Adaptive NiTROM: Online Reduced-Order Modeling

• Adaptive NiTROM is an online reduced-order modeling method that updates both the latent subspace and reduced dynamics in real time.
• The approach uses joint Riemannian optimization on the product manifold to refine oblique encoder/decoder pairs and polynomial operators for enhanced energy tracking.
• It demonstrates robustness against amplitude drift and instability in transient flows, particularly in cases like the lid-driven cavity test, and offers a reliable hybrid alternative.

Searching arXiv for the cited Adaptive NiTROM paper and closely related ROM papers. Adaptive NiTROM is an online, non-intrusive reduced-order modeling formulation in which both the latent subspace and the reduced dynamics are updated during deployment rather than fixed offline. In the formulation studied in "Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges" (Hedayat et al., 11 Feb 2026), it extends static NiTROM by refitting, on a moving data window, an oblique encoder/decoder pair and polynomial reduced operators through joint Riemannian optimization on a product manifold. The method is positioned against static projection-based reduced-order models and against Adaptive OpInf, with the central claim that static surrogates drift or destabilize once the system leaves the training manifold, whereas online adaptation can suppress amplitude drift, preserve physical coherence, and maintain bounded energy when supported by explicit online budgets and periodic full-order-model queries (Hedayat et al., 11 Feb 2026).

1. Conceptual placement within non-intrusive ROMs

Non-intrusive ROMs are projection-based reduced-order models constructed purely from data, namely snapshots of states and inputs, without access to full-order-model residuals, Jacobians, or operators. Their standard ingredients are a decoder from latent to full state, an encoder from full state to latent coordinates, and a learned latent dynamics. Within this family, Operator Inference learns structured polynomial reduced dynamics from reduced data via least squares and typically uses an orthonormal POD basis for both encoder and decoder. Static NiTROM increases expressiveness by jointly optimizing an oblique encoder/decoder pair and polynomial reduced operators against full-state trajectory mismatch in physical space. Adaptive NiTROM carries that construction into the online setting by periodically updating both subspaces and operators on a moving window (Hedayat et al., 11 Feb 2026).

Formulation	Update mechanism	Reported trade-off
Adaptive OpInf	Sequential basis/operator refits	Robust, simple, low cost
Adaptive NiTROM	Joint Riemannian optimization	Near-exact energy tracking under frequent updates; sensitive to initialization
Hybrid	OpInf update followed by NiTROM refinement	Most reliable under regime changes and minimal offline data

The motivating failure mode is explicit. Static ROMs rely on a fixed subspace and fixed reduced operators identified offline. In transient, non-normal, or regime-changing flows, low-energy but dynamically relevant features may be truncated by orthogonal projections, and the learned reduced operators may cease to represent the evolving dynamics. The reported consequences are amplitude drift, phase errors, and instability beyond the training horizon. This suggests that Adaptive NiTROM should be understood less as a replacement for offline model reduction than as a self-correcting continuation mechanism for deployment beyond the initial manifold (Hedayat et al., 11 Feb 2026).

2. Mathematical structure

The full-order state is $x(t) \in \mathbb{R}^n$, and the reduced latent state is $z(t) \in \mathbb{R}^r$ with $r \ll n$. In the reported formulation, the encoders and decoders are linear; no nonlinear autoencoder is used. Let the decoder basis be $\Phi \in \mathbb{R}^{n \times r}$ and the encoder basis be $\Psi \in \mathbb{R}^{n \times r}$. The encoder is

$\psi = \Psi^T,$

and the decoder is

$\phi = \Phi(\Psi^T \Phi)^{-1}.$

The resulting oblique projector is

$P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$

Orthogonal projection is recovered when $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$. The distinctive feature of NiTROM is that it allows $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$, so the trial and test spaces need not coincide (Hedayat et al., 11 Feb 2026).

The reduced dynamics is polynomial up to quadratic order: $z(t) \in \mathbb{R}^r$0 with

$z(t) \in \mathbb{R}^r$1

For the lid-driven cavity study, a quadratic model with no cubic terms is used. No explicit energy-preserving structure is enforced in the experiments, although the formulation allows such structure to be incorporated through constraints or penalties. Prediction proceeds by initializing with

$z(t) \in \mathbb{R}^r$2

integrating the reduced dynamics forward in time, and lifting through

$z(t) \in \mathbb{R}^r$3

Online adaptation is posed as a windowed trajectory-mismatch minimization. A sliding window collects $z(t) \in \mathbb{R}^r$4 snapshots every $z(t) \in \mathbb{R}^r$5 reduced time steps: $z(t) \in \mathbb{R}^r$6 The online objective is

$z(t) \in \mathbb{R}^r$7

where $z(t) \in \mathbb{R}^r$8. In practice, the paper uses ridge regularization for $z(t) \in \mathbb{R}^r$9, and no hard structure constraints are enforced. The reduced model is integrated in continuous time with the same $r \ll n$0 used for the full-order model; any standard ODE integrator is acceptable, and the experiments use continuous-time reduced ODEs integrated numerically (Hedayat et al., 11 Feb 2026).

