Non-Intrusive Operator Inference (OpInf)

• Non-Intrusive Operator Inference is a data-driven, projection-based modeling technique that learns reduced operators directly from snapshot data without accessing full-order operators.
• It employs least-squares regression to identify linear and quadratic operators while utilizing POD for basis extraction and regularization to ensure stability.
• OpInf is applied to turbulent flows, chaotic systems, and mechanical problems, delivering significant speed-ups and interpretable, structure-preserving reduced models.

Searching arXiv for recent and foundational papers on non-intrusive Operator Inference to support the article. Non-Intrusive Operator Inference (OpInf) is a data-driven, projection-based reduced-order modeling methodology that learns reduced operators directly from snapshot data rather than assembling them by projection of full-order operators. Across the literature, it is consistently characterized as non-intrusive because it does not require access to the full-order model operators or source code, and instead operates on state snapshots, projected coordinates, and, depending on the formulation, time-derivative, output, or boundary data (Benner et al., 2020, Gahr et al., 2024, Moore et al., 2024). In its standard form, OpInf assumes that the reduced dynamics inherit a prescribed algebraic structure—most often linear or quadratic polynomial terms—and identifies the corresponding reduced operators through least-squares regression (Sawant et al., 2021, Almeida et al., 2022). Subsequent developments extend this paradigm to systems with non-polynomial nonlinearities, quadratic manifolds, constrained mechanical systems, domain-decomposed subdomain models, and adaptive online updates (Benner et al., 2020, Geelen et al., 2022, Benner et al., 7 Jul 2025, Hedayat et al., 11 Feb 2026).

1. Definition and model class

Non-intrusive OpInf is described as a ROM technique that learns reduced operators from data rather than computing them by intrusive projection of full-order matrices (Moore et al., 2024). A common starting point is a high-dimensional dynamical system of the form

$$\dot{q}(t) = \mathbf{A}q(t) + \mathbf{H}(q(t)\otimes q(t)), \qquad y(t) = \mathbf{C}q(t) + \mathbf{G}(q(t)\otimes q(t)),$$

or, more generally,

$$\frac{d}{dt}\mathbf{s}(t) = \mathbf{f}(t,\mathbf{s}(t)),$$

with reduced coordinates obtained from a low-dimensional basis constructed from snapshots (Gahr et al., 2024, Geelen et al., 2022). In many formulations, the reduced model is postulated in the same polynomial class as the semi-discrete full-order model: $$\dot{a}(t) = \mathbf{A}_r a(t) + \mathbf{H}_r(a(t)\otimes a(t)), \qquad \hat{y}(t) = \mathbf{C}_r a(t) + \mathbf{G}_r(a(t)\otimes a(t)),$$ with no need to access the full operators $\mathbf{A}, \mathbf{H}, \mathbf{C}, \mathbf{G}$ (Gahr et al., 2024).

The literature repeatedly emphasizes that OpInf occupies an intermediate position between intrusive projection-based model reduction and black-box machine learning. Compared with intrusive Galerkin ROMs, it avoids source-level operator projection; compared with unconstrained neural surrogates, it preserves an equation-structured reduced model with interpretable operators (Gahr et al., 2024). This suggests that OpInf is best understood not as a purely statistical surrogate but as a reduced dynamical-system identification framework with a prescribed operator ansatz.

2. Canonical workflow

The standard non-intrusive OpInf workflow begins with snapshot collection from a full-order solver treated as a black box. These snapshots are assembled into a matrix, and a reduced basis is typically obtained by proper orthogonal decomposition or PCA through an SVD of the state data (Gahr et al., 2024, Almeida et al., 2022). If $V\in\mathbb{R}^{N_{\text{state}}\times r}$ denotes the basis, then the reduced coordinates are obtained by projection,

$a_i = V^\top q_i,$

or, in other notation,

$\hat{s}(t) = V^\top s(t).$

The reduced derivatives are either projected from available derivative data or approximated numerically from snapshots (Gahr et al., 2024, Benner et al., 2020).

