当前位置: X-MOL 学术J. Comput. Phys. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Deep learning nonlinear multiscale dynamic problems using Koopman operator
Journal of Computational Physics ( IF 4.1 ) Pub Date : 2021-08-24 , DOI: 10.1016/j.jcp.2021.110660
Mengnan Li , Lijian Jiang

In this paper, a deep learning method using Koopman operator is presented for modeling nonlinear multiscale dynamical problems. Koopman operator is able to transform a non- linear dynamical system into a linear system in a Koopman invariant subspace. However, it is usually very challenging to choose a set of suitable observation functions spanning the Koopman invariant subspace when only data is available for the model. It is practically important for us to predict the evolution of the state of the dynamical system from the Koopman invariant subspace. To this end, we introduce a reconstruction operator that maps the observation function space to the model's state space. Incorporating measurement data, a set of neural networks are constructed to learn the Koopman invariant subspace and the reconstruction operator. The loss function not only considers the properties of Koopman invariant subspace, but also reflects the prediction of future state, which makes the proposed method can realize the prediction of future state for a relatively long time. It may be experimentally expensive to collect the fine-scale data. It will be challenging to use limited computational resources to generate sufficient fine-scale data for neural network training. To overcome this difficulty, we use the data in a coarse-scale and learn effective coarse models for the nonlinear multiscale dynamical problems. In order to make the learned coarse model effectively capture fine-scale information, the loss functions for the neural networks are constructed using a set of multiscale basis functions, which are assumed to be given as a prior. In this case, an accurate fine-scale model can be derived by downscaling the learned coarse model. The deep learning multiscale models using Koopman operator can achieve a relatively long-time prediction for the evolution of the state of the nonlinear multiscale dynamical problems. A few numerical examples are presented to show that the effectiveness of learning multiscale models and the long-time prediction. The numerical results also demonstrate the advantage of the proposed learning method over some other similar learning methods.



中文翻译:

使用 Koopman 算子的深度学习非线性多尺度动态问题

在本文中,提出了一种使用 Koopman 算子的深度学习方法来建模非线性多尺度动力学问题。Koopman 算子能够将非线性动力系统转换为 Koopman 不变子空间中的线性系统。然而,当模型只有数据可用时,选择一组合适的跨越 Koopman 不变子空间的观测函数通常是非常具有挑战性的。从 Koopman 不变子空间预测动力系统状态的演化对我们来说非常重要。为此,我们引入了一个重构算子,将观察函数空间映射到模型的状态空间。结合测量数据,构建一组神经网络来学习 Koopman 不变子空间和重构算子。损失函数不仅考虑了 Koopman 不变子空间的性质,还反映了对未来状态的预测,使得该方法可以实现对未来状态的预测较长时间。收集精细数据在实验上可能是昂贵的。使用有限的计算资源为神经网络训练生成足够的精细数据将具有挑战性。为了克服这个困难,我们使用粗尺度数据并学习非线性多尺度动力学问题的有效粗模型。为了使学习到的粗模型有效地捕获细尺度信息,神经网络的损失函数是使用一组多尺度基函数构建的,假设这些函数是先验的。在这种情况下,可以通过缩小学习到的粗模型来推导出准确的精细模型。使用 Koopman 算子的深度学习多尺度模型可以实现对非线性多尺度动力学问题状态演化的相对长时间的预测。给出了一些数值例子来证明学习多尺度模型和长时间预测的有效性。数值结果也证明了所提出的学习方法优于其他一些类似的学习方法。给出了一些数值例子来证明学习多尺度模型和长时间预测的有效性。数值结果也证明了所提出的学习方法优于其他一些类似的学习方法。给出了一些数值例子来证明学习多尺度模型和长时间预测的有效性。数值结果也证明了所提出的学习方法优于其他一些类似的学习方法。

更新日期:2021-09-01
down
wechat
bug