Splitting-domain reformulation of symmetric strongly coupled four-spin NMR systems

The analysis of strongly coupled ¹H NMR spin systems of the AA'BB' and AA'XX' type has remained one of the most challenging problems in spectral interpretation for over six decades. The classical Pople framework provides a complete closed-form solution, but requires the prior assignment of observed peaks to specific quantum-mechanical transitions, a combinatorial problem that has rendered the method impractical for routine use. Alternative approaches based on iterative quantum-mechanical fitting exist but remain computationally demanding and sensitive to initial parameter estimates, limiting their adoption for routine analysis. Here we demonstrate that both limitations can be bypassed by reformulating the analysis in the domain of pairwise frequency differences. We show that the coupling interaction between nuclei guarantees the existence of characteristic shared splittings within this set, reducing the problem from combinatorial assignment to pattern recognition. A Set Transformer extracts the relevant splittings from the unordered difference set, and a differentiable Pople inversion layer converts them into coupling constants through exact analytical equations, enabling end-to-end training with physics-informed loss functions. The approach requires prior identification of the coupling topology and is validated on two distinct four-spin cases (ortho- and para-disubstituted benzenes), each served by a dedicated model, achieving sub-0.05 Hz mean absolute errors on synthetic data and sub-0.08 Hz on experimental spectra from six compounds spanning the full range from the AA'XX' limit to the genuinely second-order AA'BB' regime. The two models share an identical neural architecture differing only in the deterministic Pople layer, demonstrating the generality of the splitting-domain representation across coupling topologies.