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基于物理信息神经网络的多介质非线性瞬态热传导问题研究

SOLVING MULTI-MEDIA NONLINEAR TRANSIENT HEAT CONDUCTION PROBLEM BASED ON PHYSICS-INFORMED NEURAL NETWORKS

  • 摘要: 文章基于物理信息神经网络(Physics-informed Neural Network, PINN)求解多介质非线性瞬态热传导问题. 根据多介质热物性参数的不同, 将求解区域划分成多个子域, 在每个子域中单独应用PINN, 不同子域通过公共界面的通量连续性条件相联系. 利用偏微分方程、初始条件、边界条件和子域间公共界面连续性条件的残差构建损失函数. 通过自动微分算法计算偏微分方程中温度对各输入变量的偏导数. 利用链式求导法计算损失函数对权重和偏差的梯度, 再根据梯度下降法更新网络参数. 为了加速网络收敛, 在激活函数中引入训练参数, 通过调节激活函数斜率, 使网络具有自适应性. 文章探讨了PINN在求解多介质非线性瞬态热传导问题中的适用性, 并进一步讨论了不同激活函数、学习率、网络结构和损失函数中的各项权重等对PINN计算结果的影响. 计算结果表明, PINN在求解多介质非线性瞬态热传导问题时仍具有较高的可靠性和较简洁的求解流程, 且不需要对求解域进行人为的前处理, 有一定工程应用可行性. 文章通过系统的理论分析和数值验证, 充分展示了PINN解决复杂热传导问题的可靠性.

     

    Abstract: In this paper, the physical-informed Neural Networks (PINN) is used to solve the multi-media nonlinear transient heat conduction problem. According to the thermophysical parameters of multi-media, the computational domain is divided into multiple sub-domains, and a tailored PINN is applied in each sub-domain. The PINNs in the sub-domains are connected by the flux continuity of the common interface between the these sub-domains. To formulate the PINN framework, a loss function is constructed based on the residuals arising from the governing partial differential equations (PDEs), initial conditions, boundary conditions, and the continuity conditions at the interfaces between the subdomains. The methodology employs the automatic differential algorithm to accurately compute the partial derivatives of temperature with respect to various input variables present in the PDEs. The gradient of loss function with respect to the weight and deviation is calculated by the chain derivation method, and then the network parameters are updated according to the gradient descent method. In order to accelerate the convergence of the network, the training hyperparameter is introduced into the activation function and the network is adaptive by adjusting the slope of the activation function. The study evaluates the versatility and effectiveness of the PINN framework in addressing the multi-medium nonlinear transient heat conduction problem. Additionally, it investigates the influences of various factors, including different activation functions, learning rates, network structures, and the weights of components within the loss function, on the output results of the PINN. The results indicate that PINN exhibits high reliability and a straightforward solving process when addressing multi-media nonlinear transient heat conduction problems. Moreover, it does not require artificial preprocessing of the solution domain, demonstrating a significant level of practicality for engineering applications. Through systematic theoretical analysis and a series of numerical examples, the article thoroughly illustrates the robustness of PINN as a potent tool for solving complex heat conduction problems.

     

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