Operator learning is then posed as a least-squares regression problem. For a quadratic reduced model, a representative objective is

$$\min_{\mathbf{A}_r,\mathbf{H}_r} \frac{1}{n_t}\sum_{i=1}^{n_t} \left\| \dot{a}_i - \mathbf{A}_r a_i - \mathbf{H}_r(a_i\otimes a_i) \right\|_2^2 + \lambda_A\|\mathbf{A}_r\|_F^2 + \lambda_H\|\mathbf{H}_r\|_F^2,$$

with analogous regression for output operators when quantities of interest are modeled explicitly (Gahr et al., 2024). In the linear setting, the same pattern appears with only linear state and boundary/input operators (Moore et al., 2024).

A recurrent implementation detail is the use of regularization. Tikhonov penalties on reduced operators are introduced to stabilize inference, avoid overfitting, and improve conditioning, especially for nonlinear or chaotic regimes (Gahr et al., 2024, Sawant et al., 2021). Several works further treat regularization as a central design choice rather than a numerical afterthought.

3. Reduced bases, lifted coordinates, and nonlinear state representations

The classical OpInf pipeline uses a linear reduced basis, but later work broadens the state representation beyond linear subspaces. In "Operator inference for non-intrusive model reduction with quadratic manifolds" (Geelen et al., 2022), the reduced state is embedded through a quadratic manifold

$$\mathbf{s}(t) \approx \mathbf{s}_\text{ref} + \mathbf{V}\,\hat{\mathbf{s}}(t) + \overline{\mathbf{V}}\big(\hat{\mathbf{s}}(t)\otimes\hat{\mathbf{s}}(t)\big),$$

where the quadratic component lies in the orthogonal complement of the POD subspace. The reduced dynamics are still inferred by OpInf, but the manifold enriches the reconstruction without increasing the reduced coordinate dimension (Geelen et al., 2022).

A different extension appears in non-polynomial settings. "Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms" (Benner et al., 2020) assumes that the non-polynomial term is known analytically and spatially local. In that case, only the linear, quadratic, and input operators are inferred, while the local non-polynomial term is explicitly evaluated and projected: $$\dot{\hat{s}}(t) = \hat{A}\hat{s}(t) + \hat{H}(\hat{s}(t)\,{data}'\,\hat{s}(t)) + V^T f(t,V\hat{s}(t)) + \hat{B}u(t).$$ This preserves the linear least-squares character of OpInf while extending applicability beyond purely polynomial reduced models (Benner et al., 2020).

These developments indicate that the essential non-intrusive feature of OpInf is not tied to a linear latent space alone. A plausible implication is that the method is better viewed as a family of regression-based operator-identification procedures built around reduced coordinates, rather than a single fixed projection formula.

4. Stability, regularization, and structure preservation

A major concern in OpInf is that low training error does not guarantee stable or physically meaningful reduced dynamics. "Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference" (Sawant et al., 2021) addresses this by focusing on systems with quadratic nonlinearities and proposing a regularizer that penalizes quadratic terms with large norms. The paper relates the Frobenius norm of the quadratic operator to a Lyapunov-based estimate of the stability radius, motivating a regularized problem of the form

$$\frac{d}{dt}\mathbf{s}(t) = \mathbf{f}(t,\mathbf{s}(t)),$$0

It also introduces a constrained formulation enforcing structure such as symmetry and definiteness in the linear operator via semidefinite constraints (Sawant et al., 2021).

Structure preservation is especially prominent in mechanical systems. In "Application of operator inference to reduced-order modeling of constrained mechanical systems" (Benner et al., 7 Jul 2025), the inferred ROM is a second-order ODE

$$\frac{d}{dt}\mathbf{s}(t) = \mathbf{f}(t,\mathbf{s}(t)),$$1

and the state-equation operators are inferred under semidefinite constraints

$$\frac{d}{dt}\mathbf{s}(t) = \mathbf{f}(t,\mathbf{s}(t)),$$2

The paper explicitly states that stability and interpretability are guaranteed by enforcing the symmetric positive definite structure of the system operators using semidefinite programming (Benner et al., 7 Jul 2025).

