BTF-PINN: Enforcing Dirichlet Boundary Conditions Without Boundary Training

Abstract

The homogeneous Dirichlet boundary value problem captures the core difficulty of solving Dirichlet problems with non-interpolatory methods. We propose BTF-PINN (Boundary-Training-Free Physics-Informed Neural Network), an interior-only strategy for solving homogeneous Dirichlet boundary value problems that requires no boundary training, boundary penalties, or boundary-conforming parametrizations. The key idea is to embed the essential boundary condition into a newly designed boundary-free loss function. We prove the equivalence between the proposed interior-only variational formulation and the original boundary value problem, establish a sharp threshold condition for the residual weight, and develop a convergence analysis based on the coercivity of the functional. Numerical experiments on high-dimensional problems, irregular geometries, and anisotropic elliptic equations demonstrate the effectiveness of BTF-PINN. Comparisons with standard boundary-penalty PINNs further show that BTF-PINN achieves superior boundary trace accuracy.

30 pages, 13 figures, 9 tables