Solving Diffusion Problems by a Random Feature Method (East Asian Journal on Applied Mathematics, 2025)
Solving diffusion problems requires numerical methods able to capture theheterogeneity over complex geometries and are robust in terms of positivity preserving,nonlinearity, and radiation diffusion. Current deep learning methods, although mesh-free, encounter difficulties in achieving convergence and exhibit low accuracy when confronted with these specific issues. In this paper, we develop a novel method to overcomethese issues based on the recently proposed random feature method (RFM). Our contributions include: (1) for anisotropic and discontinuous coefficient problems, we rewritea diffusion problem into a first-order system and construct the corresponding loss function and approximation spaces; (2) to avoid negative solutions, we employ the squarefunction as the activation function to enforce the positivity and the trust-region least-square solver to solve the corresponding optimization problem; (3) for the radiationdiffusion problem, we enrich the approximation space of random feature functions withthe heat kernel. Various numerical experiments show that the current method outperforms the standard RFM as well as deep learning methods in terms of accuracy, efficiency,and positivity preserving.
Keywords: Random feature method; anisotropic diffusion; first-order system; positivity preserving; radiation diffusion.