A related but distinct line concerns energy-preserving quadratic operators. "On the representation of energy-preserving quadratic operators with application to Operator Inference" (Gkimisis et al., 13 Mar 2025) proves that every energy-preserving quadratic term can be equivalently formulated using a parameterization of the corresponding operator via skew-symmetric matrix blocks. It then proposes a sequential, linear least-squares formulation for inference that ensures energy preservation of the data-driven quadratic operator (Gkimisis et al., 13 Mar 2025). The abstract states that this representation is employed in non-intrusive ROM via OpInf for systems with an energy-preserving nonlinearity, with numerical results on a 2D Burgers' equation benchmark compared to classical OpInf (Gkimisis et al., 13 Mar 2025).

Taken together, these results establish that non-intrusive OpInf is not limited to unconstrained least-squares fitting. It has evolved into a structure-aware inference framework in which stability, definiteness, symmetry, and energy preservation can be encoded directly in the reduced operators (Sawant et al., 2021, Benner et al., 7 Jul 2025, Gkimisis et al., 13 Mar 2025).

5. Applications and representative problem classes

A substantial portion of the OpInf literature is application-driven. One prominent example is plasma turbulence. "Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulations" (Gahr et al., 2024) applies OpInf to the Hasegawa–Wakatani equations, whose semi-discrete full-order model has quadratic state and output structure. Training data are generated by direct numerical simulations over a turbulent window, and OpInf ROMs are then used for predictions over additional time units beyond the training horizon (Gahr et al., 2024). The paper reports that the ROMs capture important statistical features of the turbulent dynamics and generalize beyond the training time horizon while reducing the computational effort of the high-fidelity simulation by up to five orders of magnitude (Gahr et al., 2024).

Chaotic systems constitute another major test bed. "Non-Intrusive Reduced Models based on Operator Inference for Chaotic Systems" (Almeida et al., 2022) applies quadratic OpInf to Lorenz 96 and the Kuramoto–Sivashinsky equation. The quality of predictions is assessed via NRMSE and Valid Prediction Time, and the reported VPT ranges outperform state-of-the-art machine learning methods such as backpropagation and reservoir computing recurrent neural networks, as well as Markov neural operators, on the tested problems (Almeida et al., 2022).

Mechanical systems have motivated several variants. In constrained multibody and DAE settings, OpInf is used to identify the hidden-manifold ODE directly from DAE solution snapshots in compressed form (Benner et al., 7 Jul 2025). In elastodynamics and solid mechanics, second-order OpInf ROMs are inferred from displacement and acceleration data and then coupled through Schwarz methods in hybrid FOM–ROM and ROM–ROM configurations (Rodriguez et al., 6 Sep 2025, Tezaur et al., 20 Nov 2025).

Other problem classes include transport-dominated systems addressed with quadratic manifolds (Geelen et al., 2022), parametric matrix equations such as Lyapunov and Riccati equations reformulated into polynomial structure for OpInf (Wen et al., 20 Nov 2025), and dynamical systems with spatially localized features handled by hybrid OpInf–sFOM decomposition (Gkimisis et al., 8 Jan 2025).

The range of applications suggests that OpInf is particularly effective when the governing equations admit a reduced polynomial structure or can be reformulated into one. This suggests a practical boundary of the method: its success depends not only on data availability, but also on whether an informative low-dimensional operator ansatz can be specified.

6. Domain decomposition, hybrid fidelity, and adaptive extensions

Recent work extends OpInf beyond monolithic global ROMs toward modular and adaptive settings. In "Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method" (Moore et al., 2024), local OpInf ROMs are trained on subdomain-restricted data and coupled with each other or with subdomain-local FOMs through overlapping Schwarz transmission conditions. The method is explicitly described as capable of coupling together arbitrary combinations of OpInf ROMs and FOMs, and speed-ups over a monolithic FOM are reported when performing OpInf ROM coupling (Moore et al., 2024).

A related but more elaborate development is "Hybrid coupling with operator inference and the overlapping Schwarz alternating method" (Tezaur et al., 20 Nov 2025), which formulates subdomain-local polynomial OpInf ROMs for nonlinear 3D solid dynamics and couples them to FOMs and to each other with O-SAM. The abstract reports speedups of up to $$\frac{d}{dt}\mathbf{s}(t) = \mathbf{f}(t,\mathbf{s}(t)),$$3 compared to conventional FOM-FOM couplings (Tezaur et al., 20 Nov 2025). The work emphasizes that OpInf ROMs are trained from subdomain-local simulation data and boundary histories and then inserted as cheap local solvers inside the Schwarz iteration (Tezaur et al., 20 Nov 2025).

