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

Eric Parish; Anthony Gruber; Patrick Blonigan; Irina Tezaur

cs.LGMar 9, 2026

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

Authors: Eric Parish, Anthony Gruber, Patrick Blonigan, Irina Tezaur

Organizations: Sandia National Laboratories, Livermore, CA, USA · Sandia National Laboratories, Albuquerque, NM, USA

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 & tables

Name	Expression	Description
Standard	$\mathcal{N}(\boldsymbol{\eta};\mathbf{w})$	Fully connected neural network mapping reduced inputs to the reduced velocity. This is the most flexible, least constrained operator and serves as the baseline NN-OpInf formulation in [ 20 ] .
Matrix	$\hat{\boldsymbol{A}}(\boldsymbol{\eta};\mathbf{w})\hat{\boldsymbol{x}}$	A generic matrix-valued network function acting on the reduced state.
SPSD	$\hat{\boldsymbol{L}}(\boldsymbol{\eta};\mathbf{w})\hat{\boldsymbol{L}}(\boldsymbol{\eta};\mathbf{w})^{\intercal}\hat{\boldsymbol{x}}$	A Cholesky-inspired product of lower-triangular matrix operators enforcing symmetry and positive (semi-) definiteness. Designed to yield dissipative dynamics consistent with energy decay in homogeneous systems.
Skew	$\big(\hat{\boldsymbol{S}}(\boldsymbol{\eta};\mathbf{w})-\hat{\boldsymbol{S}}(\boldsymbol{\eta};\mathbf{w})^{\intercal}\big)\hat{\boldsymbol{x}}$	Strictly lower-triangular matrix operator (uniquely) enforcing skew-symmetry by construction, leading to energy-preserving dynamics in the absence of forcing.
Vector	$\mathbf{c}(\boldsymbol{\mu})\ \text{or}\ \mathbf{c}(\boldsymbol{\eta};\mathbf{w})$	Operator representing an additive offset or forcing term. Used to represent constant inputs or purely parametric forcing independent of the reduced state.
SPSD Potential	$\begin{aligned} &\nabla_{\hat{\boldsymbol{x}}}\mathcal{L}(\hat{\boldsymbol{x}}),\\ &\mathcal{L}(\hat{\boldsymbol{x}})=\hat{\boldsymbol{x}}^{\intercal}\hat{\boldsymbol{A}}(\boldsymbol{\eta};\mathbf{w})\hat{\boldsymbol{x}}\\ &\hat{\boldsymbol{A}}(\boldsymbol{\eta};\mathbf{w})=\hat{\boldsymbol{L}}(\boldsymbol{\eta};\mathbf{w})\hat{\boldsymbol{L}}(\boldsymbol{\eta};\mathbf{w})^{\intercal}\end{aligned}$	Gradient-based model computed with automatic differentiation and derived from a scalar potential built from an SPSD operator. The resulting dynamics can be used to enforce Lagrangian structure and guarantee the conservation of energy.

Table 1: Subset of operators used in NN-OpInf.

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.

Category	Model/operator	Dominant cost
Online evaluation	Linear OpInf	$\mathcal{O}(K^{2})$
Online evaluation	Quadratic OpInf	$\mathcal{O}(K^{3})$
Online evaluation	SPSD	$\mathcal{O}(K^{2})$
Online evaluation	Skew	$\mathcal{O}(K^{2})$
Online evaluation	Standard NN	$\mathcal{O}(K^{2})$
Online evaluation	Matrix NN operator	$\mathcal{O}(K^{3})$

Table 2: Summary of dominant computational costs in the analysis section. Here

N_{s}=N_{t}N_{\text{train}}

. For NN-operator costs, we assume

n_{\mathrm{n}}\propto K

, and

n_{\mathrm{in}}=K

Name	Description	Implementation
POD-Galerkin	Intrusive POD ROM (no hyper-reduction)	In-house
P-OpInf-A	Linear P-OpInf model	OpInf [ 37 ]
P-OpInf-AH	Quadratic P-OpInf model	OpInf [ 37 ]
P-OpInf-cA	Linear P-OpInf with affine forcing	OpInf [ 37 ]
P-OpInf-cAH	Quadratic P-OpInf with affine forcing	OpInf [ 37 ]
NN-OpInf-NN	Standard NN operator from Table 1	NN-OpInf [ 1 ]

Table 3: Summary of models used in the numerical experiments.

