Artificial Neural Network Method for Solving of Bratu’s Problem
DOI:
https://doi.org/10.5564/jimdt.v4i1.2658Keywords:
Differential equations, Nonstandard finite difference method, Boundary value problem, Machine learning algorithmsAbstract
The Bratu’s problem is widely used to model phenomenas such as heat transfer and combustion theory. For certain values of the parameters, there are 2 different solutions, and finding the lower solution is not difficult, and it is quite possible to apply standard mathematical methods for it calculation. However, finding the upper solution is difficult and requires the use of high-order convergent algorithms. In this study, the method of calculating the numerical solution of the Bratu’s problem using artificial neural networks is considered. When constructing the neural network, sinusoids were used as the activation function, and RMSprop (Root Mean Squared Propagation) was used as the optimization method. By doing so, its possible to calculate two solutions of the Bratu’s problem.
Хиймэл Нейроны Сүлжээг Ашиглан Нэг Хэмжээст Брату Бодлогын Шийдийг Тооцоолох нь
Хураангуй: Брату бодлогыг дулаан дамжуулалт, шаталтын процесс гэх мэт үзэгдлүүдийг загварчлахад өргөн ашигладаг. Тэгшитгэл параметрийн тодорхой утгуудад хоёр шийдтэй бөгөөд доод шийдийг олох нь төвөггүй, тооцон бодох математикийн стандарт аргуудыг хэрэглэх бүрэн боломжтой. Харин дээд шийдийг олох нь бэрхшээлтэй бөгөөд өндөр эрэмбийн нийлэлттэй алгоритмуудыг ашиглах шаардлага тулгардаг. Энэхүү судалгаанд Брату бодлогын тоон ший- дийг хиймэл нейроны сүлжээ ашиглан тооцоолох аргыг авч үзлээ. Нейроны сүлжээг байгуулахдаа идэвхжилтийн функцээр синусоидийг, оновчлолын аргаар RMSprop (Root Mean Squared Propagation) аргыг ашиглав. Ингэснээр Брату бодлогын хоёр шийдийг тооцоолох боломжтой болов.
Түлхүүр үгс: Машин сургалт, Дифференциал тэгшитгэл, Захын нөхцөлт бодлого, Стандарт бус төгсгөлөг ялгаврын арга
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