{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T00:43:21Z","timestamp":1781570601047,"version":"3.54.5"},"reference-count":44,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","funder":[{"name":"Taif University, Saudi Arabia","award":["TU-DSPP-2024-08"],"award-info":[{"award-number":["TU-DSPP-2024-08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Model. Simul. Sci. Comput."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:p> Deep learning solves the Nonlinear Evolution Equations (NLEEs) with the aid of the gradient-enhanced Physics-Informed Neural Networks (gPINNs). The PINNs have successfully been used to solve the PDEs as they successfully integrate the residual of the PDE into the neural network\u2019s loss function. gPINNs use the predictions as well as the gradients of the prediction, providing information about the behavior of the solution in general. gPINNs train faster as the information of the gradients guides the neural network better. They are more effective and stronger tools for higher-dimensional nonlinear problems. The aim of this paper is to seek a soliton solution of the nonlinear PDE known as the Phi-four equation with the aid of the gPINN method. It will be applied: A Neural Network with two to six hidden layers, with 10, 20, 30, 40 and 50 neurons in one of the hidden layers for a total of five artificial NNs configurations. Time series of mean square and absolute error will also be investigated. Some special initial and boundary conditions will also be considered to train the NN for the governing model as well as to detect the equations\u2019 errors by comparing the obtained results with the analytical solution. The results show thus a relatively good correspondence in the governing model\u2019s approximate solution, the mean square error of which is of the order [Formula: see text] to [Formula: see text] for all the simulated equations. We try this method for the first time for this model; thus new and novel results are presented here. <\/jats:p>","DOI":"10.1142\/s1793962325500424","type":"journal-article","created":{"date-parts":[[2025,5,16]],"date-time":"2025-05-16T06:12:44Z","timestamp":1747375964000},"source":"Crossref","is-referenced-by-count":5,"title":["Neural networks optimized by data-driven gradient physics for soliton pulse equation solutions"],"prefix":"10.1142","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7206-2508","authenticated-orcid":false,"given":"Syed T. 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