# Quantifying constraint hierarchies in Bayesian PINNs via per-constraint Hessian decomposition

arXiv: https://arxiv.org/abs/2510.03278 · HTML: https://mlsanity.com/paper/2510.03278/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 28 Sep 2025.

## Builds on (most cited first, 12 of 12)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks](https://mlsanity.com/paper/2212.06965/) (2022)

## Abstract (arXiv)

Bayesian physics-informed neural networks (B-PINNs) merge data with governing equations to solve differential equations under uncertainty. However, interpreting uncertainty and overconfidence in B-PINNs requires care due to the poorly understood effects the physical constraints have on the network; overconfidence could reflect warranted precision, enforced by the constraints, rather than miscalibration. Motivated by the need to further clarify how individual physical constraints shape these networks, we introduce a scalable, matrix-free Laplace framework that decomposes the posterior Hessian into contributions from each constraint and provides metrics to quantify their relative influence on the loss landscape. Applied to the Van der Pol equation, our method tracks how constraints sculpt the network's geometry and shows, directly through the Hessian, how changing a single loss weight non-trivially redistributes curvature and effective dominance across the others.