Enhancing artificial neural network learning efficiency through Singular value decomposition for solving partial differential equations

Abstract

Partial differential equations (PDEs) hold significant potential for modelling natural phenomena. It is essential to look at a practical way to solve the PDEs. Recently, Artificial Neural Networks (ANN) have emerged as promising tool for approximating PDE solutions. This approach stands out for its adaptability in hybridizing with various optimization techniques, rendering it a potent tool. One drawback of ANN, however, lies in its tendency for slow convergence. In response, we introduce the matrix decomposition method into the ANN learning process, rooted in Singular Value Decomposition (SVD). Decomposed matrix operates by establishing connections between the weights in the first and second hidden layers of the ANN, allowing the generation of singular matrix values. The process of determining the number of retained singular value components in the decomposition method involves three reduction techniques derived from SVD: Thin SVD, Compact SVD, and Truncated SVD. Six problems of second-order PDE on one- and two-dimensionality were solved using these methods, which were combined into a unique ANN learning framework. The results showed that our proposed method exhibits better performance compared to the conventional ANN or those without the decomposition method.