Optimal prediction of dynamical systems with incomplete data

Dynamical systems describe a variety of real-life problems related to evolution over time. It is often the case that the problems are so complicated that mathematical models for them either do not exist or are not accurate enough. Nowadays, there are data-driven techniques that allow us to obtain a mathematical model from measurement data over time. One of such techniques is the method of dynamic mode decomposition (DMD). It was originally proposed by Schmid (2010) as a data-processing algorithm. Then, it was realized that DMD can be immediately derived from the Koopman analysis.

The key idea of the Koopman theory is that any nonlinear dynamical system can be made linear if we extend the space of dependent (or state) variables via their nonlinear functions, called the observables. In that space, the problem becomes linear but the dimension of space is infinite. The Koopman operator provides the dynamics (trajectory) of any observable. This operator is unknown and can be approximated from the measurement data. DMD provides a linear approximation of the Koopman operator. As a result, this approach allows us to identify key modes. As such, one can obtain a low-rank approximate solution that can be used for both the analysis of the system and its prediction. Thus, DMD is a factorization and low dimensionality algorithm. There are numerous variants of DMD. One of them, called the higher-order DMD (HODMD) is based on the generalization of the Koopman operator with delayed snapshots. It has been demonstrated that for a variety of problems, HODMD is capable of significantly increasing the accuracy of prediction.

A key problem with the application of DMD to real-life problems is related to incomplete data. The measurement data is practically always limited. In addition, as a rule, this data is noisy and inaccurate. A recently developed modification of DMD based on the optimal prediction (Katrutsa et al, 2023) is capable of taking into account the effect of uncertainties in the input data in the most optimal way. In the project, this approach is supposed to be extended to HODMD with a higher accuracy of approximation in the algorithm to make it more accurate and efficient. It is supposed to be applied to the analysis of electric power grids. One of the advantages of the algorithm is that it can be potentially applied to a variety of problems since it is entirely based on the input data.

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