Finding approximate analytical solutions of differential equations using Neural Networks with self-adaptive training sets

Artificial Neural Networks are known as powerful models capable of discovering complicated patterns and are de facto standard models in deep learning. But they are also universal function approximators and, consequently, their applicability extends to finding approximate solutions for many computational problems. These applications are interesting not only for mathematicians, but also for computer scientists and engineers interested in learning new models for many classical problems (for instance, fluid dynamics modelling, dynamical systems control, etc.). We present Neural Networks based methods for solving differential equations analytically and use the underlying optimization problem's loss function to produce localized additional training data. Our method uses a reduced initial training dataset, which is gradually, non-uniformly augmented in order to reduce the model's approximation error. This method can be used to directly produce analytical solutions for differential equations or as a pre-processing method for finding optimal, non-uniform grid points for traditional grid-based methods.