Deep Neural Network extraction of Unpolarized Transverse Momentum Distributions

Building on the first-ever application of neural networks in TMD phenomenology: "Extraction of the Sivers function with deep neural networks", we now present a momentum space, physics-informed deep learning framework for the direct extraction of unpolarized transverse momentum dependent parton distributions (TMDs) from fixed target Drell-Yan data (E288, E605). Rather than transforming to impact-parameter space, we remain in k and embed a normalized integrand s(x, k; Q) whose auto-convolution produces the observed qT spectra. The extraction proceeds in two steps. Stage I learns the structure kernel S(qT , x1, x2; QM ) by regressing the cross-section with known kinematic prefactors and charge-weighted PDF combinations factored out; experimental and PDF uncertainties are propagated with Monte Carlo replicas. Stage II reconstructs s(x, k; Q) with an end-to-end differentiable k quadrature layer. Applied to Fermilab cross-section data from experiments E288 and E605, the method reproduces the measured qT spectra across Q and yields x and Q dependent TMDs that broaden with Q, with uncertainty bands that consistently propagate experimental, PDF, algorithmic and methodological components. The approach is minimally biased (no factorized Ansatze and no bT transform) and provides a transferable template for polarized TMDs and related QCD inverse problems.