A hybrid semi-analytical and neural network approach for solving nonlinear Kolmogorov and Rosenau-Hyman equations

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This paper presents a hybrid semi-analytical and physics-informed neural network framework for solving linear and nonlinear partial differential equations. The proposed approach combines the New Iterative Method (NIM) with Physics-Informed Neural Networks (PINNs): a low-order analytical approximation is generated by NIM, and a neural network is trained to learn only the remaining correction term. We choose NIM over related semi-analytical methods (ADM, HAM, HPM, VIM) because its recurrence is obtained by direct integration with no linearization parameter to tune and no Adomian-polynomial bookkeeping, which keeps the symbolic differentiation of the baseline – required at every training step – inexpensive even for third-order operators. Each NIM term is also obtained independently, so the truncation order can be treated as a hyperparameter without re-deriving the whole series. This approach simplifies the learning process, improves convergence behaviour, and increases computational efficiency. The linear Kolmogorov equation and the nonlinear Rosenau-Hyman equation are used as benchmark problems. An explicit Dirichlet boundary-condition loss term is included in the training objective, together with a zero-anchored initialization of the correction network and a validated-selection safeguard that reports whichever of the baseline or the corrected solution is more accurate on a held-out set. The numerical and graphical results show close agreement between the hybrid solution and the exact solution for both benchmark problems. For the Kolmogorov equation, the hybrid solution improves on the analytic baseline by roughly two orders of magnitude (relative \(L_2\) SX error \(7.6\times 10^{-5}\) versus \(6.8\times 10^{-3}\) ) and is more than an order of magnitude more accurate (roughly 23 times) than a standard PINN of identical size trained under the same budget. For the Rosenau-Hyman equation, the governing equation and boundary term together yield a hybrid relative \(L_2\) error of \(2.3\times 10^{-4}\) , improving on both the analytic baseline ( \(6.9\times 10^{-4}\) ) and a standard PINN ( \(1.3\times 10^{-2}\) ). Error analysis, residual distributions, parity plots, and convergence curves confirm the accuracy, stability, and robustness of the proposed method across both problems, with statistical validation across multiple independent training seeds.

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