Spectral Analysis of Hard-Constraint PINNs: The Spatial Modulation Mechanism of Boundary Functions

TL;DR

This paper provides a spectral analysis framework for hard-constraint PINNs, revealing how boundary functions act as spectral filters that influence training dynamics and convergence, with implications for designing effective boundary conditions.

Contribution

It introduces a neural tangent kernel analysis for HC-PINNs, showing how boundary functions modulate the spectral properties of the kernel and affect training convergence.

Findings

Boundary functions act as spectral filters reshaping the kernel eigenspectrum.

Spectral collapse can occur due to certain boundary functions, causing optimization stagnation.

Effective kernel rank predicts training convergence better than classical condition numbers.

Abstract

Physics-Informed Neural Networks with hard constraints (HC-PINNs) are increasingly favored for their ability to strictly enforce boundary conditions via a trial function ansatz $\tilde{u} = A + B \cdot N$, yet the theoretical mechanisms governing their training dynamics have remained unexplored. Unlike soft-constrained formulations where boundary terms act as additive penalties, this work reveals that the boundary function $B$ introduces a multiplicative spatial modulation that fundamentally alters the learning landscape. A rigorous Neural Tangent Kernel (NTK) framework for HC-PINNs is established, deriving the explicit kernel composition law. This relationship demonstrates that the boundary function $B(\vec{x})$ functions as a spectral filter, reshaping the eigenspectrum of the neural network's native kernel. Through spectral analysis, the effective rank of the residual kernel is identified as a deterministic predictor of training convergence, superior to classical condition numbers. It is shown that widely used boundary functions can inadvertently induce spectral collapse, leading to optimization stagnation despite exact boundary satisfaction. Validated across multi-dimensional benchmarks, this framework transforms the design of boundary functions from a heuristic choice into a principled spectral optimization problem, providing a solid theoretical foundation for geometric hard constraints in scientific machine learning.