Table 4 An evaluation of the accuracy of x (t) using NDsolve and ANN-PSO-NNA for case 1, with \({\varvec{\sigma}}\) = 0.1, \({\varvec{R}}=0.2\) and \({\varvec{B}}=0.3.\)

From: Neuro-computing solution for Lorenz differential equations through artificial neural networks integrated with PSO-NNA hybrid meta-heuristic algorithms: a comparative study

t

\({x(t)}_{Numerical}\)

\({\widehat{x}(t)}_{ANN}\)

\({AE(x(t))}_{ANN}\)

0

0

5.40E−06

5.4046E−06

0.1

0.009468358

0.009515209

4.6851E−05

0.2

0.017943263

0.017994147

5.0885E−05

0.3

0.025521751

0.025530937

9.1863E−06

0.4

0.0322914

0.032278219

1.3181E−05

0.5

0.038331266

0.038342661

1.1395E−05

0.6

0.043712682

0.043772499

5.9818E−05

0.7

0.048500038

0.048586562

8.6524E−05

0.8

0.052751446

0.052812704

6.1258E−05

0.9

0.056519362

0.056520039

6.7777E−07

1

0.05985115

0.059839029

1.212E−05