A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 15; pp. 16255 - 16281
• Main Authors: Mostajeran, F., Hosseini, S. M.
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.10.2023
• Subjects: Algorithms; Exact solutions; Finite element method; Model accuracy; Neural networks; Partial differential equations; Regularization
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9449

In this study, we introduce a new growing neural network algorithm that is based on wavelet neural networks and call our algorithm a growing wavelet neural network (GWNN) method. We apply our proposed scheme to train a wavelet neural network to solve chemotaxis problems with blow‐up. These problems are highly nonlinear time‐dependent systems of partial differential equations, and it is a challenge to get the pattern of the solution accurately. The proposed structure is partial retraining of the network, which increases its capacity to catch the spiky pattern of the solution. Our neural network‐based algorithm allows us to solve the nonlinear chemotaxis problems without the use of linearization techniques and regularization techniques, most of which reduce the accuracy of the model. This mesh‐free‐based method can manage a variety of blow‐up models with curved boundaries without imposing an extra cost. By proving the consistency and stability of the method, we show the convergence of GWNN solutions to analytical solutions of the chemotaxis problem. Several illustrative examples and simulation results are provided to demonstrate the correctness of the results and the robust performance of the presented algorithm. Moreover, to illustrate the effectiveness of the GWNN method, we make a comparison with two other network‐based methods.