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

Abstract

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.

Figures (9)

• Figure 1: Electric field (blue solid line) and the corresponding vector potential (orange dashed line as functions of time for the (a) monochromatic field, (b-b") few-cycle pulse with $N=4.3$ and carrier-envelope phases $\phi = 65^{\circ}, 155^{\circ}$ and $245^{\circ}$, (c-c"') ($\omega, 2\omega$) and $(\omega, 3\omega)$ bichromatic linearly polarized fields with ratio of field amplitudes $\xi = 0.8$ and relative phase $\phi$ as indicated in the panels, (d-d"') elliptically polarized fields with ellipticity $\varepsilon = 0.1, 0.3, 0.5$ and $0.7$ and (e-e") counter-rotating bicircular fields with identical amplitude, relative phase $\phi$ and commensurate frequency $s$ as indicated in the panels. For all fields, we use wavelength $\lambda = 800$nm ($\omega = 0.057$) a.u., and (peak) intensity $I = 1.0 \times 10^{14}$ W/cm$^2$.
• Figure 2: Schematic of the physics-informed neural network with 3 hidden layers. Inputs comprise of the momentum components ($p_\parallel,p_\perp$) and field parameters $\gamma$. The network outputs the real and imaginary parts of the complex saddle time, $(t_{\mathrm{R}},t_{\mathrm{I}})$. Training minimizes a physics residual $\mathcal{L}_{\text{physics}}$ derived from the saddle-point equation until the mean square error falls below a threshold accuracy; a supervised data term $\mathcal{L}_{\text{data}}$ can be included when labeled solutions are available, but is omitted here.
• Figure 3: Window configuration for the $x$-direction saddle-time solutions in a four-fold bicircular field. The scatter points represent the complex-time solutions of the saddle-point equation. The four windows, labeled W1--W4, define the prescribed output regions used in the windowed parameterization. For each window, the black dashed vertical line marks the window center, while the colored dashed lines and arrows indicate the corresponding window range. The upper and lower panels show the two sets of solution branches in the $x$-direction.
• Figure 4: Training loss versus epoch for the five best-performing configurations of the PINN in Table \ref{['tab:ablation']} [panel (a)] showing the model converges faster in early training and the real saddle point times $t_{\mathrm{R}}$ predicted by the PINN along $(p_\parallel, p_\perp=0)$ [panel (b)], showing that windowing enables consistent recovery of the desired root rather than collapsing to the nearest solution.
• Figure 5: The imaginary [panels (a),(a')] and real [panels (b), (b')] parts of the direct ATI ionization time computed at $(p_{\parallel}, 0)$ with the PINN solver detailed in Sec. \ref{['sec:method']}, for two windows [first and second rows respectively, and distinguished using crosses and circles] in one cycle of the monochromatic field given by Eq. \ref{['eq:monofield']}. Three different intensities $I=0.5\times10^{14}$W/cm$^2$, $1\times10^{14}$W/cm$^2$ and $2 \times10^{14}$W/cm$^2$ have been used, denoted by blue, orange and black lines respectively. Panels (c), (c') show the PMD for the smallest and largest intensities respectively, while panels (d) and (d') display the mean square error in the real and imaginary parts of the time for all intensities, respectively. All other field parameters are the same as in the associated panel of Fig. \ref{['fig:fields_overview']}.