Improved Random Feature Method for Continuous Solution Reconstruction from Sparse Observation

Abstract

The reconstruction of continuous solutions using all available mechanisms and data is essential for high-precision simulations and forecasts. This paper presents an improved random feature method (IRFM) that combines observational data with models for solution reconstruction. For the multi-region neuron approximation, we employ a global activation function with robust local approximation capabilities instead of the traditional piecewise method. This enhances smoothness across regions and accelerates convergence. We also derived the equivalence condition for the optimal solution of multi-constrained optimization problems, established criteria for determining the weights in the cost function and the distribution of randomly generated collocation points, reducing biases from subjective choices. Additionally, we introduce a weighted scheme for computing the cost function related to sparse observations, reducing interpolation errors and improving stability against noise. Numerical examples demonstrate that the IRFM is more accurate and converges faster than the original RFM. Its efficiency and accuracy are validated through comparisons with physics-informed neural networks, and its flexibility is shown by successful continuous solution reconstruction in complex domains.