PINN-wf: A PINN-based algorithm for data-driven solution and parameter discovery of the Hirota equation appearing in communications and finance

In this paper, we focus on the Hirota equation appearing in communications and finance. In the field of communications, the Hirota equation is used to describe the ultrashort pulse transmission in optical fibers, while model the generalized option pricing problem in finance. The data-driven solutions are derived and the parameters are calibrated through physics-informed neural networks (PINNs), where various complex initial conditions on a continuous wave background are considered and compared. PINNs define the loss function based on the strong form via partial differential equations (PDEs), while it is subject to the diminished accuracy when the PDEs enjoy high-order derivatives or the solutions contain complex functions. We hereby propose a PINN with weak form (PINN-wf), where the weak form residual of PDEs is embedded into the loss function accounting for data errors effectively. The proposed algorithm involves domain decomposition to derive the weak form function, assigning distinct test functions to each sub-domain based on the selected sample points. Two schemes of computational experiments are carried out to provide valuable insights into the dynamic characteristics of solutions to the Hirota equation. These experiments serve as a robust reference for understanding and analyzing the behavior of solutions in practical scenarios.