  The Fokker-Planck equation describes the evolution of the probability density
associated with a stochastic differential equation. As the dimension of the
system grows, solving this partial differential equation (PDE) using
conventional numerical methods becomes computationally prohibitive. Here, we
introduce a fast, scalable, and interpretable method for solving the
Fokker-Planck equation which is applicable in higher dimensions. This method
approximates the solution as a linear combination of shape-morphing Gaussians
with time-dependent means and covariances. These parameters evolve according to
the method of reduced-order nonlinear solutions (RONS) which ensures that the
approximate solution stays close to the true solution of the PDE for all times.
As such, the proposed method approximates the transient dynamics as well as the
equilibrium density, when the latter exists. Our approximate solutions can be
viewed as an evolution on a finite-dimensional statistical manifold embedded in
the space of probability densities. We show that the metric tensor in RONS
coincides with the Fisher information matrix on this manifold. We also discuss
the interpretation of our method as a shallow neural network with Gaussian
activation functions and time-varying parameters. In contrast to existing deep
learning methods, our method is interpretable, requires no training, and
automatically ensures that the approximate solution satisfies all properties of
a probability density.
