- TENG: Time-evolving natural gradients for solving partial differential equations using deep neural networks (arXiv)
Author: Zhuo Chen, Jacob McCarran, Esteban Vizcaino, Marin Soljačić, Di Luo
Abstract: Partial differential equations (PDEs) are useful for modeling dynamical systems in science and engineering. The advent of neural networks has begun a major shift in tackling these complexities, but challenges remain when it comes to accuracy, especially when it comes to initial value issues. In this paper, we generalize the time-dependent variational principle and optimization-based time integration, and exploit natural gradient optimization to obtain high accuracy in his neural network-based PDE solution, Time-Evolving Natural Gradient (TENG). I'd like to introduce_______ Our comprehensive development includes algorithms such as his TENG-Euler and its higher-order variants (e.g. TENG-Heun), tuned for increased accuracy and efficiency. The effectiveness of TENG is further validated by its performance, which outperforms current leading methods and achieves mechanical accuracy in stepwise optimization over a range of partial differential equations, including thermal equations, Allen-Cahn equations, and Burgers equations. Masu.
