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

Shahnawaz Alam, Mohammed Mudassir Uddin, Mohammed Kaif Pasha

TL;DR

CANUF addresses the core gap between trustworthy uncertainty quantification and domain-accurate constraint satisfaction in scientific AI. It unifies a Bayesian neural backbone with a differentiable constraint satisfaction layer and an automated constraint extraction module, enabling end-to-end uncertainty quantification under hard scientific constraints. The approach yields a 34.7% reduction in Expected Calibration Error over standard Bayesian neural networks and achieves 99.2% constraint satisfaction across materials, molecular, and climate benchmarks, while providing interpretable constraint-based explanations. These results demonstrate the practical potential of constraint-aware neurosymbolic methods to enhance reliability, interpretability, and automatic knowledge acquisition in scientific discovery pipelines.

Abstract

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.

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

TL;DR

CANUF addresses the core gap between trustworthy uncertainty quantification and domain-accurate constraint satisfaction in scientific AI. It unifies a Bayesian neural backbone with a differentiable constraint satisfaction layer and an automated constraint extraction module, enabling end-to-end uncertainty quantification under hard scientific constraints. The approach yields a 34.7% reduction in Expected Calibration Error over standard Bayesian neural networks and achieves 99.2% constraint satisfaction across materials, molecular, and climate benchmarks, while providing interpretable constraint-based explanations. These results demonstrate the practical potential of constraint-aware neurosymbolic methods to enhance reliability, interpretability, and automatic knowledge acquisition in scientific discovery pipelines.

Abstract

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.
Paper Structure (51 sections, 19 equations, 3 figures, 7 tables, 1 algorithm)

This paper contains 51 sections, 19 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Uncertainty Quantification in Deep Learning
  4. Neurosymbolic Integration Architectures
  5. Physics-Informed Machine Learning
  6. Constraint Learning and Acquisition
  7. Calibration Theory and Practice
  8. Proposed Methodology
  9. Problem Formulation
  10. Automated Constraint Extraction
  11. Knowledge Graph Construction
  12. Rule Template Matching
  13. Constraint Verification and Scoring
  14. Bayesian Neural Backbone
  15. Variational Inference Formulation
  16. ...and 36 more sections

Figures (3)

  • Figure 1: CANUF Architecture. The framework processes inputs through a Bayesian neural backbone, projects predictions onto constraint-satisfying regions via the CSL layer, and generates explanations from constraint violations.
  • Figure 2: Reliability diagram on Materials Project test set. CANUF predictions closely follow the diagonal, indicating well-calibrated uncertainty estimates.
  • Figure 3: Calibration degradation under distribution shift. CANUF maintains better calibration as distribution shift severity increases.