Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

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 method to exactly impose boundary conditions via distance functions, significantly boosting PINN accuracy.
• It employs geometry-aware trial functions incorporating R-functions and barycentric coordinates to satisfy Dirichlet, Neumann, and Robin conditions.
• The approach reduces computational load and improves simulation efficiency across various benchmark linear and nonlinear PDE problems.

Exact Imposition of Boundary Conditions in Physics-Informed Neural Networks

Introduction

Physics-informed neural networks (PINNs) offer powerful computational methods for solving boundary value problems (BVPs) involving partial differential equations (PDEs). A key challenge is ensuring the accurate imposition of boundary conditions. This paper introduces a method to exactly impose boundary conditions using distance functions, an enhancement that promises to increase the accuracy and efficiency of PINNs.

Methodology

The method constructs geometry-aware trial functions in artificial neural networks, employing concepts such as R-functions and generalized barycentric coordinates to create distance fields. This approach achieves precise satisfaction of Dirichlet, Neumann, and Robin boundary conditions, crucial for solving PDEs accurately.

Figure 1: Approximate distance function showcasing zero boundary values for Dirichlet conditions.

Here, trial functions multiply the PINN approximation by the constructed distance field, ensuring boundary conditions are met. This modification simplifies the loss function by eliminating terms related to boundary conditions, focusing optimization solely on interior residual errors.

Numerical Results

The method was tested on benchmark problems involving linear and nonlinear BVPs in one to four-dimensional spaces. High accuracy was achieved across diverse geometries, illustrating the potential to outperform traditional PINN methodologies and gallium-based meshfree collocation techniques.

Figure 2: Error analysis contrasting traditional PINN and enhanced distance function implementation, showing superior accuracy with the new approach.

Computational Efficiency

Utilizing approximate distance functions significantly reduced computational load and resource usage compared to exact distance functions, which tend to have interior derivative discontinuities unsuitable for trial functions in a collocation approach.

Real-World Applications

For complex geometries, this approach enables conducting meshfree analysis without domain discretization, thereby furnishing a path towards exact geometry simulations akin to isogeometric analysis. Moreover, applications are seen in physics simulations involving dynamic movements and deformations where precise boundary adherence is crucial.

Figure 3: Error distribution comparison across domains, indicating enhanced control of discrepancies near boundaries using geometry-aware trial functions.

Future Work

Further research applications include extending this methodology to 3D geometries and exploring deep Petrov-Galerkin domain-decomposition methods. Additionally, investigating more complex network architectures will help to further unlock the potential of precise boundary imposition and domain-independent solutions within high-dimensional frameworks.

Conclusion

This study pioneers a pathway to improvement in the effectiveness of PINN methods through the exact imposition of boundary conditions using distance functions. It represents a pivotal advancement in numerical simulation techniques by ensuring boundary satisfaction, accuracy, and computational efficiency, thus paving the way for robust solutions in complex geometric domains.