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.

This paper presents a novel approach termed Layer-of-Thoughts Prompting (LoT), which utilizes constraint hierarchies to filter and refine candidate responses to a given query. By integrating these constraints, our method enables a structured retrieval process that enhances explainability and automation. Existing methods have explored various prompting techniques but often present overly generalized frameworks without delving into the nuances of prompts in multi-turn interactions. Our work addresses this gap by focusing on the hierarchical relationships among prompts. We demonstrate that the efficacy of thought hierarchy plays a critical role in developing efficient and interpretable retrieval algorithms. Leveraging Large Language Models (LLMs), LoT significantly improves the accuracy and comprehensibility of information retrieval tasks.

Although various aspects of soft-constraint based norms have been explored, it is still challenging to understand preemption. Preemption is a situation where higher-level norms override lower-level norms when new information emerges. To address this, we propose a derivation state argumentation framework (DSA-framework). DSA-framework incorporates derivation states to explain how preemption arises based on evolving situational knowledge. Based on DSA-framework, we present an argumentative approach for explaining preemption. We formally prove that, under local optimality, DSA-framework can provide explanations why one consequence is obligatory or forbidden by soft-constraint based norms represented as logical constraint hierarchies.