Error estimates for physics informed neural networks approximating the Navier-Stokes equations
This work provides theoretical guarantees for PINNs in fluid dynamics, which is incremental as it extends error analysis to a complex domain-specific problem.
The paper tackles the problem of approximating the incompressible Navier-Stokes equations using physics-informed neural networks (PINNs) by proving rigorous error bounds, showing that the PDE residual can be made arbitrarily small with specific network architectures, and estimating total error based on training error, network size, and quadrature points.
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.