Scientific machine learning meets semi-analytical computation: a hybrid NIM-PINN approach for nonlinear PDEs

In this paper, a hybrid semi-analytical–deep learning framework for the solution of nonlinear partial differential equations (PDEs) is proposed that integrates the New Iterative Method (NIM) with Physics-Informed Neural Networks (PINNs). Low–order NIM expansion provides an analytic baseline that meets the required initial and boundary conditions; a neural network is trained to learn only the residual correction between the analytic baseline and the exact solution. The framework is evaluated on three problems: a linear diffusion equation, the nonlinear reaction–diffusion Newell–Whitehead–Segel (NWS) equation with a sharp transition front, and the nonlinear convective Burgers equation with a sigmoidal travelling front. All three problems are explicitly solved with Dirichlet boundary conditions, and for the baseline-corrected hybrid model, a single network architecture (three inputs, four hidden layers consisting of eighty tanh units, one output) is used in the same way as in a pure PINN control, so that all reported comparisons are architecture-matched. For all of the test examples, the mean absolute error and the relative $$L^2$$ error of the raw NIM baseline are lowered by one to two orders of magnitude. The controlled ablation over the baseline truncation order $$m_0=1,2,3$$ demonstrates the baseline error decays geometrically due to the NIM initialization error, confirming the separation of the NIM initialization error from the neural correction error. All the numerical statements in the text can be verified directly against a single source, as the single unified results table reports the mean absolute error, root-mean-square error, relative $$L^2$$ error, maximum error, PDE residual norm, and boundary residual norm for each method–problem pair. The results demonstrate that the hybrid approach is a stable and effective method to reduce the overall function space that the correction network needs to learn, but it is problem-dependent and this is explicitly discussed.