▁cha otic ▁time ▁series ▁generated ▁by ▁the ▁log istic ▁map . ▁ ▁Generation ▁of ▁the ▁time ▁series ▁from ▁this ▁equation ▁is ▁the ▁forward ▁problem . ▁The ▁examples ▁here ▁illustrate ▁the ▁inverse ▁problem ; ▁identification ▁of ▁the ▁underlying ▁dynamics , ▁or ▁fundamental ▁equation , ▁of ▁the ▁log istic ▁map ▁from ▁exempl ars ▁of ▁the ▁time ▁series . ▁The ▁goal ▁is ▁to ▁find ▁an ▁estimate ▁ ▁for ▁f . ▁ ▁Function ▁approximation ▁ ▁Un normal ized ▁radial ▁basis ▁functions ▁ ▁The ▁architecture ▁is ▁ ▁where ▁ ▁. ▁ ▁Since ▁the ▁input ▁is ▁a ▁scalar ▁rather ▁than ▁a ▁vector , ▁the ▁input ▁dimension ▁is ▁one . ▁We ▁choose ▁the ▁number ▁of ▁basis ▁functions ▁as ▁N = 5 ▁and ▁the ▁size ▁of ▁the ▁training ▁set ▁to ▁be ▁ 1 0 0 ▁exempl ars ▁generated ▁by ▁the ▁cha otic ▁time ▁series . ▁The ▁weight ▁ ▁is ▁taken ▁to ▁be ▁a ▁constant ▁equal ▁to ▁ 5 . ▁The ▁weights ▁ ▁are ▁five ▁exempl ars ▁from ▁the ▁time ▁series . ▁The ▁weights ▁ ▁are ▁trained ▁with ▁projection ▁operator ▁training : ▁ ▁where ▁the ▁learning ▁rate ▁ ▁is ▁taken ▁to ▁be ▁ 0 . 3 . ▁The ▁training ▁is ▁performed ▁with ▁one ▁pass ▁through ▁the ▁ 1 0 0 ▁training ▁points . ▁The ▁r ms ▁error ▁is ▁ 0 . 1 5 . ▁ ▁Normal ized ▁radial ▁basis ▁functions ▁ ▁The ▁normal ized ▁R BF ▁architecture ▁is ▁ ▁where ▁ ▁. ▁ ▁Again : ▁ ▁. ▁ ▁Again , ▁we ▁choose ▁the ▁number ▁of ▁basis ▁functions ▁as ▁five ▁and ▁the ▁size