Stochastic Reduced-Order Modeling Framework

• Stochastic reduced-order modeling is a mathematical approach that creates low-dimensional surrogates of high-dimensional uncertain systems while retaining key dynamical features.
• It employs methods such as projection-based techniques, non-intrusive regression, polynomial chaos expansions, and statistical closures to efficiently manage uncertainty.
• These frameworks are applied in uncertainty quantification, control, simulation acceleration, and optimization across fields like fluid dynamics and materials engineering.

A stochastic reduced-order modeling (ROM) framework encompasses a mathematically rigorous and algorithmically diverse class of model reduction techniques for dynamical systems subject to parametric, random, or stochastic uncertainty. These frameworks seek to construct surrogates or reduced dynamical systems that retain essential input–output or statistical features of the original high-dimensional stochastic models—such as stochastic partial differential equations (SPDEs), stochastic differential equations (SDEs), or random PDE-constrained optimization problems—while vastly reducing computational cost. Stochastic ROMs are now deployed in uncertainty quantification, control, design, simulation acceleration, and data-driven forecasting for applications ranging from fluid dynamics to materials engineering.

1. Core Mathematical Foundations

The stochastic ROM problem arises from parametrized models of the form

$$\frac{\partial u(x,t, \xi)}{\partial t} + \mathcal{A}(u; x, t, \xi) = f(x,t; \xi),$$

where $\xi$ encodes $d$-dimensional random or uncertain inputs, either via finite-dimensional sampling, random processes, or noise terms, and $u$ is typically high-dimensional by spatial discretization. The principal aim of stochastic ROM frameworks is to produce an explicit low-dimensional surrogate

$$\hat{u}(x,t,\xi) = \sum_{i=1}^r a_i(t,\xi)\, \phi_i(x)$$

or an analogous latent representation, where $\operatorname{dim}(r) \ll N$ (the full spatial or state dimension), while retaining the structure of the stochastic dependence on $\xi$.

Popular methodologies can be categorized as follows:

• Projection-based methods: Galerkin or Petrov–Galerkin projection onto reduced subspaces (e.g., Proper Orthogonal Decomposition, DBO/DO decompositions) derived from stochastic snapshot data (Patil et al., 2019, Patil et al., 2021, Iliescu et al., 2017).
• Non-intrusive regression/surrogate approaches: Encode high-fidelity simulation data into low-dimensional latent codes and regress these on input parameters or stochastic variables, using machine learning architectures such as autoencoders or operator inference (Abdedou et al., 2022, Fang et al., 24 Mar 2024, Yong et al., 30 Aug 2024).
• Stochastic polynomial or operator expansions: Combine a reduced spatial basis with a polynomial chaos or moment expansion in the stochastic domain (Sun et al., 2019, Jin et al., 2021, Patil et al., 2021).
• Statistical closure models: Augment deterministic ROMs with stochastic terms (drift and diffusion) learned from multi-trajectory data to capture model-form and closure uncertainties (Lu et al., 2022, Tran et al., 2020, Mou et al., 2022).

2. Reduced Basis and Stochastic Subspace Construction

The dominant algorithmic operation in many frameworks is the extraction of a reduced basis or subspace that captures the essential stochastic variation of the system:

• Proper Orthogonal Decomposition (POD): From an ensemble of full-order snapshots $\{u(x,t_l;\xi_k)\}$, construct an orthonormal basis $\{\phi_i\}_{i=1}^r$ maximizing projected energy

$$\min_{\Psi^r,\, (\Psi^r)^T \Psi^r = I} \| \mathcal{U} - \Psi^r (\Psi^r)^T \mathcal{U} \|_F^2,$$

with $\mathcal{U}$ the snapshot matrix. The leading $r$ modes are often selected to capture a prescribed energy fraction (Sun et al., 2019, Jin et al., 2021, Abdedou et al., 2022).

• Time-dependent bases and dynamically orthogonal decompositions: For time-evolving or strongly nonstationary stochastic systems, reduced subspaces themselves may evolve in time. The DBO (dynamically/bi-orthonormal) and DO (dynamically orthogonal) frameworks propagate coupled spatial and stochastic bases via constrained evolution equations derived from variational principles (Patil et al., 2019, Patil et al., 2021).
• Clustering and classification: In stochastic systems with regime switches or strong nonlinearity, data-driven frameworks cluster the ensemble of snapshot trajectories and build local, cluster-adapted reduced subspaces (CPOD) selected online via pre-classification or Bayesian classifiers (Xiong et al., 2022).

3. Parameterization of Stochastic Dependence

The ROM employs a stochastic parameterization to represent the effect of input uncertainties or intrinsic stochasticity, typically in one of three paradigms:

• Polynomial chaos expansions (PCE): ROM coefficients are expanded in an orthonormal polynomial basis $\{\Psi_\alpha(\xi)\}$, yielding a representation

$$u(x,\xi) \approx \sum_{i=1}^r \sum_{|\alpha|\leq p} c_{i,\alpha} \Psi_\alpha(\xi) \psi_i(x)$$

with coefficients learned by non-intrusive regression (least squares, LASSO) on a designed sample set (Sun et al., 2019, Jin et al., 2021).

• Nonlinear regression and manifold learning: Neural-network surrogates (e.g., two-stage convolutional autoencoders with parameter-to-latent regression) map input uncertainties to a nonlinear latent space, which is then decoded to reconstruct high-dimensional solutions at negligible cost in the online phase (Abdedou et al., 2022, Fang et al., 24 Mar 2024).
• Data-driven SDE/Fokker–Planck closures: When unresolved scales or closure errors are significant, the reduced coefficients are modeled as solutions to SDEs or Fokker–Planck equations, with drift and diffusion components calibrated to reproduce the observed statistics or PDFs of the full-order system (Lu et al., 2022, Tran et al., 2020, Mou et al., 2022).

