  We propose a novel approach for computing committor functions, which describe
transitions of a stochastic process between metastable states. The committor
function satisfies a backward Kolmogorov equation, and in typical
high-dimensional settings of interest, it is intractable to compute and store
the solution with traditional numerical methods. By parametrizing the committor
function in a matrix product state/tensor train format and using a similar
representation for the equilibrium probability density, we solve the
variational formulation of the backward Kolmogorov equation with linear time
and memory complexity in the number of dimensions. This approach bypasses the
need for sampling the equilibrium distribution, which can be difficult when the
distribution has multiple modes. Numerical results demonstrate the
effectiveness of the proposed method for high-dimensional problems.
