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

Jiakang Chen*, Sufia Hashim*, and Carla Figueira de Morisson Faria†

  • *These authors contributed equally to this work.
  • †Contact author: c.faria@ucl.ac.uk
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Phys. Rev. Research 8, 033029 – Published 7 July, 2026

DOI: https://doi.org/10.1103/ltzl-bj62

Abstract

We develop an unsupervised physics-informed neural network (PINN) to solve the saddle-point equations (SPEs) governing direct above-threshold ionization (ATI) within the strong-field approximation. This setting provides a well-understood test bed in which the saddle-point structure is known for tailored driving fields, enabling systematic validation of the proposed solver. In this approach, the network is trained by minimizing the residual of the saddle-point equations 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 unconstrained 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 traditional local gradient-based solver for a range of monochromatic, few-cycle, bichromatic, elliptical, and bicircular fields, demonstrating robust recovery of the dominant complex ionization times over wide parameter ranges. The network naturally tracks changes in ionization event dominance as laser parameters are varied, enabling continuous exploration of regimes where conventional solvers require repeated manual initialization. This renders it much less sensitive to initial guesses than conventional solvers. Using the PINN-derived solutions, we compute coherent ATI photoelectron momentum distributions and show that 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 computational framework for semiclassical strong-field calculations and provide a foundation for extending machine-learning solvers to Coulomb-corrected models or to more complex strong-field 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. The present method is transferable to other research areas in which similar saddle-point equations are solved.

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Phys. Rev. Research 8, 033029– Published 7 July, 2026
  • Received 16 March 2026
  • Accepted 21 May 2026
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