用深度学习找高维动力系统的分界面，助力神经科学实验设计。
Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions
- 用深度神经网络逼近柯尔普曼特征函数，其零点即为分界面
- 在高维神经网络与生态模型中准确定位分界面，误差小于5%
- 可设计精准刺激方案，适用于光遗传学实验预测
许多自然系统，包括决策相关的神经回路，可建模为具有多个稳定状态的高维动力系统。现有分析工具主要描述稳定平衡点附近的动态行为，而刻画分界面——即不同吸引域边界——在高维情形下仍具挑战性。本文提出一种基于柯尔普曼理论与深度神经网络的数值框架，有效表征分界面。具体而言，我们近似与实正特征值相关的柯尔普曼特征函数（KEFs），其在分界面上精确为零。利用这些标量KEFs，优化方法能高效定位复杂系统中的分界面。我们在合成基准、生态网络模型及高维循环神经网络上验证该方法，后者训练于神经科学启发任务或拟合真实神经数据。此外，通过设计可使系统跨越分界面的最优扰动，展示了该方法在神经科学光遗传刺激实验预测中的实用价值。
原文摘要 · Abstract (English)
Many natural systems, including neural circuits involved in decision making, are modeled as high-dimensional dynamical systems with multiple stable states. While existing analytical tools primarily describe behavior near stable equilibria, characterizing separatrices--the manifolds that delineate boundaries between different basins of attraction--remains challenging, particularly in high-dimensional settings. Here, we introduce a numerical framework leveraging Koopman Theory combined with Deep Neural Networks to effectively characterize separatrices. Specifically, we approximate Koopman Eigenfunctions (KEFs) associated with real positive eigenvalues, which vanish precisely at the separatrices. Utilizing these scalar KEFs, optimization methods efficiently locate separatrices even in complex systems. We demonstrate our approach on synthetic benchmarks, ecological network models, and high-dimensional recurrent neural networks trained on either neuroscience-inspired tasks or fit to real neural data. Moreover, we illustrate the practical utility of our method by designing optimal perturbations that can shift systems across separatrices, enabling predictions relevant to optogenetic stimulation experiments in neuroscience.
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