具有非线性热扩散系数的二维局部时间分数反常扩散方程的傅里叶特征信息矩阵的Fisher信息矩阵方法

IF 4 3区 工程技术 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Navnit Jha, Ekansh Mallik
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引用次数: 0

摘要

目的探讨傅里叶特征增强物理信息神经网络(pinn)对有效求解具有非线性热扩散率的二维局部时间分数反常扩散方程的影响。通过解决传统数值方法在管理分数阶导数和非线性方面的缺点,本研究解决了文献中关于复杂扩散过程的有效解决策略的重大空白。设计/方法/方法本研究采用一种定量方法,采用前馈神经网络架构结合傅里叶特征层。自动微分的实现,以确保精确的梯度计算分数阶导数。通过各种亚扩散和超扩散场景的数值模拟,通过调整分形空间参数来检查行为,证明了该方法的有效性。此外,使用Fisher信息矩阵来评估训练过程,以分析损失情况。结果表明,傅里叶特征增强的pinn有效地捕获了异常扩散方程的动力学,比传统方法获得了更高的解精度。使用Fisher信息矩阵的分析强调了超参数调优在优化网络性能中的重要性。这些发现支持了傅里叶特征提高模型表示复杂解行为的能力的假设,提供了模型结构和扩散动力学之间的关系。独创性/价值本研究提出了一种利用傅里叶特征增强pin求解分数阶反常扩散方程的新方法。这些结果有助于在热工程、材料科学和生物扩散建模等领域的计算方法的进步,同时也为未来研究神经网络中的训练动力学提供了基础。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Fisher’s information matrix approach for Fourier features physics-informed neural networks for two-dimensional local time-fractional anomalous diffusion equations with nonlinear thermal diffusivity

Purpose

This study aims to explore the influence of Fourier-feature enhanced physics-informed neural networks (PINNs) on effectively solving two-dimensional local time-fractional anomalous diffusion equations with nonlinear thermal diffusivity. By tackling the shortcomings of conventional numerical methods in managing fractional derivatives and nonlinearities, this research addresses a significant gap in the literature regarding efficient solution strategies for complex diffusion processes.

Design/methodology/approach

This study uses a quantitative methodology featuring a feed-forward neural network architecture combined with a Fourier feature layer. Automatic differentiation is implemented to ensure precise gradient calculations for fractional derivatives. The effectiveness of the proposed approach is showcased through numerical simulations across various sub-diffusion and super-diffusion scenarios, with fractal space parameters adjusted to examine behavior. In addition, the training process is assessed using the Fisher information matrix to analyze the loss landscape.

Findings

The results demonstrate that the Fourier-feature enhanced PINNs effectively capture the dynamics of the anomalous diffusion equation, achieving greater solution accuracy than traditional methods. The analysis using the Fisher information matrix underscores the importance of hyperparameter tuning in optimizing network performance. These findings support the hypothesis that Fourier features improve the model’s capacity to represent complex solution behaviors, providing the relationship between model architecture and diffusion dynamics.

Originality/value

This research presents a novel approach to solving fractional anomalous diffusion equations through Fourier-feature enhanced PINNs. The results contribute to the advancement of computational methods in areas such as thermal engineering, materials science and biological diffusion modeling, while also providing a foundation for future investigations into training dynamics within neural networks.

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来源期刊
CiteScore
9.50
自引率
11.90%
发文量
100
审稿时长
6-12 weeks
期刊介绍: The main objective of this international journal is to provide applied mathematicians, engineers and scientists engaged in computer-aided design and research in computational heat transfer and fluid dynamics, whether in academic institutions of industry, with timely and accessible information on the development, refinement and application of computer-based numerical techniques for solving problems in heat and fluid flow. - See more at: http://emeraldgrouppublishing.com/products/journals/journals.htm?id=hff#sthash.Kf80GRt8.dpuf
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