3. Manifold optimization and online update cycle

Adaptive NiTROM optimizes over a product manifold whose factors reflect the geometry of the latent spaces and the Euclidean structure of the polynomial operators. The decoder subspace $r \ll n$1 lies on the Grassmann manifold $r \ll n$2, the encoder $r \ll n$3 lies on the Stiefel manifold $r \ll n$4 with orthonormal columns, and $r \ll n$5 lies in Euclidean space. For $r \ll n$6, if $r \ll n$7 denotes the ambient gradient, the Riemannian gradient on the Stiefel manifold is

$r \ll n$8

with

$r \ll n$9

For a Grassmann representative $\Phi \in \mathbb{R}^{n \times r}$0 spanning $\Phi \in \mathbb{R}^{n \times r}$1, the Grassmann Riemannian gradient is

$\Phi \in \mathbb{R}^{n \times r}$2

Updates are performed jointly or alternately on $\Phi \in \mathbb{R}^{n \times r}$3 Euclidean factors, followed by retractions such as QR-based retraction to map iterates back to the manifold (Hedayat et al., 11 Feb 2026).

The online protocol is organized by update cadence $\Phi \in \mathbb{R}^{n \times r}$4, window size $\Phi \in \mathbb{R}^{n \times r}$5, and optimization depth $\Phi \in \mathbb{R}^{n \times r}$6. Between two adaptation events, the ROM is advanced for $\Phi \in \mathbb{R}^{n \times r}$7 steps: $\Phi \in \mathbb{R}^{n \times r}$8 At the next adaptation time, the latest ROM state is lifted and the full-order model is advanced by one step to obtain a high-quality snapshot $\Phi \in \mathbb{R}^{n \times r}$9, which updates the moving window. The adaptation stage then either warm-starts directly from the previous $\Psi \in \mathbb{R}^{n \times r}$0 or uses a hybrid initialization: compute a windowed POD basis $\Psi \in \mathbb{R}^{n \times r}$1, solve an OpInf least-squares problem for $\Psi \in \mathbb{R}^{n \times r}$2, and initialize NiTROM with $\Psi \in \mathbb{R}^{n \times r}$3. A few Riemannian gradient steps then refine $\Psi \in \mathbb{R}^{n \times r}$4 jointly (Hedayat et al., 11 Feb 2026).

The hybrid initialization is mathematically explicit. Holding $\Psi \in \mathbb{R}^{n \times r}$5 fixed and taking the encoder as $\Psi \in \mathbb{R}^{n \times r}$6, the reduced least-squares problem is

$\Psi \in \mathbb{R}^{n \times r}$7

with

$\Psi \in \mathbb{R}^{n \times r}$8

This construction matters because the paper reports that pure Adaptive NiTROM is sensitive to initialization and optimization depth, whereas the hybrid often places the manifold optimization in a better local region (Hedayat et al., 11 Feb 2026).

4. Computational budgets and cost-aware deployment

A defining feature of the framework is that adaptation is budgeted explicitly. The quantities $\Psi \in \mathbb{R}^{n \times r}$9, $\psi = \Psi^T,$0, $\psi = \Psi^T,$1, and $\psi = \Psi^T,$2 are not merely hyperparameters; they determine both accuracy and online cost. Projection, lifting, and windowing scale as $\psi = \Psi^T,$3. ROM integration between updates scales as $\psi = \Psi^T,$4 for polynomial degree $\psi = \Psi^T,$5. For Adaptive OpInf, the basis update by windowed SVD scales as $\psi = \Psi^T,$6, or $\psi = \Psi^T,$7 with incremental SVD, and the overall per-update cost is dominated by

$\psi = \Psi^T,$8

For Adaptive NiTROM, $\psi = \Psi^T,$9 Riemannian iterations, each with one forward and one adjoint reduced integration over $\phi = \Phi(\Psi^T \Phi)^{-1}.$0 snapshots, give

$\phi = \Phi(\Psi^T \Phi)^{-1}.$1

while manifold linear algebra contributes

$\phi = \Phi(\Psi^T \Phi)^{-1}.$2

The total per-update cost is therefore

$\phi = \Phi(\Psi^T \Phi)^{-1}.$3

The hybrid cost is the sum of the OpInf stage and the NiTROM refinement. In all cases, each update assumes one full-order-model step at cost $\phi = \Phi(\Psi^T \Phi)^{-1}.$4, with $\phi = \Phi(\Psi^T \Phi)^{-1}.$5, and this can be dominant in practical CFD (Hedayat et al., 11 Feb 2026).

