Systems of coupled dynamical units (e.g., oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution. Such coarse-graining procedures often require extensive experience and/or a deep understanding of the system dynamics. In this paper we present a systematic, data-driven approach to discovering “bespoke” coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of “effective” parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible extensions of this approach, including the possibility of obtaining data-driven effective partial differential equations for coarse-grained neuronal network behavior, as illustrated by the synchronization dynamics of Hodgkin–Huxley type neurons with a Chung-Lu network. Thus, we build an integrated suite of tools for obtaining data-driven coarse variables, data-driven effective parameters, and data-driven coarse-grained equations from detailed observations of networks of oscillators.

中文翻译：

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耦合振荡器的紧急空间

众所周知，耦合动力单元（例如，振荡器或神经元）的系统会表现出复杂的突发行为，这些行为可以通过粗粒度简化：发现粗变量并推导出其演化方程的过程。这种粗粒度程序通常需要丰富的经验和/或对系统动力学的深刻理解。在本文中，我们提出了一种系统的、数据驱动的方法来发现基于流形学习算法的“定制”粗变量。我们用经典的 Kuramoto 相位振荡器模型说明了这种方法，并演示了我们的流形学习技术如何成功识别与既定 Kuramoto 阶参数一一对应的粗略变量。然后，我们介绍了粗粒度方法的扩展，这使我们能够通过以数值时间积分器（初始值求解器）为模板的人工神经网络架构学习发现的粗变量的演化方程。这种方法使我们能够从稀疏流数据中学习状态变量的时间导数的精确近似值，从而发现有用的对其动态行为的近似微分方程描述。我们通过学习与已知的 Kuramoto 阶参数动力学在连续极限下的解析表达式一致的 ODE 来证明这种能力。然后我们展示了如何使用这种方法来学习通过我们的流形学习方法发现的粗略变量的动态。在这两个例子中，我们将基于神经网络的方法的结果与补充有几何谐波的典型有限差分进行比较。最后，我们展示了一系列计算示例，说明如何使用我们的流形学习方法的变体来发现具有复杂耦合的多参数模型的“有效”参数集、减少的参数组合。我们最后讨论了这种方法的可能扩展，包括获得粗粒度神经元网络行为的数据驱动的有效偏微分方程的可能性，如具有 Chung-Lu 网络的 Hodgkin-Huxley 型神经元的同步动力学所示. 因此，我们构建了一套集成的工具，用于获取数据驱动的粗变量、数据驱动的有效参数、