In this paper, block neural network (BNN) method is proposed to solve several kinds of differential equations. BNN is used to construct approximating functions and its derivatives, the improved extreme learning machine (IELM) algorithm are designed to train network weights. To evaluate the performance of the proposed method, numerical examples are performed by the presented method. Comparison of the numerical results with exact solutions validate the feasibility of the proposed method in accuracy. Results compared with other recent research works also validate the superiority of the proposed approach. Numerical results show that the proposed BNN with IELM algorithm perform well in accuracy and requires less hidden neurons.
K.S.Mcfall, An artificial neural network method for solving boundary value problems with arbitrary irregular boundaries, Ph.D. thesis, Georgia Institute of Technology (2006).
2.
Y.Shirvany, M.Hayati and R.Moradian, Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks, Commun Nonlinear Sci Numer Simul13 (2008), 2132–2145.
3.
N.Yadav, A.Yadav and K.Deep, Artificial neural network technique for solution of nonlinear elliptic boundary value problems, in Proceedings of Fourth International Conference on Soft Computing for Problem Solving, Advances in Intelligent Systems and Computing335 (2015) 113–121.
4.
I.G.Tsoulos, D.Gavrilis and E.Glavas, Solving differential equations with constructed neural Networks, Neurocomputing72 (2009), 2385–2391.
5.
N.K.Masmoudi, C.Rekik, M.Djemel and N.Derbel, Two coupled neural network based solution of the Hamilton-Jacobi-Bellman equation, Appl Soft Comput11 (2011), 2946–2963.
6.
M.Kumar and N.Yadav, Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey, Comput Math Appl62 (2011), 3796–3811.
7.
H.Chen, L.Kong and W.Leng, Numerical solution of PDEs via integrated radial basis function networks with adaptive training algorithm, Appl Soft Comput11 (2011), 855–860.
8.
R.S.Beidokhti and A.Malek, Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques, J Franklin Inst346 (2009), 898–913.
9.
I.G.Tsoulos, D.Gavrilis and E.Glavas, Neural network construction and training using grammatical evolution, Neurocomputing72 (2008), 269–277.
10.
A.Yadav and K.Deep, A new disc based particle swarm optimization, Adv Intell Soft Comput130 (2012), 23–30.
11.
M.Kumar and N.Yadav, Buckling analysis of a beam column using multilayer perceptron neural network technique, J Franklin Inst350(10) (2013), 3188–3204.
12.
N.Yadav, A.Yadav and M.Kumar, An introduction to neural network methods for differential equations, Springer Netherlands (2015).
13.
A.Barra, M.Beccaria and A.Fachechi, A new mechanical approach to handle generalized Hopfield neural networks, Neural Networks106 (2018), 205–222.
14.
Fatma B.Rizaner and A.Rizaner, Approximate solutions of initial value problems for ordinary differential equations using radial basis function networks, Neural Process Letter48 (2018), 1063–1071.
15.
G.Chen, W.Hu, J.Shen etc. An HDG method for distributed control of convection diffusion PDEs, Journal of Computational and Applied Mathematics343 (2018), 643–661.
16.
M.Leshno, V.Yalin, A.Pinkus etc, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks6 (1993), 861–867.
17.
D.Costarelli and G.Vinti, Approximation theorems for a family of multivariate neural network operators in Orlicz-type spaces, Ricerche Matematica67 (2018), 387–399.
18.
G-B.Huang, Q-Y.Zhu and C-K.Siew, Extreme learning machine: Theory and applications, Neurocomputing70 (2006), 489–501.
19.
G.B.Huang, L.Chen and C.K.Siew, Universal approximation using incremental constructive feedforward networks with random hidden nodes, IEEE Trans Neural Netw17 (2006), 879–892.
G.B.Huang and L.Chen, Enhanced random search based incremental extreme learning machine, Neurocomputing71 (2008), 3460–3468.
22.
G.B.Huang, H.Zhou, X.Ding, et al., Extreme learning machine for regression and multiclass classification, IEEE Trans Syst Man Cybern B Cybern42 (2012), 513–529.
23.
C.M.Wong, C.M.Vong, P.K.Wong, et al., Kernel-based multilayer extreme learning machines for representation learning, IEEE Transactions on Neural Networks & Learning Systems99 (2016), 1–6.
24.
I.Sgura, A finite difference approach for the numerical solution of non-smooth problems for Boundary Value ODEs, Mathematics and Computers in Simulation95 (2013), 146–162.
25.
E.Suli, Finite element methods for partial differential equations. (2005).
26.
D.Hua, Matrix Theory, Science Press. 2001.
27.
H.SadoghiYazdi, M.Pakdaman and H.Modaghegh, Unsupervised kernel least mean square algorithm for solving ordinary differential equations, Neurocomputing74 (2011), 2062–2071.
28.
H.Sun, M.Hou, Y.Yang, etc. Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm. Neural Processing Letters. (2018). 10.1007/s11063-018-9911-8
29.
S.Mall and S.Chakraverty, Single layer Chebyshev neural network model for solving elliptic partial differential equations, Neural Processing Letters45 (2017) 825–840.