4. Algorithms and Theoretical Analysis

A stochastic ROM workflow typically comprises the following algorithmic components:

• Offline (training) phase: High-resolution simulations generate an ensemble of trajectories or snapshots. POD/SVD or machine learning algorithms process this data into reduced bases or latent codes; PCE, MLP, or SDE coefficients are fitted.
• Online (predictive/UQ) phase: Fast evaluation of the ROM enables dense sampling in parameter/stochastic space—for example, millions of Monte Carlo realizations or real-time prediction.
• Error control and theoretical guarantees:
  • Output error bounds are often linked to energy captured by the reduced basis and, in PCE-based ROMs, to the truncation of the polynomial chaos expansion (Sun et al., 2019, Jin et al., 2021).
  • For SDE-based ROMs, statistical consistency and asymptotic convergence rates (e.g., $O(M^{-1/2})$ in the number of trajectories $M$) are established for parameter inference (Lu et al., 2022).
  • In nonlinear SDE settings, Lyapunov-type stability criteria and generalized Gramians enable balanced-truncation error bounds in mean square norm (Redmann, 4 Aug 2025).
  • For regularized stochastic Burgers ROMs, spatial and spectral filtering stabilizes the reduced dynamics and suppresses spurious oscillations; mean and std errors are reduced by factors of 2–3 over standard (unregularized) models (Iliescu et al., 2017).

5. Extensions: Machine Learning, Hyper-Reduction, and Uncertainty Quantification

Stochastic ROM frameworks increasingly incorporate machine learning and hyper-reduction strategies to address scalability and expressive power:

• Deep neural surrogates: Non-intrusive and mesh-free neural operators (e.g., CAE-MLP, ResNet, normalizing flows) facilitate nonlinear compression of intertwined spatial–temporal–stochastic correlations, outperforming linear POD–ANN baselines in both mean and variance metrics (Abdedou et al., 2022, Fang et al., 24 Mar 2024).
• Hyper-reduction and sparse sampling: In the stochastic finite volume context, low-rank interpolation and Q-DEIM hyper-reduction reduce both computational and memory requirements by orders of magnitude, enabling feasible UQ for high-dimensional random spaces (Qu et al., 7 Jul 2025).
• Treatment of model-form and epistemic uncertainty: Recent probabilistic ROMs randomize the projection basis itself (sampling on the Stiefel manifold with a Dirichlet prior over convex combinations of anchor bases), quantifying model-form uncertainties stemming from training data selection, parametric regimes, or projection choices and providing UQ bands for predicted statistics (Yong et al., 30 Aug 2024).
• Physics-informed and constrained closures: Multiscale, conditional-Gaussian, and energy-preserving closures add systematic corrections and maintain stability in highly nonlinear or turbulent regimes; analytic data assimilation (e.g., Kalman–Bucy filters) is supported within certain frameworks (Mou et al., 2022).

6. Benchmark Applications and Performance

Stochastic ROMs have demonstrated efficacy across a broad set of physically motivated test cases:

• Parabolic and hyperbolic SPDEs: Reduced and regularized Burgers’ equations (Iliescu et al., 2017, Lu et al., 2022), stochastic advection–diffusion–reaction (Jin et al., 2021), and compressible Euler equations (Qu et al., 7 Jul 2025).
• Unsteady Navier–Stokes flows: Application of time-dependent basis and cluster-classification ROMs to stochastic boundary conditions and inflow profiles (Patil et al., 2021, Xiong et al., 2022).
• High-dimensional UQ: Large-scale heat–driven cavity flow, heat diffusion with random conductivity, and parametric river hydraulics (Sun et al., 2019, Abdedou et al., 2022).
• Materials microstructure evolution: Langevin–Fokker–Planck reduced models for ICME, phase-field, and molecular dynamics simulations, accelerating ensemble PDF prediction by 10×–500× over full-order MC (Tran et al., 2020).
• Turbofan noise-reduction and stochastic optimization: Stochastic ROMs reduce the cost of CVaR-based, PDE-constrained design optimization by orders of magnitude via parallelized snapshot-based POD–Galerkin surrogates (Yang et al., 2016).
• Market microstructure: Reduced-form linear SDEs for limit order book dynamics treat liquidity as a low-dimensional stochastic process (Malo et al., 2010).

7. Limitations, Challenges, and Future Directions

Despite the notable computational gains and flexibility of stochastic ROMs, several limitations persist:

• Curse of dimensionality: All frameworks are challenged by high-dimensional stochastic input spaces, though hyper-reduction (Q-DEIM), low-rank regressions, and specialized basis selection mitigate this issue (Qu et al., 7 Jul 2025).
• Coverage and extrapolation: Non-intrusive surrogates and data-driven closures are sensitive to the domain and diversity of the training sample set; performance can degrade outside the regime of observed data (Abdedou et al., 2022, Fang et al., 24 Mar 2024).
• Closure modeling for strongly nonlinear/turbulent systems: Physically informed or data-driven stochastic closures are essential to capture energy transfer and uncertainty growth; the construction and theoretical justification of such closures remain active areas (Mou et al., 2022, Lu et al., 2022).
• Treatment of model-form uncertainty and epistemic effects: Randomization of projection operators and information-theoretic approaches offer explicit methods for quantifying epistemic contributions to UQ (Yong et al., 30 Aug 2024).

A plausible implication is that future research will intensify on integrating data-driven and operator-theoretic ROMs with robust physics priors, adaptive hyper-reduction techniques, and rich uncertainty quantification methodologies to address both parametric variation and model-form uncertainty in nonlinear stochastic dynamical systems.