Physics-informed neural networks improved for sharp features and boundaries
GAC-PINN: Geometry-Adaptive and Constraint-Enhanced Physics-Informed Neural Networks
Artificial Intelligence
Summary
When simulating physical systems that have sudden changes or complex interactions, usual neural networks struggle to get accurate results. The authors developed a new method called GAC-PINN that adapts to the geometry of the problem and strengthens boundary conditions to improve accuracy. Their approach uses adaptive grids, flexible boundary constraints, special feature mappings, and automatic adjustments based on the type of equations. Tests show that this new method achieves much lower errors compared to existing techniques, especially for problems with sharp transitions.
What this means in practice
- •For computational physicists: Simulate physical phenomena with steep gradients more accurately using adaptive grid and boundary methods that reduce error in neural network solutions.
- •For applied mechanics engineers: Perform high-fidelity simulations of materials and phase transitions featuring sharp interfaces using an enhanced physics-informed neural network framework.
Authors
Yanxin Zhang, Yong Zhang, Houbiao Li
Abstract
For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\pm 0.450)\times 10^{-4}), ((2.868\pm 0.947)\times 10^{-5}), and ((1.756 \pm 0.712)\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.