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

by T. De Ryck and A. D. Jagtap and S. Mishra

(Report number 2022-08)

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 PDE 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.

Keywords: deep learning, neural networks, PINNs, Navier-Stokes equations

@Techreport{DJM22_996,
  author = {T. De Ryck and A. D. Jagtap and S. Mishra},
  title = {Error estimates for physics informed neural networks approximating the Navier-Stokes equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-08},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-08.pdf },
  year = {2022}
}

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