Hibrida Metode Newton dan Gradient Descent untuk Neural Network dengan Bobot yang Sensitif

Indahsari, Gabriella Alfa (2026) Hibrida Metode Newton dan Gradient Descent untuk Neural Network dengan Bobot yang Sensitif. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Optimasi Neural Network (NN) untuk penyelesaian permasalahan matematis menghadapi tantangan terkait efisiensi komputasi dan stabilitas konvergensi. Metode Gradient Descent (GD) standar bersifat stabil namun lambat, sementara metode orde kedua menawarkan konvergensi lebih cepat tetapi rentan divergen. Dalam beberapa kasus terdapat bobot yang sensitif (sensitive weights) yang berpengaruh dominan terhadap konvergensi sehingga memerlukan penanganan tersendiri. Penelitian ini mengimplementasikan metode hibrida yang menggabungkan GD dan metode Levenberg Marquardt (LM) secara bergantian, di mana LM dikhususkan untuk mengoptimasi bobot yang sensitif. Metode ini diuji pada tiga kasus: permasalahan nilai eigen matriks Lehmer, persamaan Bratu 1D dengan PINNs, dan persamaan Black-Scholes dengan DeepBSDE. Pada kasus nilai eigen matriks 100 × 100 dan 1000 × 1000, metode hibrida berhasil mencapai nilai referensi dengan penyimpangan kurang dari 0.0001 dan waktu komputasi hingga 47 kali lebih cepat dibandingkan GD standar. Pada kasus persamaan Bratu 1D, metode hibrida menghasilkan estimasi parameter positif C = 2.0011 (referensi 2.0000) dengan waktu komputasi lebih singkat, lebih akurat dibandingkan GD standar yang hanya mencapai C = 2.0153. Pada kasus Black-Scholes, metode hibrida menghasilkan estimasi y init = 57.17 dengan loss terkecil 23.98, lebih baik dari GD standar (y init = 57.10, loss = 24.87), meskipun memerlukan waktu komputasi sekitar 7.5 menit lebih lama. Secara keseluruhan, metode hibrida terbukti lebih unggul dalam akurasi dan kecepatan konvergensi pada kasus dengan bobot sensitif berdimensi rendah.
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Neural Network (NN) optimization for solving mathematical problems faces challenges related to computational efficiency and convergence stability. The standard Gradient Descent (GD) method is stable but slow, while second-order methods offer faster convergence but are prone to divergence. In certain cases, there exist sensitive weights that have a dominant influence on convergence and therefore require separate treatment. This study implements a hybrid method that alternately combines GD and the Levenberg-Marquardt (LM) method, where LM is specifically dedicated to optimizing the sensitive weights. This method was tested on three cases: the eigenvalue problem of the Lehmer matrix, the 1D Bratu equation with PINNs, and the Black-Scholes equation with DeepBSDE. For the eigenvalue case with 100 × 100 and 1000 × 100 matrices, the hybrid method successfully achieved the reference value with a deviation of less than 0.0001 and a computation time up to 47 times faster than the standard GD. For the 1D Bratu equation case, the hybrid method produced a positive parameter estimate of C = 2.0011 (reference 2.0000) with shorter computation time, more accurate than the standard GD which only achieved C = 2.0153. For the Black-Scholes case, the hybrid method produced an estimate of y init = 57.17 with the smallest loss of 23.98, better than the standard GD (y init = 57.10, loss = 24.87), although requiring approximately 7.5 minutes longer computation time. Overall, the hybrid method proved superior in accuracy and convergence speed for cases with low-dimensional sensitive weights.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Metode Hibrida, Metode Newton, Gradient Descent, Neural Network, Bobot yang Sensitif, Hybrid Method, Newton’s Method, Gradient Descent, Neural Network, Sensitive Weights
Subjects:	Q Science > QA Mathematics > QA76.87 Neural networks (Computer Science)
Divisions:	Faculty of Mathematics and Science > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Gabriella Alfa Indahsari
Date Deposited:	31 Jul 2026 00:56
Last Modified:	31 Jul 2026 00:56
URI:	http://repository.its.ac.id/id/eprint/140170

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