\section{Conclusion}
We proposed a physics-informed learning method to approximate the PDF evolution governed by FP-PDE and bound its error using recursively learned PINN-based error functions. We proved that only two error terms are needed for arbitrarily tight bounds and introduced a more efficient first-order bound requiring just one error function, reducing computation while providing clear termination criteria. 
Our results validate the bounds' correctness and demonstrate significant computational speedups over Monte Carlo methods. 
We trained the solution and error PINNs separately, but joint training may work better, which is left for future work.

