Functions Inverse Using Neural Networks via Branch-Wise Decomposition and Newton Refinement

In this work, a unified framework (using Neural Networks) is proposed to find the inverse of mathematical functions, spanning both simple one-to-one mapping and complex multivalued relations. The approach uses standard multilayer Neural Networks (NN) to approximate the functions’ inverse and introduces a deterministic branch-wise decomposition to handle multi-valued inverses. For single-valued (one-to-one) functions, a NN is directly trained on input-output pairs to learn the inverse mapping. For multi-valued functions, the function domain is decomposed into one-to-one branches, and a dedicated NN is trained for each branch. A refinement step using Newton’s method is applied to the NN output to further improve inversion accuracy. Across a broad set of benchmark functions, the proposed approach achieved low mean absolute error (MAE) and mean squared error (MSE) in recovering the true inverse, with high round-trip consistency. Newton refinement further reduces inversion error by rapidly converging to higher precision solutions. Notably, even for multi-valued inverse functions, each branch-specific NN can accurately recover the true inverse. Accordingly, standard NN, when combined with branch-wise decomposition and Newton refinement, can serve as an effective universal approximator for the inverse of functions across a spectrum of complexities.

@article{Balamash2025,
  title     = {Functions Inverse Using Neural Networks via Branch-Wise Decomposition and Newton Refinement},
  journal   = {International Journal of Advanced Computer Science and Applications},
  volume    = {16},
  number    = {12},
  year      = {2025},
  publisher = {The Science and Information Organization},
  author    = {Abdullah Balamash},
  doi       = {10.14569/IJACSA.2025.0161294},
  url       = {https://doi.org/10.14569/IJACSA.2025.0161294}
}