Quantized Local ROM Modeling

• Quantized local ROM is a reduced-order modeling strategy that clusters state-space data to build local POD–Galerkin models for complex, nonlinear dynamical systems.
• The method uses a three-stage process—unsupervised clustering, local SVD-based reduction, and dynamic switching—to capture heterogeneous attractor geometries.
• ql-ROM demonstrates improved short-term prediction, better statistical fidelity, and computational efficiency compared to global ROMs in turbulent flow applications.

Quantized Local Reduced Order Modelling, usually abbreviated ql-ROM, is an intrusive reduced-order modelling strategy for nonlinear dynamical systems whose trajectories evolve on a low-dimensional but geometrically heterogeneous attractor. In the formulation introduced in “Quantized local reduced-order modeling in time (ql-ROM),” the method replaces a single global reduced model with a collection of local POD–Galerkin models defined around cluster centroids, and advances the dynamics by switching in time between these local charts according to a nearest-centroid assignment rule. The approach was developed for spatiotemporally chaotic systems, including the Kuramoto–Sivashinsky equation and two-dimensional Navier–Stokes Kolmogorov flow, where a single global reduced subspace can be inaccurate, unstable, or unable to reproduce long-term statistics (Colanera et al., 16 Jun 2025).

1. Definition and conceptual basis

The central premise of ql-ROM is that many dissipative nonlinear PDEs evolve toward attractors that are lower-dimensional than the ambient state space, yet are not well represented by one globally linear subspace. The motivating observations reported for such systems are heterogeneous density on the attractor, non-Gaussian statistics, intermittent regimes, and multiple local structures. In that setting, a global POD basis may approximate regions near the global mean reasonably well while performing poorly in remote or sparsely sampled regions, with the resulting global Galerkin ROM suffering loss of accuracy, numerical instability, and poor long-term statistics (Colanera et al., 16 Jun 2025).

ql-ROM addresses this by “quantizing” the manifold into $K$ clusters and attaching a separate intrusive ROM to each cluster. The qualifier “in time” refers to the fact that the active reduced coordinates, basis, centroid, and operators depend on the evolving state through a cluster-assignment map. A global model is therefore replaced by a hybrid dynamical system whose continuous evolution is local and whose model selection is discrete.

A common source of ambiguity is the word “quantized.” In ql-ROM it denotes state-space quantization by clustering, not low-bit storage, fixed-point arithmetic, or mixed-precision computation. That distinction is explicit in the broader localized model reduction literature, where localized ROMs for parameterized PDEs are discussed without any treatment of numerical quantization in the low-bit sense (Buhr et al., 2019).

2. Mathematical formulation and algorithmic structure

The starting point is a semi-discrete system

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$

Given snapshots $\{u(t_m)\}_{m=1}^M$, ql-ROM proceeds in three stages: unsupervised clustering, construction of local intrusive ROMs, and online patching through switching and coordinate changes (Colanera et al., 16 Jun 2025).

In the quantization stage, K-means with k-means++ seeding and Lloyd iterations partitions the snapshots into clusters with centroids $\{c_k\}_{k=1}^K$. The assignment map is

$\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$

and the within-cluster variance minimized by K-means is

$$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$

The number of clusters can be chosen by a Bayesian Information Criterion tailored for K-means,

$$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$

or by a problem-dependent elbow criterion in $\Delta \mathrm{BIC}/\Delta K$.

For each cluster $k$, local fluctuations are defined by subtracting the nearest centroid,

$u'_m = u_m - c_{\beta(m)}.$

The corresponding local snapshot matrix $\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$0 is decomposed by SVD,

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$1

and the leading $\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$2 columns of $\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$3 provide the local POD directions. The local reduced ansatz is

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$4

Galerkin projection yields the local reduced dynamics

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$5

which, for polynomial nonlinearities, can be written in affine-quadratic form,

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$6

The paper also writes the same structure compactly as

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$7

with signs and coefficients depending on convention.

Online switching uses the reconstructed reduced state:

$\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$8

When the active cluster changes from $\partial_t u = F(u), \qquad u \in \mathbb{R}^N.$9 to $\{u(t_m)\}_{m=1}^M$0, the reduced coordinates are transformed by

$\{u(t_m)\}_{m=1}^M$1

so that the reconstructed state remains continuous,

$\{u(t_m)\}_{m=1}^M$2

The reported implementation uses pure nearest-centroid switching and no explicit hysteresis or dwell time, although hysteresis, dwell-time rules, or spline-based smoothing are identified as possible modifications if chattering becomes problematic.

