Seminar on Qualitative Theory of Differential Equations - Tatiana Kossaczká (10.10.2024)
Thursday 10.10.2024 at 14:00, Lecture room M 223
Tatiana Kossaczká:
Deep Learning Enhanced Numerical Schemes
Abstract:
In this talk, the enhancement of numerical schemes using deep learning is addressed. In the first part, a Deep Finite Difference Method is proposed to improve standard finite difference methods for solving partial differential equations by approximating the local truncation error. Higher numerical accuracy is achieved in one- and two-dimensional examples without compromising consistency and convergence. In the second part, the fifth-order Weighted Essentially Non-Oscillatory (WENO) shock capturing scheme is improved by using a neural network to adjust the smoothness indicators, leading to increased accuracy near shocks. The new method, WENO-DS (Deep Smoothness), eliminates the need for post-processing steps. WENO-DS is shown to consistently outperform the traditional WENO scheme on Buckley-Leverett, Burgers', and Euler equations. The approach is extended to the sixth-order WENO scheme for nonlinear degenerate parabolic equations, where superior performance is demonstrated in benchmark tests. Fifth- and sixth-order accuracy is maintained in both methods and is proved theoretically.