Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs

• URL: http://arxiv.org/abs/2210.03426v1
• Date: Fri, 7 Oct 2022 09:49:18 GMT
• Title: Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs
• Authors: Birgit Hillebrecht, Benjamin Unger
• Abstract summary: We present a rigorous upper bound on the prediction error of physics-informed neural networks. We apply this to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.

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