Numerical challenges in data driven learning of nonlinear dynamics

The Dynamic Mode Decomposition (DMD) is a popular and versatile numerical method for data driven analysis of nonlinear dynamical systems. The theoretical bedrock upon which the more general Extended DMD framework (EDMD) is built is the Koopman composition operator. EDMD is based on data driven Rayleigh-Ritz extraction of spectral information of a Koopman composition operator associated with the underlying dynamical system. The nonlinear data snapshots are then represented using the eigenvectors/eigenfunctions of the operator, resulting in a modal (spatio-temporal) decomposition KMD (Koopman Mode Decomposition) of the dynamics. The KMD can also be considered as a model order reduction tool that represents the nonlinear dynamics using selected eigenpairs of the operator which is in fact an infinite dimensional linearisation of the nonlinear dynamics. It can be used to reveal coherent states and for forecasting and control. The numerical realization of the Koopman operator framework (i.e. DMD and EDMD) for computational analysis of nonlinear dynamics is an excellent illustration of the importance and power of numerical linear algebra, and an instructive case study of the software development based on the design principles introduced in the state of the art software packages such as e.g. LAPACK. The main ingredients of all variants of the DMD are the SVD and low rank approximations, the QR factorization and the orthogonal projections, structured sparsity constrained least squares approximations, and approximations of eigenvalues and eigenvectors from subspaces, fast updating in sreaming real time applications. There are many challenges in the development of robust numerical methods and this talk will address some of them. For example, in some cases numerically computed compression of the (in general non-normal) Koopman operator exhibits high non-normality, and, as a result, the eigenvectors are highly ill-conditioned and the KMD becomes numerically unstable. This depends on the operator and on the basis (functions) used to derive finite dimensional compression. The problem of ill-conditioned eigenvectors is solved using a Koopman-Schur decomposition that is entirely based on unitary transformations. The analysis in terms of the eigenvectors as modes of a Koopman operator is replaced with a modal decomposition in terms of orthonormal bases in a flag of invariant subspaces that correspond to selected eigenvalues. Further, theoretical guarantees require nontrivial input from perturbation theory that identifies relevant condition numbers, and reliable residual estimates based solely on the supplied data. In particular, the backward error analysis is connected with the shadowing theory to properly interpret the computed results.