3. Canonical PDE settings and implementation choices

The original ql-ROM study evaluates the framework on two nonlinear PDE families across multiple dynamical regimes. For the Kuramoto–Sivashinsky equation,

$\{u(t_m)\}_{m=1}^M$3

with periodic boundary conditions on $\{u(t_m)\}_{m=1}^M$4, two cases are considered: a bursting regime with $\{u(t_m)\}_{m=1}^M$5 and $\{u(t_m)\}_{m=1}^M$6, and a chaotic regime with $\{u(t_m)\}_{m=1}^M$7 and $\{u(t_m)\}_{m=1}^M$8. Spatial discretization is spectral, time integration uses ETDRK4, the full-order solver uses $\{u(t_m)\}_{m=1}^M$9 modes and $\{c_k\}_{k=1}^K$0, the initial condition is $\{c_k\}_{k=1}^K$1, and a transient of $\{c_k\}_{k=1}^K$2 is discarded. The reported ql-ROM settings are $\{c_k\}_{k=1}^K$3, $\{c_k\}_{k=1}^K$4 for the bursting case, and $\{c_k\}_{k=1}^K$5, $\{c_k\}_{k=1}^K$6 for the chaotic case (Colanera et al., 16 Jun 2025).

For the two-dimensional incompressible Navier–Stokes equations with Kolmogorov forcing,

$\{c_k\}_{k=1}^K$7

on $\{c_k\}_{k=1}^K$8 with periodic boundary conditions and forcing

$\{c_k\}_{k=1}^K$9

the study considers a quasiperiodic regime at $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$0 and a chaotic or turbulent regime at $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$1. The full-order solver is pseudospectral with $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$2 dealiasing and explicit time stepping. The simulation time step is $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$3 for $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$4 and $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$5 for $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$6, random divergence-free initial conditions are used, and a transient $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$7 is discarded. For training and testing, the reported configurations are: at $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$8, $\beta_v(u)=\arg\min_{i=1,\dots,K}\|u-c_i\|,$9 training snapshots, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$0 sampling, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$1, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$2, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$3; at $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$4, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$5 training snapshots, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$6 sampling, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$7, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$8, $$J(c_1,\dots,c_K)=\frac{1}{M}\sum_{m=1}^M \|u_m-c_{\beta(m)}\|^2.$$9 (Colanera et al., 16 Jun 2025).

Clustering is performed directly on the full state, either in physical space or in Fourier space. For spectral complex-valued data $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$0, the real embedding

$$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$1

is used together with Euclidean distances. In all reported cases, the local ranks are uniform, $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$2, and $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$3 is selected from a global reconstruction criterion with $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$4 before being retained for ql-ROM. This design preserves a direct comparison with global ROMs using the same nominal reduced dimension.

4. Accuracy, stability, and statistical fidelity

The original paper evaluates ql-ROM against global ROMs through short-term prediction, long-term statistics, and empirical stability. Short-term prediction error is measured by

$$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$5

and the prediction horizon $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$6 is defined through a thresholded relative criterion with $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$7. Across all reported regimes, ql-ROM improves short-term prediction relative to a global ROM with the same rank (Colanera et al., 16 Jun 2025).

For Kuramoto–Sivashinsky bursting, the distinction is especially sharp. With $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$8, the global ROM becomes unstable, whereas ql-ROM with $$\mathrm{BIC}=M\log J + K\log M - \frac{2}{N}\sum_{k=1}^K n_k \log(n_k/M),$$9 remains stable and reproduces both the manifold geometry and the cluster occupancy probabilities $\Delta \mathrm{BIC}/\Delta K$0. With $\Delta \mathrm{BIC}/\Delta K$1, both models reproduce the phase portrait and occupancy probabilities, but ql-ROM yields a longer prediction horizon and lower local error, with overall short-term prediction improved by roughly a factor of three. In the chaotic Kuramoto–Sivashinsky case, ql-ROM improves the prediction horizon by about a factor of two, while the global ROM exhibits spectral artifacts at high spatial wavenumbers.

For Kolmogorov flow at $\Delta \mathrm{BIC}/\Delta K$2, ql-ROM achieves lower $\Delta \mathrm{BIC}/\Delta K$3 than the global ROM, tracks the kinetic energy $\Delta \mathrm{BIC}/\Delta K$4 accurately, and captures the sequence of cluster transitions over time. At $\Delta \mathrm{BIC}/\Delta K$5, both reduced models remain numerically stable over long integrations, but ql-ROM has smaller short-term error and reconstructs kinetic-energy bursts more faithfully.

The long-time statistical comparison is central to the method’s claims. For Kuramoto–Sivashinsky, ql-ROM reproduces the kinetic-energy probability density, including skewed distributions estimated by kernel density estimation, and matches the spatial Fourier energy spectrum

$\Delta \mathrm{BIC}/\Delta K$6

whereas the global ROM exhibits aliasing and high-wavenumber deviations. For Kolmogorov flow, ql-ROM reproduces the PDFs of kinetic energy, including the tails, and aligns closely with the ring-averaged spatial spectrum $\Delta \mathrm{BIC}/\Delta K$7 across scales. In the $\Delta \mathrm{BIC}/\Delta K$8 case, the global ROM underestimates energy at the largest wavenumbers by up to two orders of magnitude.

The reported explanation for the improved stability is locality rather than added stabilization or closure: centering around local means $\Delta \mathrm{BIC}/\Delta K$9 reduces projection error, and cluster-specific operators $k$0 and $k$1 better capture the local linear and nonlinear balances. The online computational overhead is correspondingly small. Only one local ROM is integrated at a time, so the per-step cost is comparable to that of a global ROM of the same rank, with extra work limited to $k$2 centroid distance checks and occasional $k$3 basis-change multiplications on switching.

