Spectral Analysis of Hard-Constraint PINNs
Published:Dec 29, 2025 08:31
•1 min read
•ArXiv
Analysis
This paper provides a theoretical framework for understanding the training dynamics of Hard-Constraint Physics-Informed Neural Networks (HC-PINNs). It reveals that the boundary function acts as a spectral filter, reshaping the learning landscape and impacting convergence. The work moves the design of boundary functions from a heuristic to a principled spectral optimization problem.
Key Takeaways
- •HC-PINNs enforce boundary conditions via a trial function ansatz.
- •The boundary function introduces a multiplicative spatial modulation that alters the learning landscape.
- •The boundary function acts as a spectral filter, reshaping the eigenspectrum.
- •Effective rank of the residual kernel is a predictor of training convergence.
- •Widely used boundary functions can induce spectral collapse, leading to optimization stagnation.
Reference
“The boundary function $B(\vec{x})$ functions as a spectral filter, reshaping the eigenspectrum of the neural network's native kernel.”