Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2104.08426v2)

Abstract: 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 artifical 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 $\phi$, an approximate distance function to the boundary of a domain. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as $\phi$ 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. We present numerical solutions for linear and nonlinear boundary-value problems over domains with affine and curved boundaries. Benchmark problems in 1D for linear elasticity, advection-diffusion, and beam bending; and in 2D for the Poisson equation, biharmonic equation, and the nonlinear Eikonal equation are considered. The approach extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the 4D hypercube. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

• The paper introduces a geometry-aware approach that exactly imposes Dirichlet boundary conditions in PINNs via distance functions.
• It leverages constructive solid geometry and generalized barycentric coordinates to simplify the loss function and enhance training efficiency.
• The method demonstrates superior performance over traditional collocation techniques across various linear and nonlinear PDE benchmarks, including high-dimensional problems.

Overview of "Exact Imposition of Boundary Conditions with Distance Functions in Physics-Informed Deep Neural Networks"

This paper presents a novel methodology for the exact imposition of boundary conditions within the framework of physics-informed neural networks (PINNs) through the utilization of distance functions. The challenges associated with the implementation of Dirichlet boundary conditions in meshfree and particle methods have been extensively documented. This issue is equally significant in the field of PINNs when they are employed to solve partial differential equations (PDEs). The authors introduce geometry-aware trial functions within artificial neural networks to enhance the training efficacy for PDEs. The paper leverages concepts from constructive solid geometry and generalized barycentric coordinates to develop an approximate distance function, denoted as $\phi(x)$, which maintains proximity to the boundary within a domain $R^d$.

Numerical Method and Results

The researchers evaluate their proposed methodology through numerical solutions for both linear and nonlinear boundary-value problems across various geometrical domains, encompassing convex and nonconvex polygons as well as curved boundaries. The benchmark problems explored include linear elasticity, advection-diffusion, beam bending, steady-state heat equation, Laplace's equation, Kirchhoff plate bending, and the nonlinear Eikonal equation in one and two dimensions, as well as a Poisson problem with homogeneous Dirichlet boundary conditions over a four-dimensional hypercube.

Key results indicate that the proposed approach surpasses traditional PINN-based collocation methods in performance metrics. This is primarily attributed to the analytical imposition of boundary conditions which results in a simplified loss function whose contribution is solely from residual errors at interior collocation points, enhancing training efficiency and accuracy.

Contributions and Implications

The main contributions of this paper include:

1. Geometry-aware Method: Introducing a method in PINNs that uses R-functions and transfinite interpolation for the exact enforcement of boundary conditions across complex geometries.
2. Approximate Distance Fields: Utilizing concepts from mean value potential fields to craft approximate distance functions expanding to higher dimensions.
3. Training Simplification and Network Accuracy: By precisely meeting boundary conditions, the required network training effort is reduced, which facilitates convergence and leads to improved accuracy.
4. High-Dimensional Application: Illustrating the application of their method in four dimensions through a Poisson problem resolved on a hypercube.
5. Meshfree Method Enabling: Facilitating simulations on exact geometries without the need for domain discretization, thus offering a promising pathway for isogeometric analysis.

Future Directions

The paper opens several directions for further exploration. Extensions to three-dimensional geometries are of particular interest, as is the development of deep Petrov-Galerkin domain-decomposition methods to enhance the scalability and efficiency of the presented approach. Moreover, exploring alternative network architectures and optimization techniques tailored for this method could further boost performance and broaden its applicability in solving complex PDEs. The integration of distance function methodologies from fields like computer vision and solid modeling remains a compelling avenue for continued research.