Physics-informed neural networks for solving saddle-point equations in strong-field physics with tailored fields

We develop an unsupervised physics-informed neural network to solve saddle-point equations (SPEs) governing direct above-threshold ionization (ATI) within the strong-field approximation. This setting provides a well-understood testbed in which the saddle-point structure is known for tailored driving fields, enabling systematic validation of the proposed solver. The network is trained by minimizing the residual of the SPEs and requires only the definition of the driving-field shape and its parameters, such as intensity, carrier-envelope phase, ellipticity, and relative phase. We introduce a window parametrization strategy that maps network outputs to prescribed regions of the complex-time plane, guiding the optimization toward physically relevant solutions and improving convergence stability. We benchmark the PINN against a conventional solver for a range of fields, demonstrating robust recovery of the dominant complex ionization times over wide parameter ranges. The network tracks changes in ionization event dominance as laser parameters are varied, enabling exploration of regimes where conventional solvers require repeated manual initialization. Using the PINN-derived solutions, we compute coherent ATI photoelectron momentum distributions and show the symmetries of the driving fields are reflected in both the saddle-point structure and the resulting spectra. These results establish PINNs as a promising framework for semiclassical strong-field calculations and provide a foundation for extending machine-learning solvers to Coulomb-corrected models or to more complex processes, such as rescattered ATI for which the SPEs are highly nonlinear and the presence of multiple closely-spaced solutions makes conventional Newton-type root-finding highly sensitive to initial guesses, which hinders systematic investigations across laser-parameter spaces, particularly for tailored fields.