Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems

TL;DR

Streaming OpInf addresses the memory bottleneck of non-intrusive reduced-order modeling by combining incremental SVD with recursive least-squares in a streaming framework. It reformulates the OpInf learning step to operate directly on streaming data, enabling online adaptation of reduced operators without storing full datasets. The approach is validated on Burgers' equation, Kuramoto–Sivashinsky dynamics, and a large-scale turbulent channel flow, showing accuracy near batch OpInf while achieving over 99% memory reduction and substantial speedups. This work enables real-time model updates and scalable deployment of reduced-order models in online or digital-twin contexts, with a clear pathway to extensions in uncertainty quantification and non-polynomial dynamics.

Abstract

Projection-based model reduction enables efficient simulation of complex dynamical systems by constructing low-dimensional surrogate models from high-dimensional data. The Operator Inference (OpInf) approach learns such reduced surrogate models through a two-step process: constructing a low-dimensional basis via Singular Value Decomposition (SVD) to compress the data, then solving a linear least-squares (LS) problem to infer reduced operators that govern the dynamics in this compressed space, all without access to the underlying code or full model operators, i.e., non-intrusively. Traditional OpInf operates as a batch learning method, where both the SVD and LS steps process all data simultaneously. This poses a barrier to deployment of the approach on large-scale applications where dataset sizes prevent the loading of all data into memory at once. Additionally, the traditional batch approach does not naturally allow model updates using new data acquired during online computation. To address these limitations, we propose Streaming OpInf, which learns reduced models from sequentially arriving data streams. Our approach employs incremental SVD for adaptive basis construction and recursive LS for streaming operator updates, eliminating the need to store complete data sets while enabling online model adaptation. The approach can flexibly combine different choices of streaming algorithms for numerical linear algebra: we systematically explore the impact of these choices both analytically and numerically to identify effective combinations for accurate reduced model learning. Numerical experiments on benchmark problems and a large-scale turbulent channel flow demonstrate that Streaming OpInf achieves accuracy comparable to batch OpInf while reducing memory requirements by over 99% and enabling dimension reductions exceeding 31,000x, resulting in orders-of-magnitude faster predictions.

Figures (14)

• Figure 1: Flowchart of each paradigm of Streaming OpInf showing different algorithmic components, selected based on time derivative data availability and memory/computational cost considerations.
• Figure 2: Burgers' Streaming SVD Assessment. Left: Subspace angles between POD bases computed from full SVD and iSVD methods. Right: Relative projection errors of the snapshot matrix onto POD bases for all SVD methods.
• Figure 3: Burgers' RLS Assessment. Top: Relative state error averaged over all parameters at each stream. Bottom: Relative streaming operator error normalized by the total number of operator elements, averaged over all parameters at each stream. Gradient lines represent different reduced dimensions $r$.
• Figure 4: Left Column: Final relative state errors for training data across all reduced dimensions $r$ for Burgers' equation. Right Column: Final relative state errors for testing data across all reduced dimensions $r$.
• Figure 5: KSE Streaming SVD Assessment. Left: Subspace angles between POD bases computed from full SVD and iSVD methods. Right: Relative projection errors of the snapshot matrix onto POD bases computed from iSVD methods.

Theorems & Definitions (8)

• Lemma 3.1: Operator perturbation bound
• Lemma 3.2: Data matrix perturbation bounds
• Theorem 3.3: Operator error bounds
• Lemma C.1: Theorem 4.1 of wedin1973perturbation
• proof : Proof of Lemma \ref{['lem:operator-perturbation']}
• proof : Proof of Lemma \ref{['lem:perturbation-bound']}
• Lemma C.2: Norm of the pseudoinverse
• proof