Mathematical equations are often used to model biological processes. However, for many systems, determining analytically the underlying equations is highly challenging due to the complexity and unknown factors involved in the biological processes. In this work, we present a numerical procedure to discover dynamical physical laws behind biological data. The method utilizes deep learning methods based on neural networks, particularly residual networks. It is also based on recently developed mathematical tools of flow-map learning for dynamical systems. We demonstrate that with the proposed method, one can accurately construct numerical biological models for unknown governing equations behind measurement data. Moreover, the deep learning model can also incorporate unknown parameters in the biological process. A successfully trained deep neural network model can then be used as a predictive tool to produce system predictions of different settings and allows one to conduct detailed analysis of the underlying biological process. In this paper, we use three biological models-SEIR model, Morris-Lecar model and the Hodgkin-Huxley model-to show the capability of our proposed method.

中文翻译：

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从数据中深度学习生物模型：ODE 模型的应用

数学方程通常用于模拟生物过程。然而，对于许多系统，由于生物过程中涉及的复杂性和未知因素，分析地确定基本方程是非常具有挑战性的。在这项工作中，我们提出了一个数值程序来发现生物数据背后的动态物理定律。该方法利用基于神经网络，特别是残差网络的深度学习方法。它还基于最近开发的动态系统流程图学习数学工具。我们证明，使用所提出的方法，可以准确地为测量数据背后的未知控制方程构建数值生物模型。此外，深度学习模型还可以将未知参数纳入生物过程。然后，成功训练的深度神经网络模型可用作预测工具，以生成不同设置的系统预测，并允许对潜在的生物过程进行详细分析。在本文中，我们使用三个生物模型——SEIR 模型、Morris-Lecar 模型和 Hodgkin-Huxley 模型——来展示我们提出的方法的能力。