Error estimates for physics-informed neural networks approximating the Navier–Stokes equations

Abstract We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier–Stokes equations with (extended) physics-informed neural networks. We show that the underlying partial differential equation residual can be made arbitrarily small for tanh neural networks w...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2024-02, Vol.44 (1), p.83-119
Main Authors: De Ryck, Tim, Jagtap, Ameya D, Mishra, Siddhartha
Format: Article
Language:English
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Summary:Abstract We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier–Stokes equations with (extended) physics-informed neural networks. We show that the underlying partial differential equation residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drac085