分界面的生成式学习
Generative Learning of Separatrices
- Johns Hopkins All Children’s Hospital（约翰斯·霍普金斯儿童全医院）
- Johns Hopkins University（约翰斯·霍普金斯大学）
- University of Exeter（埃克塞特大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究提出结合监督分类与生成式建模的框架，通过神经网络分类器识别高不确定性区域作为初步分界面，再用基于分数的生成式模型采样，实现多稳态高维动力系统分界面的经验一致重构。
中文摘要 AI 辅助
在多稳态、高维动力系统中，识别与重构吸引域之间的分界面是计算动力学领域的一项基础挑战。这些结构决定了转变路径及其他重要的大时间尺度行为，但由于其邻域在直接模拟中很少被访问，因此通常采样不足。传统计算方法在高维系统中面临计算限制，且需要先验了解动力系统及其方程。随机或均匀采样相空间等简单采样方法通常无法定量近似分界面及其结构。我们引入并实现了一种结合监督分类与生成式建模的框架来应对这一挑战。该方法首先在感兴趣系统中，对由对应吸引域标记的均匀或随机采样初始条件训练神经网络分类器。利用训练后分类器的不确定性指标量化决策边界，该方法随后将这些高不确定性区域和分类器边界识别为初步近似分界面。随后，针对高不确定性区域的样本训练基于分数的生成式模型，最终生成与采样区域中吸引域之间流形（构成分界面）上或附近样本的经验密度一致的样本密度。该方法利用了（a）用于全局相空间划分的判别式模型和（b）用于详细几何采样的生成式模型的互补优势，形成了一种系统、迭代、数据驱动的框架，可生成（近似）分界面流形的经验一致重构结果。
英文摘要
The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.
