Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
This work addresses the need for theoretical guarantees in PINNs, a popular method for solving PDEs, though it is incremental as it builds on existing stability properties.
The authors tackled the problem of quantifying the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs by deriving rigorous upper bounds in terms of training error and sample count, with validation through numerical experiments on nonlinear PDE examples.
Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous 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.