四阶分数阶阻尼随机波动方程和显式时滞偏微分方程是两类形式特殊的偏微分方程，前者起源于弹塑性微观结构模型，后者是模拟具有遗传特征现象的必要工具，被广泛应用于如种群生态学、控制论、粘弹性材料、热记忆材料等研究领域. 本文分别利用有限差分法和延时-物理信息神经网络法对这两类方程的数值解进行了研究. 主要内容如下：

第一，针对四阶分数阶阻尼随机波动方程提出了一种保持能量耗散特性的有限差分方法，采用“分数中心求导”的方法逼近阻尼项中的空间Riesz分数阶导数，并构造了一个三层隐式Crank-Nicolson差分格式逼近时间和空间导数. 此外，分别导出了连续意义下和数值解离散情形下的能量耗散定理，并证明了所提出的三层隐式差分方法在期望意义下的收敛阶为O(∆t²+h²). 最后，通过数值实验验证了该数值方法的有效性以及理论的正确性，所得结果揭示了分数阶阻尼和随机噪声在系统演化过程中对系统总体能量与解的全局行为具有不可忽视的影响.

第二，提出了延时-物理信息神经网络法(Delay-PINN )求解包含比例型延迟、减法常量延迟或时变量延迟的显式时滞偏微分方程. 首先构造一个前向深度神经网络，并通过“分离技巧”构造时滞项的延迟训练数据集，将适定的偏微分方程求解问题转化为一个最优化问题，再应用现代自动微分工具计算神经网络的偏导数，并通过优化算法逐步调整网络的可训练参数，从而使得神经网络能够逼近原方程的精确解. 最终通过求解四个不同的显式时滞方程证明了该方法的有效性和鲁棒性，并论证了此方法可以自然地推广至求解其他显式延迟问题，如延迟初边值问题、混合延迟问题等. 

第三，阐述了有限差分方法与物理信息神经网络方法的基本数学原理，分析了这两类数值方法的误差来源以及优缺点，并从两方面说明了这两类方法的互补性. 最后，针对常见特殊类型的偏微分方程列出了适合的数值解法，为研究者在不同应用场景下数值方法的选取提供了参考.

The fourth-order stochastic fractional damped wave equation and delayed partial differential equations are two kinds of special partial differential equations, the former arise from the research of elasto-plastic-microstructures models, the later are required tools when modelling phenomena with hereditary characteristics in a wide variety of scientific and technical fields, such as in population ecology, control theory, viscoelasticity, materials with thermal memory, etc. In this work, we solve these two types of differential equation by finite difference method and Delay-physics-informed neural networks, respectively. The main research contents are as follows:

Firstly, we proposes an energy dissipative finite difference method for solving a fourth-order nonlinear wave equation driven by space fractional damping and multiplicative noise. We use a ‘fractional centered derivative’ approach to approximate the Riesz fractional derivative in damping term, and develop a three-level implicit Crank–Nicolson scheme for the temporal-spatial approximation. Subsequently, we discussed the expected value of the discrete energy and proved that the proposed method attain the convergence orders O(∆t²+h²) under the expected sense. Finally, a numerical experiment is given to verify the efficiency of the scheme and confirm the correctness of theoretical results, and it also shows that the fractional damping and noise term can influence the global behavior of this evolution system.

Secondly, we propose a new framework of physics-informed neural networks (called Delay-PINN) to approach ordinary/partial differential equations with proportional delay, subtractive delay and time variable delay. We transform the well-posed system into an optimization problem by setting delayed dataset for the delay term. By making full use of the modern Auto-Differentiation tools, we can find the optimal parameters that enabled the neural network fits the solution well. Four numerical results illustrate the efficiency and the robustness of this method. Moreover, we explained our method can be extended to other explicit delay problems naturally, such as delayed initial/boundary conditions and mixed delay problems.

Finally, we summarize the mathematical principles, error sources, advantages and disadvantages of these two methods. We illustrated the complementarity of these two kinds of methods from two aspects, and we listed suitable numerical schemes for common special types of partial differential equations, which provides reference for researchers to choose numerical algorithms in different application scenarios.

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