We propose a learning algorithm for discovering unknown parameterized dynamical systems by using observational data of the state variables. Our method is built upon and extends the recent work of discovering unknown dynamical systems, in particular those using deep neural network (DNN). We propose a DNN structure, largely based upon the residual network (ResNet), to not only learn the unknown form of the governing equation but also take into account the random effect embedded in the system, which is generated by the random parameters. Once the DNN model is successfully constructed, it is able to produce system prediction over longer term and for arbitrary parameter values. For uncertainty quantification, it allows us to conduct uncertainty analysis by evaluating solution statistics over the parameter space.

中文翻译：

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深度学习参数化方程及其在不确定性量化中的应用

我们提出了一种学习算法，用于通过使用状态变量的观测数据来发现未知的参数化动力学系统。我们的方法基于并扩展了发现未知动力学系统（尤其是使用深度神经网络（DNN）的动力学系统）的最新工作。我们提出了一种主要基于残差网络（ResNet）的DNN结构，该结构不仅可以学习控制方程的未知形式，而且可以考虑嵌入在系统中的由随机参数产生的随机效应。一旦成功构建了DNN模型，它就可以长期生成系统预测，并生成任意参数值。对于不确定性量化，它允许我们通过评估参数空间上的解决方案统计信息来进行不确定性分析。