Neural-network solution of subtracted three-body Faddeev integral equations near the Efimov limit

We apply a deep-neural-network (DNN) ansatz to the symmetrized spectator vector of the subtracted three-body Faddeev integral equation for identical bosons near the Efimov limit. The network is trained by minimizing the residual of the discretized integral equation, while the positive binding scale associated with the three-body energy is treated as a trainable parameter. Deterministic diagonalization of the same discretized kernel is used only as an a posteriori numerical benchmark. As preliminary validation, the neural-solver strategy is tested on the analytically solvable hydrogen radial problem. At unitarity, the DNN reproduces the Efimov ground-state binding scale with a DNN–deterministic deviation of $$0.022%$$ 0.022 % , while the first excited state is recovered to $$0.002%$$ 0.002 % . The deterministic solver recovers the universal Efimov sscaling ratio e2π /s₀ 515.03 e 2 π / s 0 ≃ 515.03 , and the neural method traces the bound-state branches as a function of the inverse scattering length 1/ a by continuation from the unitary solution. These results indicate that DNN-based residual minimization can provide a compact and differentiable representation of a renormalized few-body integral-equation solution in a regime governed by discrete scale invariance.