5. Relation to other local ROM traditions and to the meaning of “quantized”

ql-ROM belongs to a wider family of local reduced modelling strategies, but its notion of locality is specific. In the survey “Localized model reduction for parameterized problems,” locality refers to decomposing the physical domain into subdomains and constructing reduced spaces with support on parts of the domain, then coupling them by conforming or non-conforming assembly. That framework is tailored to parameterized PDEs, transfer operators, port spaces, a posteriori estimation, and adaptive enrichment, rather than to time-evolving switching on a state-space attractor (Buhr et al., 2019). The distinction is structural: localized MOR in that sense is domain-local, whereas ql-ROM is attractor-local.

A similar distinction applies to space-local POD for advection-dominated flows. The method of “Space-Local Reduced-Order Models” divides the physical domain into subdomains and uses local POD bases, optionally with overlapping partition-of-unity weighting, to induce sparsity, preserve energy, and improve generalization outside the training regime (Gastelen et al., 2024). That framework is local in space, not in phase space, and it does not use quantization by clustering. Randomized local model order reduction likewise targets optimal local approximation spaces for transfer operators in domain-decomposition and multiscale settings, again without state-space quantization (Buhr et al., 2017).

The term “quantized” creates further ambiguity because in other ROM literatures it often refers to discrete numerical encoding rather than state partitioning. In “Reduced-order Modeling on a Near-term Quantum Computer,” quantization means fixed-point binary encoding for QUBO and QAOA formulations of DMD-based ROMs, which is explicitly distinct from the ql-ROM use of clustering as a quantizer (Asztalos et al., 2023). “PolyQROM” concerns quantum-encoded flow fields and variational quantum circuits, using “quantized” in the Hilbert-space or quantum-computing sense rather than in the centroid-assignment sense used by ql-ROM (Fang et al., 30 Apr 2025). Within the ql-ROM literature itself, the quantizer is simply the map $k$4.

6. Extensions, applications, and unresolved issues

Subsequent work has extended ql-ROM beyond the original Kuramoto–Sivashinsky and Kolmogorov-flow demonstrations. In “Towards extreme event prediction of turbulent flows with quantized local reduced-order models,” ql-ROM is applied to the turbulent Minimal Flow Unit. There the framework uses $k$5 k-means clusters and local intrusive POD–Galerkin models with $k$6 velocity modes and $k$7 pressure modes per cluster. The reported model remains stable over long integrations, keeps the relative error below $k$8 on a $k$9-snapshot test horizon, and reproduces the PDFs of total kinetic energy and dissipation. That study also introduces a local modal energy-budget analysis, identifying a high-dissipation cluster by centroid dissipation $u'_m = u_m - c_{\beta(m)}.$0 and attributing dissipation bursts to triadic transfers from a streak mode and travelling-wave modes into highly dissipative vortical modes, consistent with the self-sustaining process of near-wall turbulence (Colanera et al., 6 Nov 2025).

A second extension couples ql-ROM to gradient-based optimization. In “Adjoint-based optimization with quantized local reduced-order models for spatiotemporally chaotic systems,” the method is embedded in variational data assimilation for the chaotic Kuramoto–Sivashinsky equation. The reported setting uses $u'_m = u_m - c_{\beta(m)}.$1 clusters and $u'_m = u_m - c_{\beta(m)}.$2 POD modes per cluster, derives adjoint jump conditions consistent with the forward change-of-basis map, and assumes that switching times are insensitive to the initial condition. In that formulation the reduced optimization successfully reconstructs the full trajectory for up to $u'_m = u_m - c_{\beta(m)}.$3 Lyapunov times given full-state observations at the final time, while providing a $u'_m = u_m - c_{\beta(m)}.$4 times speed-up relative to the full-order model (Ozan et al., 2 Mar 2026).

These extensions also sharpen the method’s limitations. ql-ROM as presented is intrusive: it requires access to the governing PDE operator for Galerkin projection. Hard switching introduces non-differentiable hybrid dynamics and possible boundary chatter. Reliable performance depends on sufficiently populated clusters, and overly fine partitions risk sparse local datasets and ill-conditioned local models. The original ql-ROM study and the later MFU and adjoint papers therefore all point toward similar future directions: non-intrusive local surrogates, online or adaptive clustering, overlapping or probabilistic switching, hysteresis or smoothing near cluster boundaries, and alternative distance metrics that better reflect the geometry of the attractor (Colanera et al., 16 Jun 2025).

Taken together, these developments define Quantized Local Reduced Order Modelling as a divide-and-conquer reduced-modelling paradigm for complex attractors: it quantizes state space into local patches, constructs intrusive reduced dynamics within each patch, and advances the solution through explicit switching and coordinate transfer. Its reported advantages over global projection-based ROMs are improved numerical stability, longer short-term prediction horizons, and more faithful reproduction of long-term statistics, while preserving the interpretability and algebraic transparency of projection-based reduced modelling.