Deep Smoothness WENO Schemes for Hyperbolic Conservation Laws

My name is Tatiana Kossaczká and I am a PhD student in the Applied and Computational Mathematics Group at the University of Wuppertal. Already during my master studies I became familiar with different numerical methods for solving partial differential equations (PDE). In my master thesis I focused on the application of the weighted essentially non-oscillatory (WENO) method in the field of computational finance. I was very happy to get the opportunity to continue my research in a PhD program supervised by Prof. Dr. Matthias Ehrhardt.

Recently, deep learning has been widely applied in the field of numerical mathematics. I was very interested in this field and thought about how deep learning could improve the existing numerical schemes. I find this approach very promising because it involves improving the existing schemes by incorporating only a small deep learning component. This allows preserving the important properties of the original schemes, such as convergence, consistency, and conservation property.

Our first application of deep learning was the improvement of the WENO scheme for solving hyperbolic conservation laws and nonlinear degenerate parabolic equations [1,3], which we later successfully applied also to computational finance problems [2]. We named the final scheme WENO-DS (Deep Smoothness). Our approach involved training a compact Convolutional Neural Network (CNN) to improve the so-called smoothness indicators of the method, which are critical parameters. By using CNN, we achieve that the final numerical scheme becomes spatially invariant, which is very important for the convergence of the scheme. In addition, CNN proved to be more computationally efficient than traditional dense neural networks in our case.

Recently, we have demonstrated the generalizability and efficiency of the extended WENO-DS scheme even in a challenging example involving a Riemann problem for the two-dimensional Euler system of gas dynamics [4]. It is found that the method performs admirably in scenarios involving combinations of contact discontinuities, shock waves, and rarefaction waves. This work was done in collaboration with Dr. Ameya Jagtap from the CRUNCH Group of Prof. Dr. George Karniadakis at Brown University.

Another example of the improvement of classical numerical schemes by deep learning that we have studied is its application to classical finite difference methods (FDM) [5]. Here, the goal is to approximate the local truncation error of the numerical method used to approximate a spatial derivative of a given PDE. In the examples presented, the improved Deep FDM outperforms the standard numerical schemes. Since we added only a small CNN component to the method, it remained computationally efficient. This approach serves as a proof of concept for improving standard FDMs, and it can be easily applied to other conventional numerical schemes.

I am very happy to have the opportunity to do research in this area, not only to improve my programming skills, but also to get a deep insight into a very interesting research area.

References
[1] T. Kossaczká, M. Ehrhardt, and M. Günther, Enhanced fifth order WENO shock-capturing schemes with deep learning. Results Appl. Math. 12 (2021): 100217.
[2] T. Kossaczká, M. Ehrhardt, and M. Günther, A deep smoothness WENO method with applications in option pricing. In: European Consortium for Mathematics in Industry, Cham: Springer (2021), pp. 417-423.
[3] T. Kossaczká, M. Ehrhardt, and M. Günther, A neural network enhanced WENO method for nonlinear degenerate parabolic equations. Phys. Fluids. 34.2 (2022): 026604.
[4] T. Kossaczká, A.D. Jagtap, and M. Ehrhardt, WENO-DS: Deep smoothness WENO scheme for two-dimensional hyperbolic conservation laws. arXiv preprint September 2023, arXiv:2309.10117.
[5] T. Kossaczká, M. Ehrhardt, and M. Günther, Deep FDM: Enhanced finite difference methods by deep learning. Franklin Open (2023): 100039.