We present a general numerical approach for learning unknown dynamical systems using deep neural networks (DNNs). Our method is built upon recent studies that identified residual network (ResNet) as an effective neural network learning structure. In this paper, we present a generalized ResNet framework and broadly define “residue” as the discrepancy between observation data and prediction made by another model, which can be an existing coarse model or reduced order model. In this case, the generalized ResNet serves as a model correction to the existing model and recovers the unresolved/missing dynamics. When an existing coarse model is not available, we present numerical strategies for fast creation of coarse models, to be used in conjunction with the generalized ResNet. These coarse models are constructed using the same data set and thus do not require additional resource. The generalized ResNet is capable of learning the underlying unknown dynamics and producing predictions with accuracy higher than the standard ResNet structure. This is demonstrated via several numerical examples, including long-term prediction of a chaotic system.

我们提出了一种使用深度神经网络（DNN）学习未知动力系统的通用数值方法。我们的方法基于最近的研究，该研究将残差网络（ResNet）确定为有效的神经网络学习结构。在本文中，我们提出了一个广义的ResNet框架，并将“残差”广泛定义为观测数据与另一个模型（可以是现有的粗略模型或降阶模型）做出的预测之间的差异。在这种情况下，广义的ResNet可以对现有模型进行模型校正，并恢复未解决/缺失的动态。当现有的粗糙模型不可用时，我们提出了快速创建粗糙模型的数值策略，可与广义ResNet结合使用。这些粗略模型是使用相同的数据集构建的，因此不需要其他资源。广义的ResNet能够学习潜在的未知动态，并能够以比标准ResNet结构更高的精度生成预测。这通过几个数值示例得到证明，包括对混沌系统的长期预测。