Variational estimation of generator invariant subspaces

We present VEGIS (Variational Estimation of Generator Invariant Subspaces), a variational method to approximate invariant subspaces of the infinitesimal generator of reversible diffusion processes. The method represents a trial subspace by neural networks and optimizes a Dirichlet-form trace objective that can be evaluated using only equilibrium samples and gradients of the network outputs. After training, the learned trial space can be diagonalized to recover generator eigenfunctions and eigenvalues or transformed by PCCA+ to obtain membership functions associated with metastable sets. In addition, we introduce a VEGIS-driven sampling strategy in which a rough approximation of the dominant slow mode is used to modify the effective diffusivity while preserving the invariant density. Numerical results on low-dimensional and molecular systems demonstrate the accuracy and flexibility of VEGIS.