NEURAL NETWORK METHOD FOR CONSTRUCTIVE VARIATIONAL PROBLEMS BY GENERALIZED MULTIPLIER METHOD

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.