一种用于时间信号分类的非自治方程发现方法
A non-autonomous equation discovery method for time signal classification
摘要：
某些神经网络架构，在无限层的限制，导致系统的非线性微分方程。基于这一思想，我们开发了一个基于非自治动力学方程的时间信号分析框架。我们认为，时间信号作为一个强制功能的动力系统，管理一个随时间变化的隐藏变量。与方程发现一样，动力系统使用函数字典表示，系数从数据中学习。该框架被应用到时间信号分类问题。我们展示了如何梯度可以有效地使用伴随方法计算，我们从动态系统的方法来建立稳定的分类器。通过各种实验，在合成和真实数据集上，我们表明，所提出的方法使用的参数比竞争方法少几个数量级，同时达到相当的精度。我们使用越来越复杂的动力系统创建了合成数据集;虽然地面真实向量场通常是多项式，但我们始终发现傅立叶字典产生最佳结果。我们还演示了所提出的方法如何产生图形的可解释性的形式的相图。
英文摘要：
Certain neural network architectures, in the infinite-layer limit, lead to systems of nonlinear differential equations. Motivated by this idea, we develop a framework for analyzing time signals based on non-autonomous dynamical equations. We view the time signal as a forcing function for a dynamical system that governs a time-evolving hidden variable. As in equation discovery, the dynamical system is represented using a dictionary of functions and the coefficients are learned from data. This framework is applied to the time signal classification problem. We show how gradients can be efficiently computed using the adjoint method, and we apply methods from dynamical systems to establish stability of the classifier. Through a variety of experiments, on both synthetic and real datasets, we show that the proposed method uses orders of magnitude fewer parameters than competing methods, while achieving comparable accuracy. We created the synthetic datasets using dynamical systems of increasing complexity; though the ground truth vector fields are often polynomials, we find consistently that a Fourier dictionary yields the best results. We also demonstrate how the proposed method yields graphical interpretability in the form of phase portraits.
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