Spectral analysis of hard-constraint PINNs: The spatial modulation mechanism of boundary functions.

Abstract

Physics-Informed Neural Networks with hard constraints (HC-PINNs) enforce boundary conditions exactly via a trial function ansatz u˜=A+B·N, yet the theoretical mechanisms governing their training dynamics have remained unexplored. This work establishes a Neural Tangent Kernel (NTK) framework for HC-PINNs, revealing that the boundary function B acts as a multiplicative spatial modulator that fundamentally reshapes the kernel eigenspectrum-a mechanism fundamentally distinct from additive penalty terms in soft-constrained formulations. Through spectral analysis of the residual NTK K_r, the effective rank r_eff is identified as a robust, deterministic predictor of training convergence that outperforms classical condition numbers. It is shown that poorly chosen boundary functions can induce spectral collapse-the concentration of the eigenvalue spectrum toward zero-leading to optimization stagnation despite exact boundary satisfaction. The framework is extended to nonlinear PDEs via local Fréchet linearization and validated on the 1D viscous Burgers equation. A systematic finite-width and optimizer consistency study quantifies the gap between idealized NTK theory and practical training with Adam/L-BFGS, demonstrating that the initial r_eff retains diagnostic value as a convergence feasibility indicator despite feature learning. Building on these theoretical insights, an automated boundary function design algorithm is proposed that selects optimal candidates based on initialization-time spectral screening, bridging the gap from analysis to methodology. Validated across 1D/2D/3D linear diffusion benchmarks, nonlinear Burgers, and against representative soft-constraint and adaptive baselines, this framework transforms boundary function design from a heuristic choice into a principled spectral optimization problem, providing a solid theoretical foundation for hard constraints in scientific machine learning.