  We solve high-dimensional steady-state Fokker-Planck equations on the whole
space by applying tensor neural networks. The tensor networks are a tensor
product of one-dimensional feedforward networks or a linear combination of
several selected radial basis functions. The use of tensor feedforward networks
allows us to efficiently exploit auto-differentiation in major Python packages
while using radial basis functions can fully avoid auto-differentiation, which
is rather expensive in high dimensions. We then use the physics-informed neural
networks and stochastic gradient descent methods to learn the tensor networks.
One essential step is to determine a proper truncated bounded domain or
numerical support for the Fokker-Planck equation. To better train the tensor
radial basis function networks, we impose some constraints on parameters, which
lead to relatively high accuracy. We demonstrate numerically that the tensor
neural networks in physics-informed machine learning are efficient for
steady-state Fokker-Planck equations from two to ten dimensions.
