  Neural networks are increasingly recognized as a powerful numerical solution
technique for partial differential equations (PDEs) arising in diverse
scientific computing domains, including quantum many-body physics. In the
context of time-dependent PDEs, the dominant paradigm involves casting the
approximate solution in terms of stochastic minimization of an objective
function given by the norm of the PDE residual, viewed as a function of the
neural network parameters. Recently, advancements have been made in the
direction of an alternative approach which shares aspects of nonlinearly
parametrized Galerkin methods and variational quantum Monte Carlo, especially
for high-dimensional, time-dependent PDEs that extend beyond the usual scope of
quantum physics. This paper is inspired by the potential of solving
Hamilton-Jacobi-Bellman (HJB) PDEs using Neural Galerkin methods and commences
the exploration of nonlinearly parametrized trial functions for which the
evolution equations are analytically tractable. As a precursor to the Neural
Galerkin scheme, we present trial functions with evolution equations that admit
closed-form solutions, focusing on time-dependent HJB equations relevant to
finance.
