Measurement noise is an integral part while collecting data of a physical process. Thus, noise removal is a necessary step to draw conclusions from these data, and it often becomes quite essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and sparsely sampled measurements. In our methodology, the main innovation can be seen in of integration of deep neural networks with a classical numerical integration method. Precisely, we aim at learning a neural network that implicitly represents the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by enforcing the constraint that the data at the next time-steps can be given by following a numerical integration scheme such as the fourth-order Runge-Kutta scheme. The proposed framework to learn a model predicting the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are not available at the same temporal grid. We demonstrate the effectiveness of the proposed method to learning models using data obtained from various differential equations. The proposed approach provides a promising methodology to learn dynamic models, where the first-principle understanding remains opaque.
翻译:在收集物理过程的数据时,测量噪音是一个不可分割的组成部分。因此,清除噪音是从这些数据中得出结论的必要步骤,因此,利用这些数据构建动态模型往往变得相当必要。我们讨论使用杂音和零星抽样测量方法学习差异方程式的方法。在我们的方法中,主要的创新表现在将深神经网络与传统的数字集成法相结合方面。准确地说,我们的目标是学习一个隐含地代表数据的信息神经网络和一个额外的神经网络,以模拟依赖变量的矢量字段。我们通过强制这两个网络相结合,即下一个时间步骤的数据可以采用数字集成计划,如四等Runge-Kutta计划来提供。在噪音测量下,用于学习矢量字段预测模型的拟议框架非常有效。该方法可以处理在同一时间网格上无法找到依赖变量的情形。我们展示了利用从各种差异方程式获得的数据学习模型的拟议方法的有效性。拟议方法提供了一种很有希望的方法,即学习动态模型,而第一个原则的理解仍然不透明。