Problem

Scientific AI applications require models that provide reliable uncertainty estimates while strictly adhering to fundamental domain constraints. However, current uncertainty quantification (UQ) methods, such as Bayesian Neural Networks, often produce predictions that are highly confident but physically implausible, violating critical scientific principles despite good calibration. Conversely, existing neurosymbolic AI approaches, which incorporate domain knowledge through constraints, typically operate deterministically without a principled way to quantify prediction confidence or propagate uncertainty. This creates a significant gap: no unified framework currently offers both calibrated uncertainty estimates and guaranteed satisfaction of hard scientific constraints, especially with automatic constraint acquisition from scientific literature.

Method

This research introduces the Constraint-Aware Neurosymbolic Uncertainty Framework (CANUF), an end-to-end differentiable architecture designed to address the aforementioned problems. CANUF integrates three core components: First, an automated constraint extraction module that mines scientific rules from literature and databases. This module uses named entity recognition, knowledge graphs, and rule templates to identify and verify relevant constraints. Second, a probabilistic neural backbone, implemented as a Bayesian neural network with variational inference, models epistemic uncertainty by maintaining a posterior distribution over network parameters. This allows for the propagation of uncertainty through predictions. Third, a differentiable constraint satisfaction layer (CSL) projects unconstrained predictions onto the feasible region defined by the extracted constraints. This projection preserves differentiability for end-to-end training and dynamically adjusts uncertainty estimates by increasing confidence for predictions within the feasible region and decreasing it for those requiring large corrections. The overall training objective combines prediction accuracy, variational inference's ELBO, a differentiable approximation of Expected Calibration Error, and penalties for constraint violations, generating interpretable explanations when predictions are significantly modified by constraints.

Results

CANUF was extensively evaluated across three scientific domains: materials science (Materials Project database for formation energy), molecular property prediction (QM9 dataset for properties like HOMO-LUMO gap), and climate modeling (CMIP6 dataset for temperature and precipitation). The framework consistently demonstrated superior performance in uncertainty calibration, achieving the lowest Expected Calibration Error (ECE) across all datasets, reducing it by an average of 34.7% compared to standard Bayesian Neural Networks, while maintaining competitive prediction accuracy. Furthermore, CANUF achieved a high constraint satisfaction rate of 99.2%, effectively guaranteeing physical plausibility. Ablation studies revealed that the differentiable constraint satisfaction layer contributed most significantly to improvements in both calibration and constraint satisfaction. The automated constraint extraction module also provided a notable benefit by discovering 12 additional valid constraints on the Materials Project dataset beyond manual specification. CANUF also exhibited robust and gracefully degrading calibration performance under distribution shifts, and its generated explanations were rated as highly comprehensible by domain experts.

Implications

The findings imply that by restricting predictions to physically plausible regions, constraint enforcement effectively reduces the complexity of the hypothesis space, leading to provably improved generalization and calibration in scientific AI. For scientists, CANUF offers actionable and trustworthy uncertainty estimates that directly incorporate domain knowledge, facilitating risk-aware decision-making and model debugging. The automated constraint extraction capability significantly reduces the manual effort typically required for integrating symbolic rules, potentially broadening the adoption of constrained learning. However, the framework has limitations, including the computational overhead of the CSL, the need for domain-specific template libraries, its current focus on algebraic inequality constraints, and its primary evaluation on regression tasks. Future work could explore approximate projection methods, transfer learning for constraint templates across domains, extensions to handle more complex constraint types (temporal, spatial, probabilistic), and applications to classification or structured prediction tasks. Integration with large language models for constraint extraction and active learning strategies also represents promising avenues for further research.