  Interface problems have long been a major focus of scientific computing,
leading to the development of various numerical methods. Traditional mesh-based
methods often employ time-consuming body-fitted meshes with standard
discretization schemes or unfitted meshes with tailored schemes to achieve
controllable accuracy and convergence rate. Along another line, mesh-free
methods bypass mesh generation but lack robustness in terms of convergence and
accuracy due to the low regularity of solutions. In this study, we propose a
novel method for solving interface problems within the framework of the random
feature method. This approach utilizes random feature functions in conjunction
with a partition of unity as approximation functions. It evaluates partial
differential equations, boundary conditions, and interface conditions on
collocation points in equal footing, and solves a linear least-squares system
to obtain the approximate solution. To address the issue of low regularity, two
sets of random feature functions are used to approximate the solution on each
side of the interface, which are then coupled together via interface
conditions. We validate our method through a series of increasingly complex
numerical examples. Our findings show that despite the solution often being
only continuous or even discontinuous, our method not only eliminates the need
for mesh generation but also maintains high accuracy, akin to the spectral
collocation method for smooth solutions. Remarkably, for the same accuracy
requirement, our method requires two to three orders of magnitude fewer degrees
of freedom than traditional methods, demonstrating its significant potential
for solving interface problems with complex geometries.
