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

TL;DR

A rigorous Neural Tangent Kernel (NTK) framework for HC-PINNs is established, deriving the explicit kernel composition law, and it is shown that widely used boundary functions can inadvertently induce spectral collapse, leading to optimization stagnation despite exact boundary satisfaction.

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.

Figures (31)

• Figure 1: Theoretical framework for HC-PINN NTK analysis.
• Figure 2: Distributions of power function family boundary functions for varying exponent $p$.
• Figure 3: Analysis of power function family characteristics and spectral properties.
• Figure 4: Correlation analysis between boundary function characteristics and spectral properties.
• Figure 5: Boundary function $B(x) = x^\alpha(1-x)^\alpha$ and its derivatives for different power exponents $\alpha$. (a) Boundary function $B(x)$, symmetric about $x=0.5$. (b) First derivative $B'(x)$, vanishing at the symmetry point. (c) Absolute second derivative $|B"(x)|$ in log scale, showing dramatic growth for $\alpha=0.5$ near boundaries.