Operator Inference (OpInf) Overview

• Operator Inference (OpInf) is a reduced-order modeling framework that learns low-dimensional operators from trajectory data using linear or constrained regression while retaining a prescribed algebraic model class.
• It employs data compression (e.g., POD/PCA) and regression over linear and quadratic features to infer operators without accessing full-order models, with special adaptations for mechanical, parametrized, and algebraic systems.
• Recent developments focus on enforcing physical and stability constraints, localizing the inference process, and extending the method to parameterized PDEs and adaptive, coupled frameworks for enhanced practical performance.

Operator Inference (OpInf) is a non-intrusive reduced-order modeling methodology that learns low-dimensional operators directly from trajectory data while retaining a prescribed algebraic model class. In the works considered here, OpInf is used to infer reduced continuous-time polynomial systems, second-order mechanical models, affine-parametric reduced operators, and even reduced models for algebraic matrix equations, typically after POD or PCA compression of state snapshots and without direct access to the full-order operators of the high-fidelity model (McQuarrie et al., 2021, Koike et al., 2024, Filanova et al., 2022, Benner et al., 7 Jul 2025).

1. Core concept and model classes

A common OpInf ansatz in the cited literature is a reduced continuous-time polynomial system of the form

$$\dot{\widehat{\mathbf q}}(t) = \widehat{\mathbf c} + \widehat{\mathbf A}\widehat{\mathbf q}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf q}(t)\otimes \widehat{\mathbf q}(t)\big) + \widehat{\mathbf B}\mathbf u(t),$$

or, in autonomous quadratic form,

$$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$

This model class appears in studies of polynomial reduced dynamics, chaotic forecasting, energy-preserving quadratic systems, and turbulence surrogates (Almeida et al., 2022, Koike et al., 2024, Gahr et al., 2024).

The method is not restricted to first-order state-space models. For mechanical systems, several papers formulate OpInf directly in second-order form,

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$

or, in a mass-normalized variant,

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$

This preserves the native second-order structure of structural and constrained mechanical models instead of rewriting them in first-order companion form (Filanova et al., 2022, Benner et al., 7 Jul 2025).

Output maps can also be learned. In the Hasegawa–Wakatani and constrained-mechanical formulations, the reduced model is paired with either a quadratic output map,

$$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$

or a linear second-order mechanical output map,

$$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$

so OpInf acts on both latent dynamics and observable prediction (Gahr et al., 2024, Benner et al., 7 Jul 2025).

These formulations define OpInf less by one fixed equation than by a recurring principle: the reduced model class is specified first, and its operators are then inferred from data by linear or constrained regression. This suggests that OpInf is best understood as a structured operator-learning framework rather than as a single reduced model architecture.

2. Data pipeline and regression structure

The standard pipeline begins with state trajectory data generated by a full-order solver. Snapshot matrices such as

$$\mathbf X= \begin{bmatrix} \mathbf x(t_1)&\cdots&\mathbf x(t_K) \end{bmatrix}, \qquad \dot{\mathbf X}= \begin{bmatrix} \dot{\mathbf x}(t_1)&\cdots&\dot{\mathbf x}(t_K) \end{bmatrix},$$

are compressed by POD or PCA, usually via SVD, to obtain a basis $\mathbf V_r$ or $\mathbf V$. Reduced coordinates are then formed by projection,

$$\hat{\mathbf x}_i=\mathbf V_r^\top \mathbf x(t_i), \qquad \dot{\hat{\mathbf x}}_i=\mathbf V_r^\top \dot{\mathbf x}(t_i),$$

and the reduced operators are inferred from these projected data (Koike et al., 2024, McQuarrie et al., 2021, Almeida et al., 2022).

For quadratic OpInf, the least-squares problem is typically posed using a data matrix built from linear and quadratic reduced features. One representative formulation is

$$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$0

with

$$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$1

where $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$2 contains the Kronecker products $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$3 (Koike et al., 2024). To eliminate monomial redundancy, some implementations replace the full Kronecker product by a compact representation based on $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$4, reducing the number of quadratic unknowns from $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$5 per row to $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$6 per row (Koike et al., 2024).

The same logic extends to richer libraries. In continuous-time polynomial OpInf for chaotic systems, the regressor may include a constant block, linear block, quadratic block, and input block,

$$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$7

with ridge-type regularization on the inferred constant, linear, quadratic, and input operators (Almeida et al., 2022).

