With the success of deep learning technologies in many scientific and engineering applications, neural network approximation methods have emerged as an active research area in numerical partial differential equations. However, the new approximation methods still need further validations on their accuracy, stability, and efficiency so as to be used as alternatives to classical approximation methods. In this talk, we first introduce the neural network approximation methods for partial differential equations, where a neural network function is introduced to approximate the PDE (Partial Differential Equation) solution and its parameters are then optimized to minimize the cost function derived from the differential equation. We then present the approximation error and the optimization error behaviors in the neural network approximate solution. To reduce the approximation error, a neural network function with a larger number of parameters is often employed but when optimizing such a larger number of parameters the optimization error usually pollutes the solution accuracy. In addition to that, the gradient-based parameter optimization usually requires computation of the cost function gradient over a tremendous number of epochs and it thus makes the cost for a neural network solution very expensive. To deal with such problems in the neural network approximation, a partitioned neural network function can be formed to approximate the PDE solution, where localized neural network functions are used to form the global neural network solution. The parameters in each local neural network function are then optimized to minimize the corresponding cost function. To enhance the parameter training efficiency further, iterative algorithms for the partitioned neural network function can be developed. We finally discuss the possibilities in this new approach as a way of enhancing the neural network solution accuracy, stability, and efficiency by utilizing classical domain decomposition algorithms and their convergence theory. Some interesting numerical results are presented to show the performance of the partitioned neural network approximation and the iteration algorithms.
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