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\title{
% Error Bounds for Physics-Informed Neural Networks for the Fokker-Planck PDEs
Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
}

% The standard author block has changed for UAI 2025 to provide
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% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
% information below.
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% Add authors
\author[1]{\href{mailto:<chko1829@colorado.edu>?Subject=Your UAI 2025 paper}{Chun-Wei Kong}{}}
\author[2]{Luca Laurenti}
\author[1]{Jay McMahon}
\author[1]{Morteza Lahijanian}
% Add affiliations after the authors
\affil[1]{%
    Dept. of Aerospace Eng. Sciences\\
    University of Colorado Boulder
}
\affil[2]{%
    Center for Systems and Control \\
    Delft University of Technology  
}
  
  \begin{document}
\maketitle

\begin{abstract}
Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.
\end{abstract}

\input{sections/intro}
\input{sections/problem}
\input{sections/learn}
\input{sections/theory}
\input{sections/tight_error_bound}
\input{sections/experiment}
\input{sections/conclusion}

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\begin{acknowledgements} % will be removed in pdf for initial submission,
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    We acknowledge the anonymous reviewers for their constructive feedback. 
    This material is based upon work supported by the Air Force Research Laboratory/RVSW under Contract No. FA945324CX026. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Air Force Research Laboratory/RVSW.
\end{acknowledgements}

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\bibliography{uai2025-template}

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\onecolumn

\title{Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs\\(Supplementary Material)}
\maketitle

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\appendix
\input{sections/appendix_proofs}
\input{sections/appendix_systems}
\input{sections/appendix_training}


\end{document}
