Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs

TL;DR

This paper develops an adaptive randomized neural network approach for solving PDEs, leveraging theoretical insights on sampling domains and integrating partition of unity for localized approximation.

Contribution

It introduces a novel adaptive PIRaNN method combining theory-driven sampling with partition of unity for efficient PDE solutions.

Findings

Theoretical link between sampling domain size and function smoothness.

Adaptive PIRaNN effectively captures localized solution features.

Numerical experiments confirm strong approximation capabilities.

Abstract

This paper establishes an approximation theorem for randomized neural networks (RaNNs) whose hidden-layer parameters are uniformly sampled from a prescribed bounded domain. Our analysis shows that, for RaNNs of the form $\mathop{\sum}_i W_i \sigma(A_i, b_i)$, the size of the sampling domain required to achieve optimal approximation is intrinsically linked to the smoothness of the target function and the number of neurons. Motivated by this theoretical insight, we integrate a partition of unity (PoU) with RaNNs to develop an adaptive physics-informed randomized neural network (PIRaNN) method for solving partial differential equations with limited local regularity. The proposed adaptive strategy refines the PoU based on a posteriori error indicators, enabling the network to efficiently capture localized solution features. Numerical experiments validate the theoretical results and demonstrate the strong approximation capabilities of RaNNs, confirming the effectiveness of the adaptive PIRaNN method on a range of benchmark problems.