A recurring numerical issue is derivative availability. Some papers assume that $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$8 and $$\dot{\widehat{\mathbf x}}(t) = \widehat{\mathbf A}\widehat{\mathbf x}(t) + \widehat{\mathbf H}\big(\widehat{\mathbf x}(t)\otimes \widehat{\mathbf x}(t)\big).$$9 are directly available from the full solver, as in mechanical and DAE settings, while others state explicitly that reduced derivatives may be computed by finite differences or spline interpolation (Benner et al., 7 Jul 2025, Filanova et al., 2022, Almeida et al., 2022). The cited works repeatedly identify derivative quality as practically important, especially in continuous-time OpInf for chaotic or turbulent dynamics.

3. Structure preservation and stability-oriented variants

A central theme in recent OpInf work is that unconstrained least-squares fitting does not automatically preserve physical structure. This is explicit in energy-preserving quadratic systems, where standard OpInf may fit the data well yet fail to recover the algebraic identity

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$0

For reduced operators with the symmetry convention $$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$1, the energy-preserving condition is equivalent to

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$2

and EP-OpInf enforces these constraints directly in the regression problem (Koike et al., 2024).

The same line of work also introduces a compressed quadratic representation $$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$3 and a constrained least-squares formulation in which the energy-preserving constraints are written in the compressed coordinates. On viscous Burgers’ equation and the Kuramoto–Sivashinsky equation, EP-OpInf retained the exact energy-preserving structure to machine precision while remaining comparable in reconstruction accuracy to intrusive POD-Galerkin and standard OpInf (Koike et al., 2024).

A complementary development appears in the analysis of energy-preserving quadratic operators via skew-symmetric parameterizations. Building on an earlier proposal by Goyal et al., 2023, one paper proves that every energy-preserving quadratic term can be equivalently formulated using a parameterization via skew-symmetric matrix blocks, develops an algorithm to compute such a representation from an arbitrary energy-preserving operator, and proposes a sequential formulation by linear least-squares problems for OpInf so that the inferred quadratic operator remains energy-preserving (Gkimisis et al., 13 Mar 2025). The reported 2D Burgers benchmark indicates accurate inferred dynamics and operators faithful to the underlying physical properties of the system (Gkimisis et al., 13 Mar 2025).

A second stability-oriented line regularizes the quadratic operator norm itself. For autonomous quadratic ROMs, a Lyapunov-based radius estimate

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$4

motivates the penalty $$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$5, since the estimate scales inversely with the quadratic operator norm. This yields physics-informed regularized OpInf and, with the additional semidefinite constraint

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$6

a structure-preserving variant that enforces symmetric negative definiteness of the linear operator (Sawant et al., 2021).

Mechanical and constrained-mechanical variants pursue analogous goals with different structures. For constrained DAEs, one formulation enforces

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$7

through semidefinite programming, with the stated purpose of preserving stability and interpretability of the learned mass, damping, and stiffness operators (Benner et al., 7 Jul 2025). Taken together, these developments suggest that structure preservation in OpInf serves both as a physical inductive bias and as a way to retain operator meaning after non-intrusive learning.

4. Parametric and operator-space generalizations

One major extension embeds parameter dependence directly into the reduced model. In affine-parametric OpInf for time-dependent PDEs, the reduced dynamics are written as

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$8

so the known affine parameter dependence of the PDE is embedded directly into the regression matrix rather than handled by post hoc interpolation (McQuarrie et al., 2021).

A different parametric strategy trains separate ROMs at sampled parameter points and interpolates the reduced operators afterward. In the PECVD purging study, a global POD basis was constructed from all training snapshots, a separate reduced model was inferred at each training parameter, and SciPy’s LinearNDInterpolator was used to interpolate reduced operator entries across the two-dimensional parameter space (Kang et al., 4 Apr 2025). Because all operators act in one shared reduced coordinate system, no basis alignment or Grassmann interpolation is required (Kang et al., 4 Apr 2025).

OpInf has also been adapted to algebraic matrix equations. For parameter-dependent Lyapunov and Riccati equations, the matrix equations are vectorized and rewritten in a polynomial structured representation,

$$\widehat{M}\,\ddot{\widehat{x}}(t)+\widehat{D}\,\dot{\widehat{x}}(t)+\widehat{K}\,\widehat{x}(t)=\widehat{B}\,u(t),$$9

after which a reduced model

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$0

is learned from solution snapshots by linear least squares (Wen et al., 20 Nov 2025). Here the state itself, rather than a time derivative, becomes the regression target, which is a substantial change in the role OpInf plays.

