Koopman Operator Learning for Nonlinear PDEs
Abstract: Koopman operators globally linearize nonlinear dynamical systems, and their spectral information provides a powerful tool for understanding the underlying system. Data-driven methods such as Dynamic Mode Decomposition (DMD) have exploited this idea and are widely used on nonlinear PDEs. Yet the supporting theory is almost entirely restricted to finite-dimensional dynamical systems. This talk takes the first steps towards a rigorous framework for learning Koopman operators of infinite-dimensional dynamical systems. In infinite dimensions, one must tread carefully: even the existence of a Koopman operator for simple PDEs is not guaranteed. For nonlinear Hamiltonian PDEs, however, significant progress is possible. We prove that, for initial data drawn from a Gaussian measure, the dynamics admit a strongly continuous Koopman semigroup, whose spectral content yields a simple model for the PDE solution. We also give error bounds for numerical approximations of these operators via conditional expectations. Together, these results shed some theoretical light on the topic and suggest practical algorithms.