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    We developed a rigorous upper bound for physics-informed neural network (PINN) prediction errors. This bound uses only a priori information about the dynamical system, not the true solution, aiding PDE-governed model analysis.

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    Area of Science:

    • Computational Mathematics
    • Machine Learning
    • Scientific Computing

    Background:

    • Prediction error quantification is often overlooked in neural network (NN) research.
    • Existing methods for NNs, both data-driven and physics-informed, lack rigorous error bounds.
    • Physics-informed neural networks (PINNs) offer a promising approach for solving partial differential equations (PDEs).

    Purpose of the Study:

    • To introduce a rigorous a posteriori upper bound for the prediction error of PINNs.
    • To provide a method for error quantification that does not require knowledge of the true solution.
    • To demonstrate the applicability of the proposed error bound to various PDE-governed systems.

    Main Methods:

    • Derivation of a theoretical upper bound for PINN prediction error.
    • The bound relies solely on a priori information about the dynamical system governed by a PDE.
    • Application and validation of the error bound on benchmark PDE problems.

    Main Results:

    • A novel, computable upper bound on PINN prediction error is presented.
    • The error bound is demonstrated to be effective across diverse PDEs, including transport, heat, Navier-Stokes, and Klein-Gordon equations.
    • The method quantifies uncertainty without needing the ground truth solution.

    Conclusions:

    • The proposed a posteriori error bound offers a significant advancement in the reliable application of PINNs.
    • This work addresses a critical gap in the methodological investigation of PINNs by providing robust error quantification.
    • The developed technique enhances the trustworthiness and interpretability of PINN models in scientific applications.