Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse Problems

In recent years, there has been a growing interest in data-driven approaches in physics, such as extended dynamic mode decomposition (EDMD). The EDMD algorithm focuses on nonlinear time-evolution systems, and the constructed Koopman matrix yields the next-time prediction with only linear matrix-product operations. Note that data-driven approaches generally require a large dataset. However, assume that one has some prior knowledge, even if it may be ambiguous. Then, one could achieve sufficient learning from only a small dataset by taking advantage of the prior knowledge. This paper yields methods for incorporating ambiguous prior knowledge into the EDMD algorithm. The ambiguous prior knowledge in this paper corresponds to the underlying time-evolution equations with unknown parameters. First, we apply the proposed method to forward problems, i.e., prediction tasks. Second, we propose a scheme to apply the proposed method to inverse problems, i.e., parameter estimation tasks. We demonstrate the learning with only a small dataset using guiding examples, i.e., the Duffing and the van der Pol systems.

Extraction of nonlinearity in neural networks and model compression with Koopman operator

Nonlinearity plays a crucial role in deep neural networks. In this paper, we first investigate the degree to which the nonlinearity of the neural network is essential. For this purpose, we employ the Koopman operator, extended dynamic mode decomposition, and the tensor-train format. The results imply that restricted nonlinearity is enough for the classification of handwritten numbers. Then, we propose a model compression method for deep neural networks, which could be beneficial to handling large networks in resource-constrained environments. Leveraging the Koopman operator, the proposed method enables us to use linear algebra in the internal processing of neural networks. We numerically show that the proposed method performs comparably or better than conventional methods in highly compressed model settings for the handwritten number recognition task.