Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs
This work addresses the challenge of solving PDEs with localized features for computational mathematics and engineering, representing an incremental improvement by integrating partition of unity with randomized neural networks.
The paper tackled the problem of approximating functions and solving partial differential equations (PDEs) with limited local regularity by developing an adaptive physics-informed randomized neural network method, achieving strong approximation capabilities validated through numerical experiments on benchmark problems.
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 $\mathop{\sum}_i W_i Ï(A_i, b_i)$, 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.