The paper also reports wall-clock timings averaged at $\phi = \Phi(\Psi^T \Phi)^{-1}.$6 on an Apple M3 Pro: ROM step $\phi = \Phi(\Psi^T \Phi)^{-1}.$7 ms, FOM one-step $\phi = \Phi(\Psi^T \Phi)^{-1}.$8 ms, windowed SVD $\phi = \Phi(\Psi^T \Phi)^{-1}.$9 ms, OpInf least squares $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$0 ms, and one NiTROM iteration $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$1 ms. This suggests that the practical attractiveness of Adaptive NiTROM depends strongly on keeping $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$2 modest or using the hybrid mode, since a few well-initialized manifold iterations may be preferable to a deeper optimization from a poor warm start (Hedayat et al., 11 Feb 2026).

The article’s methodological argument is broader than asymptotic complexity. Predictive claims for ROMs are said to require clear separation of training, adaptation, and deployment regimes, together with explicit reporting of online budgets and full-order-model queries. The required bookkeeping includes the number of updates,

$P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$3

the total number of FOM queries, which equals $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$4, and both per-update and total online wall-clock cost. In this framework, “adaptive” does not mean free of high-fidelity information; it means that the model is periodically anchored by one FOM step per update (Hedayat et al., 11 Feb 2026).

5. Reported performance on the transiently perturbed lid-driven cavity

The principal test problem is a transiently perturbed lid-driven cavity flow at $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$5. The full-order system is the 2D incompressible Navier–Stokes equations on a $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$6 staggered grid with $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$7 unknowns, forced near the upper-right corner by $P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$8 with

$P = \Phi(\Psi^T \Phi)^{-1}\Psi^T, \qquad P^2 = P.$9

The FOM time step is $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$0 using two-step Adams–Bashforth, and the output is the full velocity field. Performance is assessed through the energy

$$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$1

the global field error

$$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$2

and qualitative inspection of velocity slices and vorticity or velocity fields. The ROM configuration uses reduced dimension $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$3, quadratic reduced dynamics with no cubic terms, and linear encoder/decoder maps only (Hedayat et al., 11 Feb 2026).

Three online scenarios are studied. Case 1, labeled rich training, trains on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$4 and tests on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$5. Case 2, regime change, trains on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$6 and tests on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$7. Case 3, minimal training, trains on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$8 and tests on $$\operatorname{span}(\Phi)=\operatorname{span}(\Psi)$$9. Across these cases, static Galerkin, static OpInf, and static NiTROM reproduce training oscillations but their energy grows beyond training and they diverge under regime change or minimal training. Adaptive OpInf robustly suppresses amplitude drift, though with mild over-damping and occasional small numerical wiggles in fields. Adaptive NiTROM with frequent updates can achieve near-exact energy tracking and coherent fields with a small phase lag, but under sparse updates or abrupt regime change it can be initialization-limited and may offer limited improvement over static models when $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$0 is small. The hybrid OpInf-initialized NiTROM is reported as the most reliable under regime changes and minimal offline data, producing bounded energy, coherent fields, and reduced artifacts (Hedayat et al., 11 Feb 2026).

The specific parameter settings make the sensitivity concrete. In Case 1, a frequent-update configuration $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$1, $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$2, $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$3 yields near-exact energy tracking for Adaptive NiTROM. A sparse configuration $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$4, $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$5, $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$6 causes Adaptive NiTROM to struggle, while a hybrid with $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$7 restores near-perfect energy. In Cases 2 and 3, the setting $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$8, $$\operatorname{span}(\Phi)\neq \operatorname{span}(\Psi)$$9, $z(t) \in \mathbb{R}^r$00 leads Adaptive OpInf to underestimate amplitude, Adaptive NiTROM to offer limited gains alone, and the hybrid to best balance amplitude and phase while removing OpInf wiggles. For Case 1, the test horizon $z(t) \in \mathbb{R}^r$01 with $z(t) \in \mathbb{R}^r$02 yields $z(t) \in \mathbb{R}^r$03 online steps, so $z(t) \in \mathbb{R}^r$04 requires $z(t) \in \mathbb{R}^r$05 FOM queries and $z(t) \in \mathbb{R}^r$06 requires $z(t) \in \mathbb{R}^r$07 (Hedayat et al., 11 Feb 2026).

