Model discovery based on existing data has been one of the major focuses of mathematical modelers for decades. Despite tremendous achievements of model identification from adequate data, how to unravel the models from limited data is less resolved. In this paper, we focus on the model discovery problem when the data is not efficiently sampled in time. This is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. This recursive approach can retrieve the governing equation in a simple and efficient manner, and it can significantly improve the approximation accuracy by increasing the recursive stages. In particular, our proposed approach shows superior power when the existing data are sampled with a large time lag, from which the traditional approach might not be able to recover the model well. Several widely used examples of dynamical systems are used to benchmark this newly proposed recursive approach. Numerical comparisons confirm the effectiveness of this recursive neural network for model discovery.

中文翻译：

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用递归深度神经网络发现控制方程

数十年来，基于现有数据的模型发现一直是数学建模者的主要关注点之一。尽管从足够的数据中识别模型取得了巨大的成就，但如何从有限的数据中解开模型还没有解决。在本文中，我们专注于当数据没有及时有效采样时的模型发现问题。由于有限的实验可及性和劳动力/资源限制，这很常见。具体来说，我们为数据驱动的模型发现引入了递归深度神经网络 (RDNN)。这种递归方法可以以简单有效的方式检索控制方程，并且可以通过增加递归阶段显着提高逼近精度。特别是，当现有数据以大的时间延迟采样时，我们提出的方法显示出卓越的能力，传统方法可能无法从中很好地恢复模型。几个广泛使用的动态系统示例用于对这种新提出的递归方法进行基准测试。数值比较证实了这种递归神经网络对模型发现的有效性。