming $n_{\mathrm{h}}=3$, $n_{\mathrm{n}}=K$, and $n_{o,a}=1$ (RELU).
• Figure 2: Training cost ratios relative to linear and quadratic P-OpInf baselines as a function of reduced dimension $K$, assuming $n_{\text{epochs}}=10{,}000$, $n_{\mathrm{h}}=3$, $n_{\mathrm{n}}=K$, and $N_t N_{\text{train}}=10{,}000$ snapshot pairs.
• Figure 3: Burgers' equation. Relative errors for various OpInf formulations.
• Figure 4: Nonlinear CDR example. FOM solution for $\sigma = 2.551$, $\nu = 7.111 \times 10^{-4}$, $\theta = 0.8282$, and $\eta = 2.0217$.
• Figure 5: Nonlinear CDR example in a reproductive configuration. Relative errors for various OpInf formulations (left) and solutions for the finest ROMs at $t = 1.0$ (right).

Theorems & Definitions (3)

• Proposition 5.1: Convex linear-operator cases
• proof
• Remark 5.1: When convexity is lost