解析物理约束对贝叶斯神经网络不确定性的影响机制
Quantifying constraint hierarchies in Bayesian PINNs via per-constraint Hessian decomposition
- 通过无矩阵拉普拉斯方法分解各约束对后验海森矩阵的贡献
- 发现调整单个损失权重会非平凡地重分配其他约束的曲率与主导性
- 适合关注模型不确定性解释与物理约束设计的研究者
贝叶斯物理信息神经网络（B-PINNs）将数据与控制方程结合，用于在不确定条件下求解微分方程。然而，由于物理约束对网络的影响尚不明确，解释其不确定性与过度自信需谨慎；过度自信可能反映的是约束强加的精确性，而非校准错误。为厘清各物理约束如何塑造网络行为，本文提出一种可扩展的、无矩阵的拉普拉斯框架，将后验海森矩阵分解为各约束的贡献，并提供量化其相对影响的指标。应用于范德波尔方程时，该方法揭示了约束如何塑造网络几何结构，并直接通过海森矩阵显示：改变单一损失权重会非平凡地重新分配其他约束的曲率与有效主导性。
原文摘要 · Abstract (English)
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.
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