We propose an efficient retraining strategy for a parameterized Reduced Order Model (ROM) that attains accuracy comparable to full retraining while requiring only a fraction of the computational time and relying solely on sparse observations of the full system. The architecture employs an encode-process-decode structure: a Variational Autoencoder (VAE) to perform dimensionality reduction, and a transformer network to evolve the latent states and model the dynamics. The ROM is parameterized by an external control variable, the Reynolds number in the Navier-Stokes setting, with the transformer exploiting attention mechanisms to capture both temporal dependencies and parameter effects. The probabilistic VAE enables stochastic sampling of trajectory ensembles, providing predictive means and uncertainty quantification through the first two moments. After initial training on a limited set of dynamical regimes, the model is adapted to out-of-sample parameter regions using only sparse data. Its probabilistic formulation naturally supports ensemble generation, which we employ within an ensemble Kalman filtering framework to assimilate data and reconstruct full-state trajectories from minimal observations. We further show that, for the dynamical system considered, the dominant source of error in out-of-sample forecasts stems from distortions of the latent manifold rather than changes in the latent dynamics. Consequently, retraining can be limited to the autoencoder, allowing for a lightweight, computationally efficient, real-time adaptation procedure with very sparse fine-tuning data.

翻译：本文提出一种参数化降阶模型的高效重训练策略，该策略在仅依赖全系统稀疏观测数据的情况下，能以远低于完全重训练的计算成本达到与之相当的精度。该架构采用编码-处理-解码结构：利用变分自编码器实现降维，并通过Transformer网络演化潜在状态以建模动力学行为。降阶模型以外部控制变量（Navier-Stokes框架中的雷诺数）为参数，Transformer通过注意力机制同时捕捉时间依赖性与参数效应。概率化变分自编码器支持轨迹集合的随机采样，通过前两阶矩提供预测均值与不确定性量化。在有限动力学体系上进行初始训练后，模型仅需稀疏数据即可适应样本外参数区域。其概率化表述天然支持集合生成，我们将其嵌入集合卡尔曼滤波框架，通过数据同化实现基于最小观测量的全状态轨迹重构。研究进一步表明，对于所考虑的动力学系统，样本外预测误差的主要来源是潜在流形的畸变而非潜在动力学的改变。因此，重训练可局限于自编码器部分，从而形成仅需极稀疏微调数据的轻量级计算高效实时自适应方案。