We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network is used as function approximator and its output is used to construct the function library, allowing to perform the sparse regression within the neural network. This construction makes it extremely robust to noise, applicable to small data sets, and, contrary to other deep learning methods, does not require a training set. We benchmark our approach on several physical problems such as the Burgers', Korteweg-de Vries and Keller-Segel equations, and find that it requires as few as $\mathcal{O}({10}^2)$ samples and works at noise levels up to 75%. Motivated by these results, we apply DeepMoD directly on noisy experimental time-series data from a gel electrophoresis experiment and find that it discovers the advection-diffusion equation describing this system.

我们介绍DeepMoD，这是一种基于深度学习的模型发现算法。DeepMoD在可能的函数及其导数库上使用稀疏回归来发现时空数据集下方的偏微分方程。神经网络用作函数逼近器，其输出用于构造函数库，从而可以在神经网络内执行稀疏回归。这种构造使其对噪声极其鲁棒，适用于小型数据集，并且与其他深度学习方法相反，不需要训练集。我们对几种物理问题（例如Burgers方程，Korteweg-de Vries方程和Keller-Segel方程）进行基准测试，发现所需的方法少至$\mathcal{Ø}（{10}^2）$采样并在高达75％的噪声水平下工作。基于这些结果，我们将DeepMoD直接应用于来自凝胶电泳实验的嘈杂实验时间序列数据，发现它发现了描述该系统的对流扩散方程。