Solving Partial Differential Equations with Analytic Random Feature Method

As a machine learning method for solving partial differential equations, the random feature method enjoys both the flexibility of neural networks and the fidelity with spectral convergence. However, the partition-of-unity method used in existing works of the random feature method refrains the method from straightforwardly solving partial differential equations of order greater than two, and extra equations for continuity conditions or for order reduction are required. In this paper, we propose the analytic random feature method to tackle this issue, by applying analytic unit partition-of-unity functions in the random feature method to achieve arbitrary smoothness. Namely, we study a group of partition-of-unity functions constructed using analytic sigmoidal functions, which have exponential decay rate, and can be truncated to the same window like by the existing approach. Numerical tests validate the proposal in solving high-order partial differential equations, and that the proposed analytic unit partition-of-unity functions can replace the default b function used by existing works, both with improved precision when the same number of unknowns are solved.