Abstract
In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.
| Original language | English |
|---|---|
| Article number | 6224185 |
| Pages (from-to) | 1356-1367 |
| Number of pages | 12 |
| Journal | IEEE Transactions on Neural Networks and Learning Systems |
| Volume | 23 |
| Issue number | 9 |
| DOIs | |
| Publication status | Published - 2012 |
Bibliographical note
Funding Information:Manuscript received July 7, 2011; revised May 14, 2012; accepted May 15, 2012. Date of publication June 22, 2012; date of current version August 1, 2012. This work was supported in part by the Research Council KUL GOA/11/05 Ambiorics, GOA/10/09 MaNet, CoE EF/05/006 Optimization in Engineering (OPTEC), IOF-SCORES4CHEM, several Ph.D and Post-Doctoral and Fellowship Grants, the Flemish Government (FWO: Ph.D. and Post-Doctoral Grants), under Project G0226.06 (cooperative systems and optimization), Project G0321.06 (Tensors), Project G.0302.07 (SVM/Kernel), Project G.0320.08 (convex MPC), Project G.0558.08 (Robust MHE), Project G.0557.08 (Glycemia2), Project G.0588.09 (Brain-Machine), Project G.0377.12 (structured models) research communities (WOG: ICCoS, ANMMM, MLDM), Project G.0377.09 (Mechatronics MPC) IWT: Ph.D. Grants, Eureka-Flite+, SBO LeCoPro, SBO Climaqs, SBO POM, O&O-Dsquare, the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 2007–2011), EU (ERNSI), FP7-HD-MPC (INFSO-ICT-223854), COST intelliCIS, FP7-EMBOCON (ICT-248940), Contract Research (AMI-NAL), Helmholtz (viCERP), ACCM, Bauknecht, Hoerbiger, and the ERC under Advanced Grant A-DATADRIVE-B.
Funding
Manuscript received July 7, 2011; revised May 14, 2012; accepted May 15, 2012. Date of publication June 22, 2012; date of current version August 1, 2012. This work was supported in part by the Research Council KUL GOA/11/05 Ambiorics, GOA/10/09 MaNet, CoE EF/05/006 Optimization in Engineering (OPTEC), IOF-SCORES4CHEM, several Ph.D and Post-Doctoral and Fellowship Grants, the Flemish Government (FWO: Ph.D. and Post-Doctoral Grants), under Project G0226.06 (cooperative systems and optimization), Project G0321.06 (Tensors), Project G.0302.07 (SVM/Kernel), Project G.0320.08 (convex MPC), Project G.0558.08 (Robust MHE), Project G.0557.08 (Glycemia2), Project G.0588.09 (Brain-Machine), Project G.0377.12 (structured models) research communities (WOG: ICCoS, ANMMM, MLDM), Project G.0377.09 (Mechatronics MPC) IWT: Ph.D. Grants, Eureka-Flite+, SBO LeCoPro, SBO Climaqs, SBO POM, O&O-Dsquare, the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, Dynamical systems, control and optimization, 2007–2011), EU (ERNSI), FP7-HD-MPC (INFSO-ICT-223854), COST intelliCIS, FP7-EMBOCON (ICT-248940), Contract Research (AMI-NAL), Helmholtz (viCERP), ACCM, Bauknecht, Hoerbiger, and the ERC under Advanced Grant A-DATADRIVE-B.
Keywords
- Closed-form approximate solution
- collocation method
- least squares support vector machines (LS-SVMs)
- ordinary differential equations (ODEs)