An adaptive sampling approach for efficient detection of bifurcation boundaries in parametrized fluid flow problems is presented herein. The study extends the machine-learning approach of Silvester~(J. Comput. Phys., 553 (2026), 114743), where a classifier network was trained on preselected simulation data to identify bifurcated and nonbifurcated flow regimes. In contrast, the proposed methodology introduces adaptivity through a flow-based deep generative model that automatically refines the sampling of the parameter space. The strategy has two components: a classifier network maps the flow parameters to a bifurcation probability, and a probability density estimation technique (KRnet) for the generation of new samples at each adaptive step. The classifier output provides a probabilistic measure of flow stability, and the Shannon entropy of these predictions is employed as an uncertainty indicator. KRnet is trained to approximate a probability density function that concentrates sampling in regions of high entropy, thereby directing computational effort towards the evolving bifurcation boundary. This coupling between classification and generative modeling establishes a feedback-driven adaptive learning process analogous to error-indicator based refinement in contemporary partial differential equation solution strategies. Starting from a uniform parameter distribution, the new approach achieves accurate bifurcation boundary identification with significantly fewer Navier--Stokes simulations, providing a scalable foundation for high-dimensional stability analysis.

翻译：本文提出了一种自适应采样方法，用于高效检测参数化流体流动问题中的分岔边界。本研究扩展了Silvester（J. Comput. Phys., 553 (2026), 114743）的机器学习方法，该方法通过在预选模拟数据上训练分类器网络来识别分岔与非分岔流态。与之相比，本文提出的方法通过基于流模型的深度生成模型引入自适应性，从而自动优化参数空间的采样。该策略包含两个组成部分：一个将流动参数映射到分岔概率的分类器网络，以及用于在每个自适应步骤中生成新样本的概率密度估计技术（KRnet）。分类器输出提供了流动稳定性的概率度量，其预测结果的香农熵被用作不确定性指标。KRnet经过训练以逼近一个概率密度函数，该函数将采样集中在高熵区域，从而将计算资源导向动态演化的分岔边界。这种分类与生成建模之间的耦合建立了一个反馈驱动的自适应学习过程，类似于当代偏微分方程求解策略中基于误差指示的网格细化方法。从均匀参数分布出发，新方法仅需显著更少的Navier-Stokes模拟即可实现精确的分岔边界识别，为高维稳定性分析提供了可扩展的基础。