Titel: Kernel-based Approximation of Dynamical Systems and Quantum Problems
Abstract: Reproducing kernel Hilbert spaces (RKHS) are powerful model classes which have been applied to a wide range of problems. In this talk, I will present recent progress on the use of kernel-based methods for the approximation of evolution operators of dynamical systems based on ergodic sampling. In the limit of infinite data, the resulting method provides a Galerkin projection of evolution operators on (possibly infinite-dimensional) RKHSs. In the second part, I will recall the connection between a general class of differential operators and stochastic dynamical systems, and explain how data-driven methods can in principle be used to approximate a broad range of differential operators. This also includes the Schrödinger operator for electronic quantum systems. For this context, I will present recent results on incorporating the specific symmetries of quantum systems into kernel-based approximation schemes.
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