Learning in Sinusoidal Spaces with Physics-Informed Neural Networks

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing differential equations, to ensure its output is consistent with fundamental physics laws. However, it turns out to be difficult to train an accurate PINN model for many problems in practice. In this paper, we address this issue through a novel perspective on the merits of learning in sinusoidal spaces with PINNs. By analyzing asymptotic behavior at model initialization, we first prove that a PINN of increasing size (i.e., width and depth) induces a bias towards flat outputs. Notably, a flat function is a trivial solution to many physics differential equations, hence, deceptively minimizing the residual term of the augmented loss while being far from the true solution. We then show that the sinusoidal mapping of inputs, in an architecture we label as sf-PINN, is able to elevate output variability, thus avoiding being trapped in the deceptive local minimum. In addition, the level of variability can be effectively modulated to match high-frequency patterns in the problem at hand. A key facet of this paper is the comprehensive empirical study that demonstrates the efficacy of learning in sinusoidal spaces with PINNs for a wide range of forward and inverse modelling problems spanning multiple physics domains.