2000 character limit reached
The lowest-order Neural Approximated Virtual Element Method
Published 30 Nov 2023 in math.NA and cs.NA | (2311.18534v1)
Abstract: We introduce the Neural Approximated Virtual Element Method, a novel polygonal method that relies on neural networks to eliminate the need for projection and stabilization operators in the Virtual Element Method. In this paper, we discuss its formulation and detail the strategy for training the underlying neural network. The efficacy of this new method is tested through numerical experiments on elliptic problems.
- L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, “Basic principles of virtual element methods,” Mathematical Models and Methods in Applied Sciences, vol. 23, no. 01, pp. 199–214, 2013. [Online]. Available: https://doi.org/10.1142/S0218202512500492
- F. Credali, S. Bertoluzza, and D. Prada, “Reduced basis stabilization and post-processing for the virtual element method,” 9 2023.
- A. Russo and N. Sukumar, “Quantitative study of the stabilization parameter in the virtual element method,” 2023.
- S. Berrone, A. Borio, F. Marcon, and G. Teora, “A first-order stabilization-free virtual element method,” Applied Mathematics Letters, vol. 142, p. 108641, aug 2023.
- A. Borio, C. Lovadina, F. Marcon, and M. Visinoni, “A lowest order stabilization-free mixed virtual element method,” 2023.
- M. L. Trezzi and U. Zerbinati, “When rational functions meet virtual elements: The lightning virtual element method,” 2023.
- S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, “Scientific machine learning through physics-informed neural networks: Where we are and what’s next,” Journal of Scientific Computing, vol. 92, no. 88, 2022.
- S. Berrone, G. Teora, and F. Vicini, “Improving high-order vem stability on badly-shaped elements,” Mathematics and Computers in Simulation, vol. 216, pp. 367–385, 2024. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0378475423004287
- L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo, “Virtual element methods for general second order elliptic problems on polygonal meshes,” Math. Models Methods Appl. Sci., vol. 26, no. 04, pp. 729–750, 2015.
- S. Berrone, C. Canuto, and M. Pintore, “Variational physics informed neural networks: the role of quadratures and test functions,” Journal of Scientific Computing, vol. 92, no. 3, pp. 1–27, 2022.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.