Panel of Figure 3 — Figure 3: Burgers’ equation. Relative errors for various OpInf formulations.

Panel of Figure 4 — Figure 4: Nonlinear CDR example. FOM solution for $\sigma=2.551$ , $\nu=7.111\times 10^{-4}$ , $\theta=0.8282$ , and $\eta=2.0217$ .

Panel of Figure 5 — 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).

Panel of Figure 6 — Figure 6: Nonlinear CDR example in a parametric configuration. Relative errors for various OpInf formulations (left) and solutions for the finest ROMs at $t=2.5$ (right). Solutions across the training set are shown on the top, while the testing set is shown on the bottom. In the testing regime, the P-OpInf-cAH ROM went unstable for $K>20$ .

Panel of Figure 7 — Figure 7: 2D nonlinear heat conduction. Relative errors for various OpInf formulations. We note that the P-OpInf-AH ROM at $K=4$ was unstable in the future-state configuration.

Panel of Figure 8 — Figure 8: Premixed $\mathrm{H}_{2}$ air-flame example. Schematic of geometry and boundary conditions.

Panel of Figure 9 — Figure 9: Premixed $\mathrm{H}_{2}$ air-flame example. FOM solutions for temperature at $t=0.02$ (left), $t=0.04$ (center), and $t=0.06$ (right) for $(\mu_{1},\mu_{2})=(5.55625\times 10^{12},8.437583\times 10^{3})$ .

Panel of Figure 10 — Figure 10: Premixed $\mathrm{H}_{2}$ air-flame example. Relative errors for various OpInf formulations as a function of ROM dimension.

Panel of Figure 11 — Figure 11: Torsion problem. Relative error versus basis dimension.

Figure 12: Torsion problem. Displacement solutions at

t=2.5\times 10^{-3}

s (

K=40

). The coloring is the displacement magnitude.

Figure 13: Torsion problem. Displacement solutions at

t=3.75\times 10^{-3}

s (

K=40

). The coloring is the displacement magnitude.

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Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition

Changhong Mou, Binghang Lu, Guang Lin

Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT, 84322, USA. · School of Electrical and Computer Engineering, Purdue University, 610 Purdue Mall, West Lafayette, IN, 47907, USA. · Department of Mathematics, Purdue University, 610 Purdue Mall, West Lafayette, IN, 47907, USA.+1

Proper Orthogonal Decomposition Orthogonal Representation Learning Reduced-Order Modeling Neural Operators

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AI for science (AI4Science) models often suffer from discretization: learned representations remain tied to the training grid, limiting transfer across resolutions, solvers and applications. We introduce Neural Proper Orthogonal Decomposition (Neural-POD), a plug-and-play neural operator that learns nonlinear, orthogonal basis functions directly in function space and can be integrated in both projection-based reduced order models and operator-learning frameworks such as DeepONet. Neural-POD replaces SVD-derived, resolution-dependent linear modes with continuous, resolution-invariant bases learned via sequential residual minimization, analogous to Gram-Schmidt orthogonalization. The framework supports training under task-specific norms (e.g.,

L^2

L^1

), improves out-of-distribution generalization to unseen parameter regimes, and captures nonlinear structure in complex systems. Because the learned bases are interpretable and reusable, Neural-POD serves as a general representation module for AI4Science workflows. We demonstrate Neural-POD on Burgers' and Navier-Stokes equations.

Abstract

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