Operator inference (OpInf) learns low-dimensional reduced-order models of high-dimensional dynamical systems by solving a linear least-squares problem for the reduced linear and polynomially nonlinear operators, while treating analytically known non-polynomial nonlinear terms on the right-hand side; it requires only snapshot data from a possibly black-box full-order solver. Benner, Goyal, Kramer, Peherstorfer, and Willcox (2020) proved that for systems with spatially local non-polynomial nonlinearities the learned operators converge to their intrusive POD-Galerkin counterparts as the reduced dimension grows and the data time step shrinks — but without a convergence rate — and reported three unresolved empirical phenomena: the operator error need not decrease monotonically at low reduced orders (attributed in part to least-squares conditioning), learned models can lose stability at higher reduced orders when training data are insufficient, and preserving the coupling structure of coupled systems empirically stabilizes nonlinear learned models without any theoretical justification. Work since 2020 has supplied method-level answers to the engineering items among the paper's open issues (measurement noise, regularization, embedding of physical constraints), but the theoretical core is untouched: no explicit convergence rates or error bounds for realistic trajectory data exist, there is no characterization of the relationship between data richness, reduced dimension, and stability, and the nonlinear coupled-structure stabilization effect remains unjustified beyond the linear case. The problem is therefore the missing predictive theory of accuracy and stability for non-intrusively learned nonlinear reduced models.

Background

Large-scale simulations of systems governed by partial differential equations produce full-order models (FOMs) with millions of state variables, which motivates model order reduction: replacing the FOM by a reduced-order model (ROM) of dimension rr that reproduces its dynamics at a small fraction of the cost. The classical intrusive route computes a reduced basis by proper orthogonal decomposition (POD) — a singular-value decomposition of a matrix of state snapshots that extracts the dominant spatial patterns — and then projects the discretized FOM operators onto that basis by Galerkin projection. This requires access to the assembled discretized operators, which is impossible when the governing equations are hidden inside a proprietary or legacy solver; only simulation snapshots may be available. Operator inference (OpInf), introduced by Peherstorfer and Willcox (2016) for systems with polynomial nonlinearities, is a non-intrusive alternative: it learns the reduced operators directly from snapshot data by a linear least-squares fit, because for polynomial dynamics the reduced equations are linear in the unknown reduced operators. Benner, Goyal, Kramer, Peherstorfer, and Willcox (2020) extended OpInf to semi-discrete systems (ordinary-differential-equation models obtained by discretizing a PDE in space while leaving time continuous) of the form s˙(t)=As(t)+H(s(t)⊗s(t))+f(t,s(t))+Bu(t)\dot{s}(t) = A s(t) + H(s(t)\otimes s(t)) + f(t, s(t)) + B u(t), where ⊗\otimes denotes the Kronecker product so that the quadratic operator HH collects all second-order terms in the state, and the non-polynomial term ff is spatially local (each component depends only on state components at the same or nearby spatial grid points) and is given in analytic form; the known term ff is moved to the right-hand side of the least-squares system, so the remaining reduced operators A^,H^,B^\hat A, \hat H, \hat B (hats mark reduced quantities) are still learned by a linear least-squares problem, and the reduced nonlinear term is evaluated efficiently by discrete empirical interpolation (DEIM), a method that approximates a nonlinear vector field from a few interpolated components. This avoids the lifting transformations of Qian, Kramer, Marques, and Willcox (2019) and Khodabakhshi and Willcox (2022), which recast certain non-polynomial systems into polynomial form at the price of enlarging the state. The paper's Theorem 1 shows that, under mild assumptions on the time-stepping scheme and for sufficiently small data time step, the learned operators converge to the operators of the intrusive POD-Galerkin ROM as the reduced dimension rr approaches the FOM dimension nn — a consistency result with no convergence rate. The numerical experiments then expose three gaps between this theorem and practice: (i) the error between learned and projected operators does not decrease monotonically at low reduced orders (the tubular reactor example, attributed in part to the conditioning of the least-squares problem); (ii) learned ROMs become unstable beyond a moderate reduced order (r=14r=14 in the Chafee–Infante diffusion–reaction example) when the training data stay fixed, i.e., stability inheritance from the FOM fails with insufficient data; and (iii) for the coupled batch-chromatography system, ROMs that do not preserve the block coupling structure of the components can be unstable while structure-preserving ROMs are stable, an observation for which — in contrast to the linear theory of Reis and Stykel (2007) — no theoretical justification exists for nonlinear systems. The conclusions section additionally lists measurement noise, problem-dependent regularization, and the inclusion of physical constraints in the least-squares problem as future research directions. Subsequent literature has supplied method-level answers to these engineering directions: physics-informed regularizers and constrained formulations (Sawant, Kramer, and Peherstorfer 2023), stability-by-construction parameterizations of quadratic models (Goyal, Pontes Duff, and Benner 2024, 2025), embedded state constraints (Kim and Kramer 2025), noise-aware active sampling and rollout-based training (Uy, Wang, Wen, and Peherstorfer 2023; Uy, Hartmann, and Peherstorfer 2023), and coupling- or block-structure-preserving multiphysics formulations (Zastrow et al. 2026; Zhang et al. 2026). However, the guarantees in this literature are confined to specific model classes (quadratic, Hamiltonian as in Gruber and Tezaur 2023, or Parish, Gruber, Blonigan, and Tezaur's composable structure-preserving parameterizations) or to actively designed training data (Rosenberger, Sanderse, and Stabile 2025), and the 2024 review by Kramer, Peherstorfer, and Willcox still describes error estimates for OpInf as a research pathway rather than an obtained result. What remains is the theory gap itself.

Problem Statement

The problem is the missing predictive theory of accuracy and stability for operator inference with non-polynomial nonlinear terms, posed by the unresolved issues of Benner, Goyal, Kramer, Peherstorfer, and Willcox (2020). Retaining the framework at its source generality — semi-discrete dynamical systems whose non-polynomial nonlinearities are spatially local and analytically known, learned from trajectory snapshots produced by a possibly black-box full-order solver — the problem asks for results of three linked kinds that together replace the paper's convergence-without-rate theorem and its empirical stabilization heuristics by quantifiable guarantees. (1) Accuracy: explicit convergence rates, or sharp non-asymptotic error bounds, for the inferred reduced operators relative to the intrusive POD-Galerkin operators as functions of the reduced dimension rr, the data time step, and the amount and excitation of the training trajectories; these should account for the conditioning of the least-squares operator-inference problem and explain the empirically observed non-monotone operator error at low reduced orders. (2) Stability: a theoretical characterization of when learned reduced models inherit stability from the full-order model, i.e., of the relationship between training-data richness, reduced dimension, and stability that accounts for the observed loss of stability at higher reduced orders under fixed data. (3) Coupled systems: a theoretical justification, beyond the linear theory of Reis and Stykel (2007), for the empirically observed stabilizing effect of preserving the coupling structure of coupled nonlinear systems in learned reduced models. These three components reflect the distinct unresolved theoretical issues listed in the source paper and together constitute the missing predictive theory of accuracy and stability for non-intrusively learned nonlinear reduced models. Method-level answers to the engineering items of the source (regularizers, constraint embeddings, noise-robust training variants) already exist in the literature and do not by themselves answer this problem; what is asked is when and why such learned models are accurate and stable.

References

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