Deep NURBS—admissible physics-informed neural networks

IF 8.7 2区 工程技术 Q1 Mathematics
Hamed Saidaoui, Luis Espath, Raúl Tempone
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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.

Abstract Image

深度 NURBS 可容许物理信息神经网络
在本研究中,我们为物理信息神经网络(PINNs)提出了一种新的数值方案,该方案能够在任意几何形状的情况下精确、廉价地求解偏微分方程(PDEs),同时强力强制执行狄利克特边界条件。所提出的方法将定义物理域和 Dirichlet 边界条件所需的可容许 NURBS 参数化(在微积分变化意义上可容许,即满足边界条件)与 PINN 求解器相结合。因此,在这个新颖的深度 NURBS 框架中,边界条件可以自动满足。此外,我们在参数空间中进行采样,并映射到物理域。这种参数采样可以作为一种重要度采样方案,因为在几何形状较为复杂的区域,点会比较集中。在考虑任意几何形状(包括非 Lipschitz 域)时,我们使用二维椭圆 PDE 验证了我们的新方法。与经典的 PINN 求解器相比,Deep NURBS 估计器在所有研究问题上都具有极高的精度。此外,对于所研究的大多数 PDEs,只需使用一个神经网络隐层就能获得理想的精度。这种新颖的方法被认为是为更有效地解决高维问题铺平了道路,因为它允许用更现实的物理信息统计学习框架来解决 PDEs。
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来源期刊
Engineering with Computers
Engineering with Computers 工程技术-工程:机械
CiteScore
16.50
自引率
2.30%
发文量
203
审稿时长
9 months
期刊介绍: Engineering with Computers is an international journal dedicated to simulation-based engineering. It features original papers and comprehensive reviews on technologies supporting simulation-based engineering, along with demonstrations of operational simulation-based engineering systems. The journal covers various technical areas such as adaptive simulation techniques, engineering databases, CAD geometry integration, mesh generation, parallel simulation methods, simulation frameworks, user interface technologies, and visualization techniques. It also encompasses a wide range of application areas where engineering technologies are applied, spanning from automotive industry applications to medical device design.
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