Feedforward neural network for solving particular fractional differential equations
Fractional differential equations (FDEs) model real-world phenomena capturing memory effects. However, existing numerical methods are mostly traditional, prompting the need for innovative approaches. Artificial neural networks (ANNs), a machine learning tool, have exhibited promising capabilities...
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| Format: | Thesis |
| Language: | English |
| Published: |
2024
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/118402/ http://psasir.upm.edu.my/id/eprint/118402/1/118402.pdf |
| Summary: | Fractional differential equations (FDEs) model real-world phenomena capturing memory
effects. However, existing numerical methods are mostly traditional, prompting the
need for innovative approaches. Artificial neural networks (ANNs), a machine learning
tool, have exhibited promising capabilities in solving differential equations. This
research aims to develop a scheme based on a feedforward neural network (FNN) with
a vectorized algorithm (FNNVA) for solving FDEs in the Caputo sense (FDEsC) using
selected first-order optimization techniques: simple gradient descent (GD), momentum
method (MM), and adaptive moment estimation method (Adam). Then, a single hidden
layer of FNN based on Chelyshkov polynomials with an extreme learning machine algorithm
(SHLFNNCP-ELM) is constructed for solving FDEsC. Next, a scheme based
on an extended single hidden layer of FNN using a second-order optimization technique
known as the Broyden–Fletcher–Goldfarb–Shanno method (ESHLFNN-BFGS)
is designed to solve FDEs in the Caputo-Fabrizio sense (FDEsCF). This study also
focuses on solving fractal-fractional differential equations in the Caputo sense with a
power-law kernel (FFDEsCP) using FNN in two hidden layers with a vectorized algorithm
alongside Adam (FNN2HLVA-Adam). In the first scheme, a vectorized algorithm
and automatic differentiation are implemented to minimize computational costs.
Numerical results indicated that FNNVA with Adam in one or two hidden layers, 5
or 10 nodes, and an appropriate learning rate offers superior accuracy compared to
FNNVA with GD and FNNVA with MM. The second approach relies on Chelyshkov
basis functions for approximation and utilizes the extreme machine learning algorithm
for weight determination, achieving high accuracy and low computational time. The
third scheme employs the BFGS solver during the learning process, attained satisfactory
numerical results with fewer iterations. The final scheme utilizes a two hidden
layer FNNVA, with Adam optimization, using suitable number of nodes and value of
learning rates to handle problems involving memory and fractal concepts. The numerical
solutions obtained are consistent with reference solutions. In conclusion, all
proposed schemes deliver more accurate results compared to existing methods while
maintaining low computational costs. |
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