Fourier domain physics-informed neural network

Ultrafast optics is driven by a myriad of complex nonlinear dynamics. The ubiquitous presence of governing equations in the form of partial integro-differential equations (PIDE) necessitates the need for advanced computational tools to understand the underlying physical mechanisms. From the experimental perspective, signal-to-noise ratio and availability of measurable data account for a bottleneck in numerical and data-driven modeling methods. In this paper, we extend the application of the physics-informed neural networks (PINNs) architecture to include prior knowledge in both the physical and Fourier domains. We demonstrate our Fourier domain PINN (FD-PINN) in two distinct forms. The continuous time FD-PINN predicts accurate solutions to a generalized pulse propagation equation, which includes the complete delayed nonlinear response, in the data-starved and noisy regime. We extend the architecture to the discrete time FD-PINN to recover the delayed-response physics from spatially separated measurement points. Our architecture ensures high fidelity predictive modeling and hidden physics recovery for applications such as image reconstruction, pulse characterization and shaping, as well as hidden parameter discovery. The benefits of the FD-PINN for ultrafast nonlinear optics make it immediately experimentally deployable. FD-PINN represents the next generation of tools to study optical phenomena, both through modeling and measurements for both forward and inverse problems.