Deep NURBS—admissible physics-informed neural networks

Saidaoui, Hamed; Espath, Luis; Tempone, Raúl

Abstract

In this study, we propose a new numerical scheme for physics-informed neural networks (PINNs) that enables precise and inexpensive solutions for partial differential equations (PDEs) in case of arbitrary geometries while strongly enforcing Dirichlet boundary conditions. The proposed approach combines admissible NURBS parametrizations (admissible in the calculus of variations sense, that is, satisfying the boundary conditions) required to define the physical domain and the Dirichlet boundary conditions with a PINN solver. Therefore, the boundary conditions are automatically satisfied in this novel Deep NURBS framework. Furthermore, our sampling is carried out in the parametric space and mapped to the physical domain. This parametric sampling works as an importance sampling scheme since there is a concentration of points in regions where the geometry is more complex. We verified our new approach using two-dimensional elliptic PDEs when considering arbitrary geometries, including non-Lipschitz domains. Compared to the classical PINN solver, the Deep NURBS estimator has a remarkably high accuracy for all the studied problems. Moreover, a desirable accuracy was obtained for most of the studied PDEs using only one hidden layer of neural networks. This novel approach is considered to pave the way for more effective solutions for high-dimensional problems by allowing for a more realistic physics-informed statistical learning framework to solve PDEs.

Citation

Saidaoui, H., Espath, L., & Tempone, R. (2024). Deep NURBS—admissible physics-informed neural networks. Engineering with Computers, https://doi.org/10.1007/s00366-024-02040-9

• Journal Article Type: Article
• Acceptance Date: Jul 25, 2024
• Online Publication Date: Aug 5, 2024
• Publication Date: Aug 5, 2024
• Deposit Date: Aug 6, 2024
• Publicly Available Date: Aug 6, 2025
• Journal: Engineering with Computers
• Print ISSN: 0177-0667
• Electronic ISSN: 1435-5663
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1007/s00366-024-02040-9
• Public URL: https://nottingham-repository.worktribe.com/output/38105716
• Publisher URL: https://link.springer.com/article/10.1007/s00366-024-02040-9
• Additional Information: This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://link.springer.com/article/10.1007/s00366-024-02040-9; ; ; ; The authors have no conflicts to disclose.; : All simulations were ran in an iMac 3,8 GHz 8-core Intel Core i7, 128 GB 2667 MHz DDR4.