A Deep Learning Approach for the solution of Probability Density Evolution of Stochastic Systems

The paper defines the GDEE, its IC with a Dirac delta, and a PINN loss that is the L2-norm of the PDE and IC residuals, and then states that ideally a zero loss implies the exact solution, while listing high-level assumptions including uniqueness, boundedness, and Lipschitz dependence on the IC; however, it does not supply a rigorous proof or a function-space treatment of the delta IC in that loss. In contrast, the candidate solution gives a correct distributional argument: zero loss implies a weak solution with the given IC and, by uniqueness (or characteristics since v is x-independent), it matches the true p_{XΘ}; conversely, the exact solution yields zero loss once the IC term is understood in a weak sense (e.g., via mollifiers or a negative Sobolev norm). This fills the paper’s gaps concerning the delta IC and the equivalence between zero loss and exact solution for the linear transport GDEE. See the paper’s definitions of the GDEE, IC, and loss, and its “ideally” claim and assumption list for context.