Subordination-based neural network approach for non-Markovian Fokker-Planck equations

The non-Markovian Fokker-Planck equation (FPE) provides an effective description of anomalous diffusion within the classical continuous time random walk framework. However, its nonlocal memory kernel leads to strong non-Markovian behaviors, making the non-Markovian FPE difficult to solve. Conventional numerical approaches must rely on intricate discretized implementations of fractional-calculus differentiation to approximate the fractional operators, leading to substantial computational and storage costs. To overcome these challenges and obtain the probability density function (PDF) across the entire temporal regime, we propose a subordination-based neural network (SNN) that leverages an equivalent subordinated integral representation of non-Markovian dynamics. This representation transforms the original non-Markovian FPE into a Markovian FPE formulated in an operational time where all memory effects are encapsulated by the subordination mechanism, enabling the solution via a deep neural network and direct transfer to FPEs with different memory kernels that share the same auxiliary Markovian FPE, without retraining. We validate the method for non-Markovian FPEs under nonlinear external forces, general boundary conditions, and with arbitrary memory kernels. With four representative examples, the SNN maintains high accuracy in both the central and tail regions of the PDFs, as well as in additional statistical quantities, e.g., survival probability and mean velocity, that are relevant to practical applications. The SNN approach provides a promising route for future extensions to higher-dimensional systems.