Parametrized Physics-Informed Neural Networks for the 1D Fokker--Planck Equation of Ornstein--Uhlenbeck Processes

I present an investigation into parameterized Physics-Informed Neural Networks (PINNs) for approximating the one-dimensional Fokker--Planck equation (FPE) of the Ornstein–Uhlenbeck process. By considering a continuous family of Gaussian initial conditions parameterized by their mean and standard deviation, this approach learns a generalized mapping to the resulting time-dependent marginal distributions. To facilitate training and reduce parameter dependency, the problem is reformulated in a non-dimensional space, and all results are reported in terms of the corresponding non-dimensional parameters $m^{*}_0$ and $s^{*}_0$, namely the non-dimensional mean and standard deviation of the initial condition respectively. The training process follows a two-phase scheme consisting of Adam optimization with parameter resampling followed by L-BFGS refinement on a fixed parameter set. I compare three architectural configurations: a vanilla baseline (V-P2INN), a version utilizing residual connections (Res-P2INN), and a variant (FiLM-Res-P2INN) that employs a Feature-wise Linear Modulation in place of embedding concatenation. All architectures achieve median $L_2$ relative errors below 1% on unseen parameter configurations, indicating the feasibility of the parameterized approach for this class of stochastic processes. Beyond feasibility, the architectural comparison reveals distinct performance profiles across the parameter space. Residual connections substantially reduce high-error outliers. FiLM-Res-P2INN shows a consistent advantage for narrow initial conditions with small $s^{*}_0$ under both log-uniform and uniform sampling of $s^{*}_0$ in the training distributions of parameter pairs. However, when trained on a log-uniform distribution of $s^{*}_0$, FiLM-Res-P2INN overfits to the overrepresented low $s^{*}_0$ regime --a behavior not observed for Res-P2INN.