A more recent systems-oriented extension uses standard OpInf as the numerical core of a language-driven PDE-solving framework. In OpInf-LLM, a shared POD basis is built across training parameters, reduced operators are fitted at those parameters, and unseen-parameter operators are obtained by regression or interpolation in operator space while an LLM handles task parsing and tool orchestration (Wang et al., 2 Feb 2026). This suggests that OpInf’s explicit operator structure is also useful outside conventional ROM deployment, particularly when one wants parameterized reduced models to be called reliably inside larger agentic workflows.

5. Mechanical, DAE, and matrix-structured formulations

Mechanical systems motivate some of the most specialized OpInf variants. For linear second-order mechanical dynamics,

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$1

one paper argues that OpInf should infer a reduced model directly in this second-order form rather than after a first-order lifting, since the second-order formulation preserves physical interpretability and avoids state-dimension doubling (Filanova et al., 2022). The corresponding reduced regression uses projected displacement, velocity, and acceleration snapshots and can be written compactly as

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$2

with $$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$3 (Filanova et al., 2022).

When complete external force data are available, the same paper proposes a force-informed constrained version that infers explicit reduced $$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$4, $$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$5, and $$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$6 by solving a semidefinite program subject to

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$7

implemented in YALMIP with SeDuMi (Filanova et al., 2022). The paper also proves asymptotic closeness of the inferred operators to the intrusive POD-reduced operators under assumptions on time discretization convergence and full rank of the data matrix (Filanova et al., 2022).

Constrained mechanical systems governed by DAEs lead to a related but distinct extension. For proper index-2 and index-3 mechanical DAEs,

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$8

one approach uses the hidden-manifold viewpoint: the DAE admits an underlying second-order ODE realization on the constraint manifold, and OpInf is applied directly to that reduced ODE in the compressed subspace (Benner et al., 7 Jul 2025). Basis construction depends on constraint type: displacement snapshots are used for position constraints, velocity snapshots for velocity constraints, so that the reduced basis itself lies in the relevant manifold (Benner et al., 7 Jul 2025).

The same general idea reappears in matrix-equation settings, although there the learned object is not a time-evolution law but a reduced algebraic operator relation. This suggests that, within the bounds of the prescribed operator class, OpInf can be understood as learning reduced operators for equations as well as for dynamical systems.

6. Localized, coupled, and adaptive OpInf

Several papers move beyond a single global reduced model. For systems with spatially localized slow singular value decay, one framework decomposes the domain into an OpInf subdomain with fast singular value decay and an sFOM subdomain with slow singular value decay. On the ROM side, OpInf learns reduced self-dynamics together with coupling terms driven by interface full-order states, while the difficult subdomain remains full-order but sparsely inferred row-by-row (Gkimisis et al., 8 Jan 2025). A Gershgorin-based regularization with a closed-form solution is introduced to promote stable OpInf and sFOM models (Gkimisis et al., 8 Jan 2025).

Another line embeds OpInf into Schwarz domain decomposition. In overlapping Schwarz coupling, local OpInf ROMs are used as interchangeable subdomain solvers and are coupled either to finite-element subdomains or to other OpInf subdomains through iterative interface exchange (Moore et al., 6 Oct 2025, Tezaur et al., 20 Nov 2025). The local reduced models include explicit boundary-input operators so that Schwarz boundary data enter as learned forcing terms rather than as intrusive reduced operators (Tezaur et al., 20 Nov 2025).

Non-overlapping Schwarz variants extend this idea to alternating Dirichlet–Neumann and Robin–Robin transmission conditions. In that setting, local second-order OpInf ROMs are trained with interface traction and displacement data, and, for Robin–Robin coupling, additional reduced operators are inferred for Robin stiffness and force effects (Rodriguez et al., 6 Sep 2025). The paper’s stated novelty is the first application of non-overlapping Schwarz alternating methods to coupling non-intrusive OpInf ROMs both with FOMs and with other OpInf ROMs (Rodriguez et al., 6 Sep 2025).

Adaptive OpInf pushes the same localization idea into time. Instead of fixing the basis and reduced operators offline, Adaptive OpInf uses a moving data window

$$\ddot{\hat{\mathbf{x}}}(t)+\widehat{E}_M\dot{\hat{\mathbf{x}}}(t)+\widehat{K}_M\hat{\mathbf{x}}(t)=\widehat{B}_M\mathbf{u}(t).$$9

updates the POD basis from the current window, and re-fits reduced operators by regularized least squares at periodic adaptation events (Hedayat et al., 11 Feb 2026). Because the method requires periodic one-step FOM queries during deployment, the same paper emphasizes that predictive claims for adaptive ROMs must be cost-aware and must report online adaptation budgets explicitly (Hedayat et al., 11 Feb 2026).

