Deep-OSG：半群中算子的深度学习
Deep-OSG: Deep Learning of Operators in Semigroup
AI总结：
本文提出Deep-OSG框架，首次将半群性质嵌入数据驱动学习过程，通过新神经网络架构和损失函数学习可变时间步长的演化算子族，显著提升了长期预测的准确性、鲁棒性并降低数据依赖。
摘要：
本文提出了一种用于学习半群中算子的新型深度学习方法，其应用是利用在不同时间滞后下收集的时间序列数据，对未知的自治动力系统进行建模。本文是之前流图学习（FML）工作的续篇，这些前期工作侧重于以固定时间步长学习单一演化算子。本文旨在学习具有可变时间步长的一族演化算子，它们构成了一个自治系统的半群。半群性质非常关键，它将系统在不同时间尺度上的演化行为联系起来，但在之前的工作中并未被考虑。我们首次提出了一个将半群性质嵌入到数据驱动学习过程中的框架，通过一种新颖的神经网络架构和新的损失函数来实现。该框架高度可行，可与任何合适的神经网络结合，并适用于学习一般的自治ODEs和PDEs。我们给出了严格的误差估计和方差分析，以理解我们方法的预测精度和鲁棒性，展示了模型中半群感知的显著优势。此外，我们的方法允许任意选择预测的时间步长，并确保预测结果良好地自洽且一致。广泛的数值实验表明，嵌入半群性质显著降低了深度学习模型对数据的依赖性，并极大地提高了长期预测的准确性、鲁棒性和稳定性。
英文摘要：
This paper proposes a novel deep learning approach for learning operators in semigroup, with applications to modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous flow map learning (FML) 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
