Distance-based attention physics-informed neural networks

Physics-informed neural networks (PINNs) provide a flexible, mesh-free framework for solving partial differential equations. However, their effectiveness decreases in irregular domains because geometric information is encoded only indirectly through collocation points. This limitation weakens boundary condition enforcement and prevents direct computation of boundary-derived physical quantities. Traditional numerical methods rely on an explicit conditioning of the solution space by the domain geometry, together with interactions between the governing equations and boundary constraints. Inspired by this computational framework, a distance-based attention physics-informed neural network (DBA-PINN) is proposed in this study to provide an explicit geometric conditioning and to guide the interactions between interior and boundary physics. The proposed formulation introduces a normalized approximate distance field (nADF), defined as a projection-based scalar field that provides a smooth representation of proximity to the boundary without requiring any explicit signed-distance computation. Although the nADF is learned implicitly through a neural network, it serves as an explicit geometric descriptor within the overall architecture and provides a geometry-driven attention mechanism that modulates the blending between residual-dominant and boundary-dominant subnetworks. This mechanism effectively reproduces the geometry-driven interactions between interior and boundary physics. This distance-based attention mechanism also improves boundary accuracy and stabilizes training across complex domains. The neural parameterization of the nADF is fully differentiable, enabling direct computation of boundary-normal vectors through automatic differentiation and the evaluation of boundary-related quantities, such as wall shear stress. Numerical experiments on several complex geometries demonstrate that the DBA-PINN improves flow-field prediction accuracy, enhances boundary fidelity, and yields more stable training dynamics.