Topology-Constrained Saddle-Point Reconstruction of Strong-Field Photoelectron Momentum Spectra

In strong-field ionization, complex-time saddle points are closely related to electron emission times and quantum-orbit interference patterns. However, not all solutions to the saddle-point equation carry physical contributions. Saddle coalescence can also invalidate the isolated-saddle approximation. Within the strong-field-approximation framework, we propose a topology-constrained approach to reconstruct photoelectron momentum spectra. A physics-informed neural network is employed to produce candidate roots, which are subsequently refined via a Newton-type iteration. A topological selector is defined by the non-zero support of intersection numbers to filter unphysical solutions, while local coherent continuation for pole-saddle pairs handles saddle-coalescence regions. We test our method on hydrogen atoms subjected to six-cycle elliptically polarized laser fields and compare the reconstructed spectra with direct numerical strong-field-approximation results. For 300 nm and 500 nm driving fields, the normalized relative L2 errors on the linear probability scale reach 3.793% and 3.520%, respectively. Our method improves the physical interpretability of saddle-based spectrum reconstruction within the strong-field approximation and is promising for other problems in computational physics involving saddle-point equations.