Deep learning nonlinear multiscale dynamic problems using Koopman operator

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 nonlinear 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. (C) 2021 Elsevier Inc. All rights reserved.