  The paper is about the computation of the principal spectrum of the Koopman
operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions
of the Koopman operator are the ones with the corresponding eigenvalues equal
to the eigenvalues of the linearization of the nonlinear system at an
equilibrium point. The main contribution of this paper is to provide a novel
approach for computing the principal eigenfunctions using a path-integral
formula. Furthermore, we provide conditions based on the stability property of
the dynamical system and the eigenvalues of the linearization towards computing
the principal eigenfunction using the path-integral formula. Further, we
provide a Deep Neural Network framework that utilizes our proposed
path-integral approach for eigenfunction computation in high-dimension systems.
Finally, we present simulation results for the computation of principal
eigenfunction and demonstrate their application for determining the stable and
unstable manifolds and constructing the Lyapunov function.
