Reducing boundary conditions for the Thomas–Fermi equation using a hybrid q-deformation and neural network method

This study introduces a hybrid \(q\) -calculus and artificial neural network framework for the Thomas–Fermi equation of free positive ions. Its contribution is a construction in which \(q\) -deformation combines the terminal derivative and terminal-value conditions, representing three classical boundary conditions by two constraints in the deformed problem. A boundary-compatible trial solution embeds the transformed constraints, allowing training to focus on the \(q\) -differential residual. For fixed \(q<1\) , the result is a deformed approximation with a classical limit as \(q→ 1\) . Experiments across nine ionization degrees and five seeds compared the \(q\) -ANN with a benchmark, a soft-constrained PINN, and an undeformed hard-constrained classical ANN. The \(q\) -ANN produced lower RMSE than the PINN in 42 of 45 matched runs and a lower median RMSE ( \(0.004127\) versus \(0.015044\) ), although two outlying outcomes caused mean RMSE to favor the PINN and paired tests did not establish a difference. Against the hard-constrained classical ANN, the \(q\) -ANN produced lower RMSE in 44 of 45 comparisons, indicating that the observed performance is not attributable solely to exact constraint embedding. The pointwise \(q\) -ANN was approximately \(102\) times slower than the vectorized PINN. The framework provides a new boundary-transformation strategy while retaining initialization sensitivity and computational cost as limitations.