A SMOOTH-GAUSSIAN ENSEMBLE NETWORK FORPARABOLIC TURNING POINT PROBLEMS INSINGULAR PERTURBATION

Abstract

Singularly perturbed parabolic partial differential equations
with turning points pose numerical challenges due to steep gradients and layers dependent on the small parameter ε. Traditional methods require fine or adaptive meshes to accurately resolve these layers. To overcome this, we propose the Smooth-Gaussian Ensemble Network, a novel computational approach that approximates solutions using a weighted sum of two
separately trained feedforward neural networks. One network captures the globally smooth behavior, while the other focuses on sharp local features near turning points. Spatially varying Gaussian weights guide the ensemble, concentrating around regions with expected layer behavior. Training utilizes data from known analytical solutions, enabling thorough validation. Numerical experiments demonstrate the method’s ability to maintain high accuracy for small ε values without adaptive meshing. This framework provides an efficient and reliable alternative for handling complex layer dynamics in singular perturbation problems.