A recurring point is that update cadence $z(t) \in \mathbb{R}^r$08, lookback $z(t) \in \mathbb{R}^r$09, and optimization depth $z(t) \in \mathbb{R}^r$10 interact nontrivially. Adaptive OpInf is robust over a wide range of $z(t) \in \mathbb{R}^r$11, whereas Adaptive NiTROM is accurate mainly for small $z(t) \in \mathbb{R}^r$12 unless optimization depth is increased or hybrid warm-starting is used. The effective horizon $z(t) \in \mathbb{R}^r$13 must capture at least one characteristic timescale, and too small an $z(t) \in \mathbb{R}^r$14 leads to poor conditioning and weak performance. The paper notes that in the cavity test, NiTROM often converged with smaller $z(t) \in \mathbb{R}^r$15 than OpInf for the same $z(t) \in \mathbb{R}^r$16 (Hedayat et al., 11 Feb 2026).

6. Practical guidance, misconceptions, and reporting norms

The practical guidance is parameterized rather than universal. For 2D CFD problems of the reported size, the recommendation is to start with $z(t) \in \mathbb{R}^r$17–$z(t) \in \mathbb{R}^r$18 modes and verify energy tracking and field coherence. Adaptive models may tolerate slightly smaller $z(t) \in \mathbb{R}^r$19 than static ROMs because they update online. The update cadence $z(t) \in \mathbb{R}^r$20 should be chosen so the ROM remains close to the true trajectory between updates; visible drift, energy blow-up, or phase error indicates that $z(t) \in \mathbb{R}^r$21 should be decreased. The lookback $z(t) \in \mathbb{R}^r$22 should be large enough that $z(t) \in \mathbb{R}^r$23 spans at least one dominant physical period or timescale. For pure Adaptive NiTROM, larger $z(t) \in \mathbb{R}^r$24 is needed if $z(t) \in \mathbb{R}^r$25 is large; in hybrid mode, small $z(t) \in \mathbb{R}^r$26, approximately $z(t) \in \mathbb{R}^r$27–$z(t) \in \mathbb{R}^r$28, is typically sufficient and keeps cost reasonable (Hedayat et al., 11 Feb 2026).

Method choice is similarly conditional. Adaptive OpInf is preferred when budgets are tight, updates are infrequent, and the dynamics remain near prior regimes. Adaptive NiTROM is preferred when frequent updates are possible and near-exact energy tracking is required, provided that warm starts are good and $z(t) \in \mathbb{R}^r$29 is adequate. The hybrid OpInf–NiTROM is presented as the best overall option when regime changes are present or offline data are scarce. This suggests that the main practical distinction is not merely expressiveness versus simplicity, but robustness of the online optimization landscape under limited adaptation budgets (Hedayat et al., 11 Feb 2026).

Several common misconceptions are addressed by the experiments. First, Adaptive NiTROM is not intrinsically structure-preserving in the reported implementation: no explicit energy-preserving structure is enforced, even though stability or physics penalties can be added through $z(t) \in \mathbb{R}^r$30. Second, online adaptation is not free of high-fidelity dependence: each update assumes one high-quality FOM snapshot obtained by advancing the FOM one step from the lifted ROM state. Third, more optimization is not automatically better: excessively large $z(t) \in \mathbb{R}^r$31 may trade responsiveness for local overfitting, and updates may be rejected if they increase hold-out loss or violate energy bounds. The recommended safeguards are ridge regularization $z(t) \in \mathbb{R}^r$32, optional stability or physics penalties, step-size control, bounded $z(t) \in \mathbb{R}^r$33, and acceptance checks based on a hold-out subset of the window or on energy boundedness (Hedayat et al., 11 Feb 2026).

For reproducibility, the reporting standard is explicit. Training, adaptation, and deployment regimes should be separated clearly; offline training window and costs should be reported; online window parameters $z(t) \in \mathbb{R}^r$34, $z(t) \in \mathbb{R}^r$35, $z(t) \in \mathbb{R}^r$36, and reduced dimension $z(t) \in \mathbb{R}^r$37 should be specified; and total adaptation events, FOM queries, per-update costs, and total online costs should be included. Sensitivity studies over $z(t) \in \mathbb{R}^r$38, $z(t) \in \mathbb{R}^r$39, $z(t) \in \mathbb{R}^r$40, and the basis-update strategy, such as windowed SVD versus incremental SVD, are part of the expected evidence. Within that framework, Adaptive NiTROM is best viewed as a joint manifold-based mechanism for online correction of latent geometry and reduced dynamics, whose success depends as much on initialization, update cadence, and budget accounting as on the expressive power of oblique projections and trajectory-level fitting (Hedayat et al., 11 Feb 2026).