Efficient operator learning method for nonlinear radiation diffusion

Nonlinear radiation diffusion equations, essential for modeling energy transport in applications like inertial confinement fusion and astrophysical systems, present significant computational challenges due to their strong nonlinearity and tight coupling. Traditional numerical methods, relying on time-marching and Picard iterations, are computationally intensive and lack adaptability to varying conditions. This paper proposes two data-driven operator learning architectures, named nonlinear radiation diffusion neural surrogate models (NRD-NSMs), that integrate Fourier neural operators (FNO) and deep operator networks to learn mappings from equation conditions to solutions of the radiation diffusion equations at specific time snapshots. The Type-1 NRD-NSM uses FNO to generate basis functions and a fully connected network (FCN) for coefficients, while the Type-2 NRD-NSM employs element-wise feature combination to merge outputs from multiple encoders, followed by an FNO-based processor and an FCN decoder. Our results demonstrate relative L2 errors on the order of 10−2 across all experiments, establishing these models as accurate surrogates. Notably, the Type-2 NRD-NSM generally achieves higher accuracy than the Type-1 NRD-NSM while maintaining comparable runtime. The proposed models achieve remarkable speedups, up to 50 times on the central processing unit and 2000 times on the graphics processing unit compared to the finite element method, by eliminating iterative processes. Numerical results underscore the potential of NRD-NSMs as fast, accurate, and generalizable surrogates for traditional solvers, enabling efficient simulations in complex physical systems.