Non-overlapping Schwarz coupling has also been investigated. "Transmission Conditions for the Non-Overlapping Schwarz Coupling of Full Order and Operator Inference Models" (Rodriguez et al., 6 Sep 2025) studies Dirichlet–Neumann and Robin–Robin transmission conditions for FOM–FOM, FOM–OpInf, and OpInf–OpInf coupling. The abstract notes that Robin–Robin coupling often yields faster convergence than alternating Dirichlet–Neumann, though improper parameter selection can induce spurious oscillations at subdomain interfaces (Rodriguez et al., 6 Sep 2025).

Adaptive OpInf moves the method beyond static offline-trained ROMs. "Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges" (Hedayat et al., 11 Feb 2026) formalizes Adaptive OpInf as sequential basis and operator refits over an online data window. The paper states that static Galerkin, static OpInf, and static NiTROM drift or destabilize when forecasting beyond training, whereas Adaptive OpInf robustly suppresses amplitude drift with modest cost (Hedayat et al., 11 Feb 2026). Its proposed framework introduces explicit online data windows, adaptation windows, and computational budgets, making adaptation itself part of the ROM design (Hedayat et al., 11 Feb 2026).

These strands collectively redefine non-intrusive OpInf as a modular technology. Rather than a single reduced system learned once offline, it can function as a local solver, a component in multi-model coupling, or an online-updated predictor.

7. Limitations, misconceptions, and current directions

A common misconception is that non-intrusive OpInf is equivalent to arbitrary black-box system identification. The literature instead treats it as strongly equation-structured: the reduced model class is chosen in advance, usually with linear and quadratic operators, and inference is linear in the unknown operator entries (Gahr et al., 2024, Almeida et al., 2022). This structural prior is central to both interpretability and computational efficiency.

Another misconception is that non-intrusiveness means zero physical information. Several works explicitly incorporate physics through the model structure, analytic evaluation of known nonlinear terms, regularization targeted at quadratic operators, or hard constraints enforcing definiteness, symmetry, or skew-symmetry (Benner et al., 2020, Sawant et al., 2021, Gkimisis et al., 13 Mar 2025, Benner et al., 7 Jul 2025). Non-intrusive OpInf is therefore data-driven in operator identification, but not agnostic about system form.

The method also has well-documented limitations. Ill-conditioning and instability arise when the inferred operators are weakly identified, when derivative estimates are noisy, or when the training window does not adequately span future dynamics (Sawant et al., 2021, Hedayat et al., 11 Feb 2026). Static linear POD bases can filter out small-scale or high-frequency dynamics, which is acceptable for statistical quantities of interest in some problems but limits pointwise predictive accuracy in others (Gahr et al., 2024). High-order polynomial terms increase parameter counts rapidly, making regularization, sparsity, or manifold-based compression increasingly important (Geelen et al., 2022).

Current directions reflect these challenges. They include energy-preserving and skew-symmetric parameterizations for quadratic nonlinearities (Gkimisis et al., 13 Mar 2025), adaptive basis and operator updates (Hedayat et al., 11 Feb 2026), nonlinear manifold embeddings (Geelen et al., 2022), domain-decomposed coupling (Moore et al., 2024, Rodriguez et al., 6 Sep 2025, Tezaur et al., 20 Nov 2025), and neural generalizations such as "NN-OpInf: an operator inference approach using structure-preserving composable neural networks" (Parish et al., 9 Mar 2026), which replaces polynomial blocks by structured neural operator modules while preserving local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation (Parish et al., 9 Mar 2026).

In this broader trajectory, non-intrusive OpInf has developed from a reduced least-squares regression technique into a general framework for learning structured reduced dynamics from data. Its distinguishing feature remains unchanged: reduced operators are learned directly from trajectories, without projecting inaccessible full-order operators, yet with a model form that remains recognizably tied to the underlying governing equations (Benner et al., 2020, Gahr et al., 2024).