2024/08/30 by Haoning Dang, Fei Wang, Dang, Haoning +2 · 1 citation
Computer Science · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2408.17225
openalex publication_date 2024/08/30 · openalex created_date 2024/10/18 · openalex updated_date 2026/07/28
Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally, RaNNs often struggle to approximate PDE solutions with sharp gradients or discontinuities when using smooth activations and shallow architectures. In this paper, we propose an Adaptive-Growth Randomized Neural Network (AG-RaNN) to address these challenges. We first design a frequency-based initialization for a shallow RaNN. Using the residual as an error indicator, we then adaptively grow the network in width (neuron growth) and depth (layer growth) to improve the accuracy of the numerical solution. The weights and biases of new neurons are constructed rather than trained, which enhances the approximation power without additional nonlinear optimization. To handle discontinuities, we further introduce a domain splitting strategy. We also establish a unified error analysis covering approximation, statistical, and optimization errors. Extensive numerical experiments demonstrate the efficiency and accuracy of AG-RaNN.