We study the problem of identifying dynamically distinct basins of attraction in high dimensional time-homogeneous Markov processes using only trajectory sampling. This problem is fundamental in the analysis of metastable dynamical systems, where the process rapidly mixes within basins while transitions between basins occur rarely on the timescale of interest, or even when the state space is reducible. Existing approaches typically rely on spatial discretization or spectral analysis of estimated transition operators, which can become unreliable in high dimensional settings or when the underlying basin geometry is highly nonlinear. We propose a discriminative approach to basin identification based on marginal trajectory distribution comparison. We prove a simple risk separation result: if two initial states belong to the same basin, the Bayes-optimal classifier distinguishing their marginal trajectory distributions achieves risk close to 1/2, whereas if they lie in distinct basins, the optimal risk is close to zero. This observation reduces basin detection to a two-sample discrimination problem between marginal trajectory distributions. Motivated by this principle, we develop a neural algorithm that receives a set of candidate basin representatives and iteratively merges them by estimating classification risk with a neural network that approximates the Bayes classifier. We evaluate the method on various metastable systems. These include synthetic systems constructed by embedding low-dimensional dynamics into high dimensional noisy ambient spaces. In these settings, standard spectral and clustering-based methods often fail, while our approach accurately recovers the underlying basin structure. These results display a shortcoming of existing methods and highlight trajectory discrimination as an effective tool for identifying dynamical basins in high dimensional stochastic systems.

翻译：我们研究仅利用轨迹采样在高维时间齐次马尔可夫过程中识别动力学上不同的吸引盆地的问题。该问题是亚稳态动力系统分析的基础，在此类系统中，过程在盆地内快速混合，而在感兴趣的时间尺度上盆地间转移罕见，甚至状态空间可约化。现有方法通常依赖于空间离散化或估计转移算子的谱分析，这些方法在高维设置或盆地几何结构高度非线性时可能变得不可靠。我们提出一种基于边缘轨迹分布比较的盆地识别判别方法。我们证明了一个简单的风险分离结果：如果两个初始状态属于同一盆地，区分它们边缘轨迹分布的贝叶斯最优分类器风险接近1/2；而若它们处于不同盆地，最优风险接近零。这一观察将盆地检测简化为边缘轨迹分布间的两样本判别问题。基于此原理，我们开发了一种神经算法，该算法接收一组候选盆地代表，并通过近似贝叶斯分类器的神经网络估计分类风险迭代合并它们。我们在多种亚稳态系统上评估该方法，包括通过将低维动力学嵌入高维含噪环境空间构建的合成系统。在这些设置中，标准谱方法和聚类方法常失效，而我们的方法能准确恢复底层盆地结构。这些结果揭示了现有方法的缺陷，并凸显轨迹判别作为识别高维随机系统动力学盆地的有效工具。