Sparse Bayesian Based Method to Solve Dirichlet Problem

Partial differential equations are closely related to many practical problems. Since they are usually difficult to obtain analytical solutions of partial differential equations, their numerical solution is a hot topic for many scholars. Sparse Bayesian learning is a hot research spot of machine learning in recent years. Relevance Vector Machine (RVM) is a learning machine based on sparse Bayesian learning, which shows good performance in solving approximation problems. In this paper, based on the reproducing kernel theory and relevance vector machine, a new method for solving discrete and non-discrete dirichlet problems is proposed. Based on the reproducing kernel theory, the reproducing kernel space satisfying the equation conditions is constructed, and the equation solving problem is transformed into the corresponding objective optimization problem. Then, the problem is solved in the framework of correlation vector machine, and the approximate solution is obtained by sparse representation of linear combination of reproducing kernels. Compared with the traditional method, this method does not need to draw the grid. Compared with the general machine learning method, this method in this paper has excellent ability to avoid over-fitting based on the processing of hyperparameters. It can calculate the posterior probability distribution of the results to provide uncertainty quantification. It does not need to set additional regularization parameters in the cross-validation process, which reduces the complexity of the model and improves the model’s generalization ability, interpretability and efficiency. Sparsity analysis shows the sparse nature of the proposed method and the possibility of extending it to high-dimensional problems. Comparison experiments with the commonly used finite difference method, neural network method and support vector machine method show that the proposed method has better sparsity, robustness and effectiveness.