Long-time prediction of future states has been challenging in data-driven modeling of nonlinear dynamical systems as the prediction error accumulates over the prediction horizon. One of the potential reasons is the lack of robustness for the data-driven model. In this study we present a recurrent neural network (RNN) framework with an adaptive training strategy to model nonlinear dynamical systems from data for long-time prediction of future states. Specifically, we exploit the recurrence of network to improve the model robustness by explicitly incorporating the multi-step prediction with error accumulation into model training. Furthermore, we introduce an adaptive training strategy, where the prediction horizon gradually increases from a small value to facilitate the RNN training. We demonstrate the proposed approach on a family of Duffing oscillators, including autonomous and non-autonomous systems with various attractors, and discuss its advantages and limitations.

在非线性动力学系统的数据驱动建模中，对未来状态的长期预测一直充满挑战，因为预测误差会在预测范围内累积。潜在原因之一是数据驱动模型缺乏鲁棒性。在这项研究中，我们提出了一种具有自适应训练策略的递归神经网络（RNN）框架，可以根据数据对非线性动力学系统进行建模，以对未来状态进行长期预测。具体来说，我们通过将具有错误累积的多步预测明确纳入模型训练中来利用网络的递归来提高模型的鲁棒性。此外，我们引入了一种自适应训练策略，其中预测范围从较小的值逐渐增加，以促进RNN训练。