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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. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of partial differential equations (PDEs). We provide upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for partial differential equations (PDEs). We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented. [ABSTRACT FROM AUTHOR]