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.

翻译：基于投影的降阶模型通常作为静态代理模型部署，这限制了其在系统离开训练流形后的实际效用。本文形式化地研究自适应非侵入式降阶模型，该模型能够在线更新潜在子空间与降阶动力学。基于静态非侵入式降阶模型（特别是算子推断法与近期提出的基于轨迹的非侵入式降阶模型优化方法）的核心思想，我们提出三种构建方案：自适应算子推断法（顺序基/算子重拟合）、自适应NiTROM方法（编码器/解码器与多项式动力学的联合黎曼优化），以及一种混合方案——使用算子推断法更新初始化NiTROM。我们阐述了在线数据窗口、自适应窗口与计算预算的设计机制，并分析了计算成本规模。在瞬态扰动顶盖驱动腔流算例中，静态Galerkin/算子推断/NiTROM方法在超出训练区间的预测中均出现漂移或失稳现象。相比之下，自适应算子推断法能以适中成本稳健抑制幅值漂移；自适应NiTROM方法在频繁更新条件下可实现近乎精确的能量追踪，但其对初始化方式与优化深度较为敏感；混合方案在工况突变与离线数据极少的场景下最具可靠性，能生成物理一致的流场并保持能量有界。我们认为降阶模型的预测性声明必须具备成本意识与透明度，需明确区分训练/自适应/部署阶段，并显式报告在线计算预算与全阶模型查询量。本研究为构建自校正的非侵入式降阶模型提供了实用框架，使其在动力学特性显著偏离初始流形时仍能保持有效性。