[Paper Review] Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

N. Sukumar, Ankit Srivastava|arXiv (Cornell University)|Apr 17, 2021

Model Reduction and Neural Networks119 references316 citations

TL;DR

This paper proposes a physics-informed deep neural network method that exactly enforces Dirichlet, Neumann, and Robin boundary conditions using approximate distance functions derived from R-functions and mean value potentials. By embedding these geometry-aware functions into the network's trial solution, the method eliminates boundary condition errors, simplifies training to rely solely on interior residual loss, and achieves superior accuracy over standard PINNs on complex 2D and 4D domains.

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.

Motivation & Objective

• To address the persistent challenge of inexact boundary condition enforcement in physics-informed neural networks (PINNs), especially on complex geometries.
• To develop a meshfree, geometry-aware approach that ensures exact satisfaction of essential and natural boundary conditions a priori.
• To simplify PINN training by eliminating boundary loss terms, relying only on interior residual error in the loss function.
• To enable isogeometric-like analysis on exact geometry without domain discretization or mesh generation.
• To extend the method to high-dimensional problems, including the 4D hypercube, demonstrating scalability.

Proposed method

• Constructs approximate distance functions (ADFs) using R-functions (via R-equivalence composition) and mean value potential fields to represent the signed distance to domain boundaries.
• Uses transfinite interpolation based on inverse-distance weighting to generalize the ADFs for inhomogeneous Dirichlet, Neumann, and Robin conditions.
• Imposes boundary conditions exactly by multiplying the PINN approximation with the ADF, ensuring kinematic admissibility in Ritz methods and pointwise satisfaction in collocation.
• Employs strong form collocation and Ritz minimization to solve PDEs, with loss function consisting only of interior residual error.
• Applies automatic differentiation to compute high-order derivatives of the neural network for PDE residual evaluation.
• Extends the approach to 4D by using R-equivalence to construct ADFs over the hypercube, enabling meshfree solution of the Poisson equation.

Experimental results

Research questions

• RQ1 Can approximate distance functions based on R-functions and mean value potentials be used to exactly enforce diverse boundary conditions in PINNs on complex geometries?
• RQ2 Does eliminating boundary loss terms by a priori satisfying boundary conditions improve training efficiency and solution accuracy in PINNs?
• RQ3 How does the proposed method perform on nonlinear and high-order PDEs, including the Eikonal and biharmonic equations?
• RQ4 Can the method be extended to high-dimensional problems, such as the 4D Poisson equation, without suffering from the curse of dimensionality?
• RQ5 How do different ADF constructions (R-equivalence vs. mean value potential) compare in terms of accuracy and computational cost?

Key findings

• The proposed method consistently outperforms standard PINN collocation methods in accuracy across all benchmark problems, including linear and nonlinear PDEs.
• Training loss and PDE residual loss coincide because boundary conditions are satisfied a priori, eliminating the need for weighted boundary loss terms and improving convergence.
• For the Eikonal equation on a nonconvex domain, the R-equivalence ADF achieved higher accuracy than the mean value potential ADF, though the latter was more efficient for many-edge polygons.
• On the 4D hypercube, the method successfully solved the Poisson equation with homogeneous Dirichlet conditions using a 2–100–100–1 network, demonstrating scalability to higher dimensions.
• The use of ADFs based on R-functions and mean value potentials enables meshfree, isogeometric-like simulations on exact geometry without domain discretization.
• The approach enables robust solution of PDEs on curved and multiply-connected domains without mesh generation, with exact boundary enforcement improving solution fidelity.