One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In recent years, deep learning methods have shown their superiority for high-dimensional PDEs where traditional methods fail. However, for low dimensional problems, it remains unclear whether these methods have a real advantage over traditional algorithms as a direct solver. In this work, we propose the random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms. RFM is based on a combination of well-known ideas: 1. representation of the approximate solution using random feature functions; 2. collocation method to take care of the PDE; 3. penalty method to treat the boundary conditions, which allows us to treat the boundary condition and the PDE in the same footing. We find it crucial to add several additional components including multi-scale representation and adaptive weight rescaling in the loss function. We demonstrate that the method exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. In addition, we find that RFM is particularly suited for problems with complex geometry, where both traditional and machine learning-based algorithms encounter difficulties.

Solving partial differential equations is of primary interest in computational sciences and engineering. There exists a long list of numerical methods, including classical methods such as finite difference and finite elements, and machine learning-based methods. When solving low dimensional problems, it remains unclear whether machine learning-based methods have a real advantage over traditional algorithms. The current work develops a random feature method that seems to share the merits of both traditional and machine learning-based methods. It competes well with traditional methods in terms of both accuracy and efficiency. At the same time, it inherits the flexibility of a machine learning-based method in the sense that is mesh-free and is particularly suited for problems with complex geometry.