Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networks

Physics-informed Neural Networks (PINNs) have recently gained popularity in the scientific community due to their effective approximation of partial differential equations (PDEs) using deep neural networks. However, their application has been generally limited to interpolation scenarios, where predictions rely on inputs within the support of the training set. In real-world applications, extrapolation is often required, but the out of domain behavior of PINNs is understudied. In this paper, we provide a detailed investigation of PINNs' extrapolation behavior and provide evidence against several previously held assumptions: we study the effects of different model choices on extrapolation and find that once the model can achieve zero interpolation error, further increases in architecture size or in the number of points sampled have no effect on extrapolation behavior. We also show that for some PDEs, PINNs perform nearly as well in extrapolation as in interpolation. By analyzing the Fourier spectra of the solution functions, we characterize the PDEs that yield favorable extrapolation behavior, and show that the presence of high frequencies in the solution function is not to blame for poor extrapolation behavior. Finally, we propose a transfer learning-based strategy based on our Fourier results, which decreases extrapolation errors in PINNs by up to $82 \%$.