# Constraint-Aware Neurosymbolic Uncertainty Quantification with Bayesian Deep Learning for Scientific Discovery

arXiv: https://arxiv.org/abs/2601.12442 · HTML: https://mlsanity.com/paper/2601.12442/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 40 references that are in the map.
- First posted on arXiv 18 Jan 2026.

## Builds on (most cited first, 24 of 24)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [SciBERT: A Pretrained Language Model for Scientific Text](https://mlsanity.com/paper/1903.10676/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [The Next Decade in AI: Four Steps Towards Robust Artificial Intelligence](https://mlsanity.com/paper/2002.06177/) (2020)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Scallop: A Language for Neurosymbolic Programming](https://mlsanity.com/paper/2304.04812/) (2023)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [NeuPSL: Neural Probabilistic Soft Logic](https://mlsanity.com/paper/2205.14268/) (2022)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Single-model uncertainty quantification in neural network potentials does not consistently outperform model ensembles](https://mlsanity.com/paper/2305.01754/) (2023)
- [A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction](https://mlsanity.com/paper/2503.19466/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Relational Neurosymbolic Markov Models](https://mlsanity.com/paper/2412.13023/) (2024)
- [On the Promise for Assurance of Differentiable Neurosymbolic Reasoning Paradigms](https://mlsanity.com/paper/2502.08932/) (2025)

## Built on by (most cited first, 1 of 1)

- [Concise and Logically Consistent Conformal Sets for Neuro-Symbolic Concept-Based Models](https://mlsanity.com/paper/2605.18202/) (2026)

## Abstract (arXiv)

Scientific Artificial Intelligence (AI) applications require models that deliver trustworthy uncertainty estimates while respecting domain constraints. Existing uncertainty quantification methods lack mechanisms to incorporate symbolic scientific knowledge, while neurosymbolic approaches operate deterministically without principled uncertainty modeling. We introduce the Constraint-Aware Neurosymbolic Uncertainty Framework (CANUF), unifying Bayesian deep learning with differentiable symbolic reasoning. The architecture comprises three components: automated constraint extraction from scientific literature, probabilistic neural backbone with variational inference, and differentiable constraint satisfaction layer ensuring physical consistency. Experiments on Materials Project (140,000+ materials), QM9 molecular properties, and climate benchmarks show CANUF reduces Expected Calibration Error by 34.7% versus Bayesian neural networks while maintaining 99.2% constraint satisfaction. Ablations reveal constraint-guided recalibration contributes 18.3% performance gain, with constraint extraction achieving 91.4% precision. CANUF provides the first end-to-end differentiable pipeline simultaneously addressing uncertainty quantification, constraint satisfaction, and interpretable explanations for scientific predictions.