剩余神经网络的属性用于模拟分数微分方程
Sneha Agarwal1, Lakshmi Narayan Mishra1
1Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632 014, Tamil Nadu, India.
Heliyon
|December 17, 2024
概括
本研究介绍了用于近似Erdélyi-Kober分数导数的残余神经网络,建立了一个参数上限. 这种新的方法是使用变量代方法来解决微分方程的验证.
科学领域:
- 数字分析 数字分析
- 分数微积分的计算.
- 机器学习 机器学习
背景情况:
- 分数衍生品,特别是埃尔德利-科贝尔类型的衍生品,带来了重大的分析挑战.
- 接近方法对于解决复杂的分数微分方程至关重要.
研究的目的:
- 探索剩余神经网络 (RNN) 的应用,用于近似Erdélyi-Kober分数导数.
- 为这些RNN的参数设定上限.
- 使用变量代公式验证RNN方法.
主要方法:
- 开发和应用残余神经网络来建模Erdélyi-Kober分数导数.
- 使用变量代方法来导出准确的解决方案进行验证.
- 采用从变量代方法的结构来指导RNN估计.
主要成果:
- 证明了残余神经网络在近似Erdélyi-Kober分数导数中的有效性.
- 建立了网络参数的理论上限.
- 通过变量代方法获得的精确解决方案验证了近似准确性.
结论:
- 剩余神经网络为估计埃尔德利-科伯分数导数提供了强大而有效的工具.
- 建立的参数界限确保了近似的稳定性和可靠性.
- 这项工作将机器学习和微积分计算用于解决微分方程的桥梁.
相关概念视频
01:26Linear Approximation in Frequency Domain
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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
85
01:27State Space Representation
162
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Consider an RLC circuit, a...
162
01:21Linear Approximation in Time Domain
62
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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01:31Classification of Systems-II
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Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
133
01:18Second Order systems II
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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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01:21First Order Systems
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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
83