Artificial neural network solver for Fokker-Planck and Koopman eigenfunctions

For a stochastic differential equation (SDE) that is an It\^{o} diffusion or Langevin equation, the Fokker-Planck operator governs the evolution of the probability density, while its adjoint, the infinitesimal generator of the stochastic Koopman operator, governs the evolution of system observables, in the mean. The eigenfunctions of these operators provide a powerful framework to analyze SDEs, and have shown to be particularly useful for systems of stochastic oscillators. However, computing these eigenfunctions typically requires solving high-dimensional PDEs on unbounded domains, which is numerically challenging. Building on previous work, we propose a data-driven artificial neural network solver for Koopman and Fokker-Planck eigenfunctions. Our approach incorporates the differential operator into the loss function, improving accuracy and reducing dependence on large amounts of accurate training data. We demonstrate our approach on several numerical examples in two, three, and four dimensions.