NEURAL NETWORK METHOD FOR CONSTRUCTIVE VARIATIONAL PROBLEMS BY GENERALIZED MULTIPLIER METHOD
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摘要: 边界条件的施加是求解偏微分方程定解问题的重要步骤。神经网络方法求解偏微分方程定解问题时，将原问题转化为对应的构造变分问题后，损失函数是包含控制方程与边界条件的泛函。采用经典罚函数法及其改进方法施加边界条件时，罚因子的取值直接影响计算精度和求解效率；直接采用Lagrange乘子法施加边界条件，计算结果可能偏离原问题最优解。为破解此局限性，使用广义乘子法施加边界条件。基于神经网络获得原问题的预测解，再使用广义乘子法构建神经网络的损失函数并计算损失值，利用梯度下降法进行参数寻优，判断损失值是否满足要求；不满足则更新罚因子与乘子后再进行求解直至损失满足要求。数值算例的计算结果表明：与采用经典罚函数法、L1精确罚函数法和Lagrange乘子法施加边界条件构造的神经网络相比，该文提出的方法具有更好的数值精度和更高的求解效率，且求解过程更加稳定。Abstract: The imposition of boundary conditions is an essential step in solving the definite problem of partial differential equations. When the definite problem of partial differential equations is resolved by neural network, the original problem should be transformed to its corresponding constructive variational problem, and the loss function is a functional consisting of the governing equations and the boundary conditions. If the boundary conditions are imposed by the classical penalty function method and its improvements, the value of the penalty factor will affect the solution accuracy and the computational efficiency. If the boundary conditions are directly imposed by the Lagrange multiplier method, the computational results may deviate from the optimal solution of the original problem. To overcome these limitations, the generalized multiplier method is employed in the imposition of boundary conditions. The predicted solution of the original problem is obtained from the neural network. The generalized multiplier method is used to construct the loss function of the neural network and calculate the loss. The gradient descent method is utilized to perform parameter optimization. Afterwards, the loss function is calculated. The penalty factor and multiplier are updated, and the resolution is repeated till the loss is acceptable. The results of numerical examples verify that the proposed method has better solution accuracy, higher computational efficiency and more stable solution process than those neural networks in which the boundary conditions are applied by the classical penalty function method, L1 exact penalty function method, and Lagrange multiplier method.
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图 1 两种常用的神经网络激活函数及其导数
Figure 1. Two widely used activation function in neural network and their derivatives
图 5 四种神经网络模型挠度结果的绝对误差平均值
Figure 5. Mean value of the absolute deflection errors resulted by four neural network models
图 6 本文算法求解的薄板挠度云图
Figure 6. Deflect distribution of thin plate resulted by the proposed algorithm
图 8 采用不同神经网络计算的平面弯曲梁挠度曲线
Figure 8. Deflection curve of plane bending beam by different neural network models
图 9 采用不同神经网络计算的误差-耗时曲线
Figure 9. Error and time-consuming curves by different neural network models
图 10 不同结构神经网络求解的误差-耗时曲线
Figure 10. Error and time-consuming curves by different neural network structures
图 11 不同配点数与隐藏层神经元数量组合求解的误差
Figure 11. Error resulted by different combinations of collocation points and neurons in the hidden layer
表 1 Kirchhoff薄板算例罚因子、乘子更新值
Table 1. The updating of penalty factor and multipliers in Kirchhoff thin plate example
更新
次数罚因子M 乘子 k1 k2 k3 k4 k5 k6 k7 k8 0 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1 2 0.750 0.749 0.752 0.749 0.749 0.748 0.747 0.749 2 4 0.679 0.678 0.683 0.678 0.678 0.677 0.676 0.678 3 8 0.649 0.648 0.655 0.650 0.447 0.647 0.646 0.648 4 16 0.627 0.640 0.636 0.638 0.626 0.641 0.626 0.643 5 32 0.622 0.635 0.632 0.627 0.621 0.629 0.620 0.640 
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表 2 简支梁算例罚因子、乘子更新值
Table 2. The updating of penalty factor and multipliers in simple supported beam example
更新次数 罚因子M 乘子 k1 k2 k3 k4 0 5 1.00 1.00 1.00 1.00 1 5 0.36 0.64 0.97 1.35 2 5 0.14 0.35 0.98 1.47 3 25 0.06 0.16 0.95 1.58 4 125 0.01 0.05 0.96 1.62
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