Two-scale Neural Networks for Singularly Perturbed Dynamical Systems with Multiple Parameters
Provides a scale-aware neural network approach for solving singularly perturbed dynamical systems with multiple small parameters, which is a challenging class of problems in scientific computing.
Extended a two-scale neural-network method from scalar problems to dynamical systems with multiple small parameters, using the geometric mean as an effective scale parameter. Numerical experiments showed satisfactory accuracy in capturing sharp transitions across coupled systems with multiple high-contrast parameters.
We extend our two-scale neural-network method for scalar singularly perturbed problems with one small parameter to dynamical systems with multiple small parameters. To accommodate multiple small parameters, we use a single effective scale parameter defined as the geometric mean of all parameters. We thus augment the network input with a scale-aware feature, enabling it to capture sharp solution transitions intrinsically. Numerical experiments across a range of dynamical systems demonstrate that the proposed framework can handle coupled systems with multiple and high-contrast small parameters and obtain satisfactory accuracy in capturing solution features induced by small parameters.