Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems
This work addresses the problem of limited data availability in physics modeling for researchers, offering an incremental improvement by integrating prior knowledge into existing methods.
The paper tackles the challenge of data-driven physics modeling with small datasets by incorporating ambiguous prior knowledge, such as underlying equations with unknown parameters, into the extended dynamic mode decomposition (EDMD) algorithm for Koopman operators, demonstrating its application to forward prediction and inverse parameter estimation tasks using examples like the Duffing and van der Pol systems.
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.