ВИКОРИСТАННЯ БАЗОВОГО РОЗРІДЖЕННОГО НАВЧАННЯ ДЛЯ ВИЗНАЧЕННЯ ПАРАМЕТРІВ МОДЕЛІ
Keywords:ЗВИЧАЙНІ ДИФЕРЕНЦІАЛЬНІ РІВНЯННЯ, БАЗОВЕ РОЗРІДЖЕННЕ НАВЧАННЯ, НЕДОСТАТНЬО ВИЗНАЧЕНІ СИСТЕМИ, ЧИСЛОВІ РЯДИ
We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without a priori knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We show that employing sparse representations provides more accurate ODE reconstruction compared to least-squares reconstruction techniques for a given amount of time series data. We test and validate the ODE reconstruction method on known 1D,2D, and 3D systems of ODEs. The 1D system possesses two stable fixed points; the 2D system possesses an oscillatory fixed point with closed orbits; and the 3D system displays chaotic dynamics on a strange attractor.
2. E. N. Lorenz, \Deterministic nonperiodic ow," Journal of the Atmospheric Sciences 20 (1963) 130.
3. C. O. Weiss and J. Brock, \Evidence for Lorenz-type chaos in a laser," Physical Review Letters 57 (1986) 2804.
4. K. M. Cuomo, A. V. Oppenheim, and S. H. Strogatz, \Synchronization of Lorenz-based chaotic circuits with applications to communications," IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Pro-cessing 40 (1993) 626.
5. H. Aref, \The development of chaotic advection," Physics of Fluids 14 (2002) 1315.
6. M. E. Csete and J. C. Doyle, \Reverse engineering of biological complexity," Science 295 (2002) 1664.