Taken together, these works suggest that OpInf is increasingly used as a modular local solver: a reduced model can now be attached to a subdomain, an interface, or an online adaptation window, rather than only to a monolithic global state space.

7. Applications, empirical behavior, and limitations

The empirical range of OpInf is broad. In chaotic forecasting, continuous-time quadratic OpInf was applied to Lorenz–96 and the Kuramoto–Sivashinsky equation, where the reported Valid Prediction Time statistics for Lorenz–96 were $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$0 on average for $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$1 and $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$2 for $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$3, with corresponding maxima $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$4 and $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$5, and the authors compared these favorably with recurrent-network baselines reported by Vlachas et al. (Almeida et al., 2022). For Hasegawa–Wakatani plasma turbulence, OpInf ROMs trained on $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$6 time units were used for prediction over $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$7 additional time units, captured statistical features of particle flux and resistive dissipation, and reduced computational effort by up to five orders of magnitude (Gahr et al., 2024).

In parametric CFD for semiconductor manufacturing, nine training ROMs inferred from $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$8 of the data were interpolated across a $$\hat y(t)=\hat{\mathbf C}\hat q(t)+\hat{\mathbf G}\big(\hat q(t)\otimes \hat q(t)\big),$$9 parameter grid; the reported maximum average relative state error over all tested cases was $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$0, and the reported online speedup factor relative to CFD was $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$1 (Kang et al., 4 Apr 2025). In constrained mechanical DAEs, OpInf reduced an index-3 example from $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$2 to $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$3 dimensions and an index-2 example from $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$4 to $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$5, while maintaining accurate outputs on both training and unseen harmonic test inputs (Benner et al., 7 Jul 2025). In hybrid Schwarz coupling for nonlinear 3D solid dynamics, overlapping OpInf–OpInf coupling achieved speedups of up to $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$6 relative to the corresponding FOM–FOM coupling (Tezaur et al., 20 Nov 2025). In OpInf-LLM, the reported overall execution success rate was $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$7, with relative $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$8 errors $$y(t)=\widehat{C}_v \dot{\widehat{x}}(t)+\widehat{C}_p \widehat{x}(t),$$9 for the heat equation, $$\mathbf X= \begin{bmatrix} \mathbf x(t_1)&\cdots&\mathbf x(t_K) \end{bmatrix}, \qquad \dot{\mathbf X}= \begin{bmatrix} \dot{\mathbf x}(t_1)&\cdots&\dot{\mathbf x}(t_K) \end{bmatrix},$$0 for Burgers’ equation, and $$\mathbf X= \begin{bmatrix} \mathbf x(t_1)&\cdots&\mathbf x(t_K) \end{bmatrix}, \qquad \dot{\mathbf X}= \begin{bmatrix} \dot{\mathbf x}(t_1)&\cdots&\dot{\mathbf x}(t_K) \end{bmatrix},$$1 for cavity flow (Wang et al., 2 Feb 2026).

The limitations are equally consistent across the cited works. Continuous-time OpInf depends on derivative data or derivative estimation, and several papers note the sensitivity of learned models to derivative quality, regularization choice, and training-data design (Almeida et al., 2022, Gahr et al., 2024, Sawant et al., 2021). Global linear bases can be ineffective when spatially localized transport produces slow singular value decay, which is one of the main motivations for localized OpInf–sFOM and domain-decomposition variants (Gkimisis et al., 8 Jan 2025). Parameterized formulations can be ill-posed when the parameter-feature matrices lack rank, and one matrix-equation study states explicitly that deriving practical sampling criteria remains open (Wen et al., 20 Nov 2025). Adaptive OpInf improves robustness but requires periodic high-fidelity access and explicit online budget accounting (Hedayat et al., 11 Feb 2026).

These papers therefore present a technically coherent picture. OpInf is consistently non-intrusive in the sense of learning reduced operators from data rather than projecting inaccessible full-order operators, but it is not uniformly plug-and-play. Its practical success depends on operator-class selection, basis quality, excitation richness, regularization, and—when structure preservation matters—carefully imposed algebraic constraints. This suggests that contemporary OpInf is less a single algorithm than a family of data-driven reduced-operator identification procedures, differentiated by the structures they preserve and by the numerical regimes in which they are deployed.