Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined in an unbounded domain requires efficient numerical methods that accurately resolve the dependence of the PDE on that variable over several orders of magnitude. Unbounded domain problems arise in various application areas and solving such problems is important for understanding multi-scale biological dynamics, resolving physical processes at long time scales and distances, and performing parameter inference in engineering problems. In this work, we combine two classes of numerical methods: (i) physics-informed neural networks (PINNs) and (ii) adaptive spectral methods. The numerical methods that we develop take advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs (s-PINNs) over standard PINNs in approximating functions, solving PDEs, and estimating model parameters from noisy observations in unbounded domains.