Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges

TL;DR

This work tackles the limitations of static reduced-order models by introducing adaptive, non-intrusive ROMs that online-update both the latent subspace and reduced dynamics. Building on Operator Inference and NiTROM, it proposes three formulations—Adaptive OpInf, Adaptive NiTROM, and a hybrid Adaptive OpInf--NiTROM—and demonstrates their performance on a lid-driven cavity flow, showing that adaptive methods can suppress energy drift and maintain physical fidelity beyond the training manifold. The hybrid approach consistently delivers the most robust results across regime changes and minimal offline training, while adaptive OpInf offers robust, computationally efficient performance and NiTROM provides high accuracy when updates are frequent. The study emphasizes cost-aware reporting and outlines practical directions for accelerating online updates, triggering adaptations, and validating adaptive ROMs in digital twin settings, thereby advancing self-correcting, non-intrusive surrogates for evolving dynamics.

Abstract

Projection-based Reduced Order Models (ROMs) are often deployed as static surrogates, which limits their practical utility once a system leaves the training manifold. We formalize and study adaptive non-intrusive ROMs that update both the latent subspace and the reduced dynamics online. Building on ideas from static non-intrusive ROMs, specifically, Operator Inference (OpInf) and the recently-introduced Non-intrusive Trajectory-based optimization of Reduced-Order Models (NiTROM), we propose three formulations: Adaptive OpInf (sequential basis/operator refits), Adaptive NiTROM (joint Riemannian optimization of encoder/decoder and polynomial dynamics), and a hybrid that initializes NiTROM with an OpInf update. We describe the online data window, adaptation window, and computational budget, and analyze cost scaling. On a transiently perturbed lid-driven cavity flow, static Galerkin/OpInf/NiTROM drift or destabilize when forecasting beyond training. In contrast, Adaptive OpInf robustly suppresses amplitude drift with modest cost; Adaptive NiTROM is shown to attain near-exact energy tracking under frequent updates but is sensitive to its initialization and optimization depth; the hybrid is most reliable under regime changes and minimal offline data, yielding physically coherent fields and bounded energy. We argue that predictive claims for ROMs must be cost-aware and transparent, with clear separation of training/adaptation/deployment regimes and explicit reporting of online budgets and full-order model queries. This work provides a practical template for building self-correcting, non-intrusive ROMs that remain effective as the dynamics evolve well beyond the initial manifold.

Figures (33)

• Figure 1: Schematic of a ROM with encoder $\psi$, decoder $\varphi$ and latent-space dynamics $\mathbf{f}_r(\mathbf{z},\mathbf{u})$.
• Figure 2: Schematic for the adaptation setup. Highlighted steps show FOM snapshots kept in the moving window.
• Figure 3: Transient 2-D lid-driven cavity flow. The leftmost field depicts the vorticity field from the steady state solution of the system at $Re=8300$, and the remaining plots show vorticity fluctuations, showing how the dynamics evolve in time.
• Figure 4: Illustration of the three training/test window configurations. Each panel shows the temporal evolution of the system energy, with the blue-shaded region denoting the offline (training) window $[t_0,t_1]$ and the red-shaded portion showing the online (test) window $(t_1,t_2]$. The training interval progressively shortens from case 1 to case 3.
• Figure 5: Performance of $10$-dimensional linear vs. quadratic static NiTROM models against the training trajectory. We notice that the linear model is unable to track the oscillatory motion of the energy, while the quadratic ROM captures the true trajectory.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4