This paper proposes a novel deep learning approach for approximating evolution operators and modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous works [T. Qin, K. Wu, and D. Xiu, J. Comput. Phys., 395:620--635, 2019], [K. Wu and D. Xiu, J. Comput. Phys., 408:109307, 2020], and [Z. Chen, V. Churchill, K. Wu, and D. Xiu, J. Comput. Phys., 449:110782, 2022], which focused on learning single evolution operator with a fixed time step. This paper aims to learn a family of evolution operators with variable time steps, which constitute a semigroup for an autonomous system. The semigroup property is very crucial and links the system's evolutionary behaviors across varying time scales, but it was not considered in the previous works. We propose for the first time a framework of embedding the semigroup property into the data-driven learning process, through a novel neural network architecture and new loss functions. The framework is very feasible, can be combined with any suitable neural networks, and is applicable to learning general autonomous ODEs and PDEs. We present the rigorous error estimates and variance analysis to understand the prediction accuracy and robustness of our approach, showing the remarkable advantages of semigroup awareness in our model. Moreover, our approach allows one to arbitrarily choose the time steps for prediction and ensures that the predicted results are well self-matched and consistent. Extensive numerical experiments demonstrate that embedding the semigroup property notably reduces the data dependency of deep learning models and greatly improves the accuracy, robustness, and stability for long-time prediction.

翻译：本文提出了一种新颖的深度学习方法，用于利用不同时间延迟采集的时间序列数据逼近演化算子并建模未知自治动力系统。该工作是先前研究[T. Qin, K. Wu, and D. Xiu, J. Comput. Phys., 395:620--635, 2019]、[K. Wu and D. Xiu, J. Comput. Phys., 408:109307, 2020]及[Z. Chen, V. Churchill, K. Wu, and D. Xiu, J. Comput. Phys., 449:110782, 2022]的延续，这些工作专注于学习固定时间步长的单一演化算子。本文旨在学习具有可变时间步长的演化算子族，这些算子构成自治系统的半群。半群性质极为关键，它关联了系统在不同时间尺度上的演化行为，但此前的工作并未考虑这一性质。我们首次提出一种将半群性质嵌入数据驱动学习过程的框架，通过新颖的神经网络架构和新的损失函数实现。该框架具有高度可行性，可与任意合适的神经网络结合，适用于学习通用的自治常微分方程和偏微分方程。我们给出了严格的误差估计和方差分析以理解方法的预测精度与鲁棒性，展示了半群感知性在模型中的显著优势。此外，我们的方法允许任意选择预测时间步长，并确保预测结果具有良好自洽性与一致性。大量数值实验表明，嵌入半群性质能显著降低深度学习模型的数据依赖性，并大幅提升长期预测的精度、鲁棒性和稳定性。