Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks
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DOI：
10.1016/j.cma.2021.114333
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发表时间：
2021-04
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影响因子：
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通讯作者：
N. Sukumar;Ankit Srivastava
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文献类型：
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作者：
N. Sukumar;Ankit Srivastava
在本文中，我们介绍了一种基于距离场的新方法，可以在物理信息深度神经网络中精确施加边界条件。在无网格和粒子方法中满足狄利克雷边界条件的挑战是众所周知的。这个问题也与用于解决偏微分方程的物理通知神经网络（PINN）的开发相关。我们在人工神经网络中引入几何感知试验函数，以改进偏微分方程深度学习的训练。为此，我们使用构造立体几何（R 函数）和广义重心坐标（均值势场）的概念来构造 ψ (x)，即到 R d 中域边界的近似距离函数。为了精确地施加齐次狄利克雷边界条件，试验函数被视为 phi (x) 乘以 PINN 近似，并且通过超限插值对其进行推广，用于先验满足复杂几何上的非齐次狄利克雷（本质）、诺依曼（自然）和罗宾边界条件。在此过程中，我们消除了配置方法中与满足边界条件相关的建模误差，并确保在 Ritz 方法中逐点满足运动学允许性。有了这个新的 ansatz，神经网络的训练就被简化了：对损失函数的唯一贡献来自于需要满足控制方程的内部配置点的残差。使用强形式搭配和 Ritz 最小化来计算数值解。为了传达主要思想并评估该方法的准确性，我们提出了凸和非凸多边形域以及具有弯曲边界的域上的线性和非线性边值问题的数值解。线弹性、平流扩散和梁弯曲的一维基准问题；在二维稳态热方程中，考虑了拉普拉斯方程、双调和方程（基尔霍夫板弯曲）和非线性Ekonal方程。使用 R 函数构建近似距离函数可以扩展到更高的维度，我们通过在四维超立方体上解决具有齐次狄利克雷边界条件的泊松问题来展示其用途。所提出的方法始终优于基于 PINN 的标准搭配方法，这强调了在 PINN 中构造损失函数时精确（先验）满足边界条件的重要性。这项研究提供了一种在无需域离散化的情况下对精确几何体进行无网格分析的途径。
In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artificial neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ (x), an approximate distance function to the boundary of a domain in R d. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ (x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.