# DeepSaDe: Learning Neural Networks that Guarantee Domain Constraint Satisfaction

arXiv: https://arxiv.org/abs/2303.01141 · HTML: https://mlsanity.com/paper/2303.01141/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 2 Mar 2023.

## Builds on (most cited first, 22 of 22)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Towards Stable and Efficient Training of Verifiably Robust Neural Networks](https://mlsanity.com/paper/1906.06316/) (2019)
- [Formal Security Analysis of Neural Networks using Symbolic Intervals](https://mlsanity.com/paper/1804.10829/) (2018)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [From Statistical Relational to Neuro-Symbolic Artificial Intelligence](https://mlsanity.com/paper/2003.08316/) (2020)
- [MurTree: Optimal Classification Trees via Dynamic Programming and Search](https://mlsanity.com/paper/2007.12652/) (2020)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [Counterexample-Guided Learning of Monotonic Neural Networks](https://mlsanity.com/paper/2006.08852/) (2020)
- [MLIC: A MaxSAT-Based framework for learning interpretable classification rules](https://mlsanity.com/paper/1812.01843/) (2018)
- [ART: Abstraction Refinement-Guided Training for Provably Correct Neural Networks](https://mlsanity.com/paper/1907.10662/) (2019)
- [A Scalable Two Stage Approach to Computing Optimal Decision Sets](https://mlsanity.com/paper/2102.01904/) (2021)
- [Self-Correcting Neural Networks For Safe Classification](https://mlsanity.com/paper/2107.11445/) (2021)
- [SaDe: Learning Models that Provably Satisfy Domain Constraints](https://mlsanity.com/paper/2112.00552/) (2021)

## Built on by (most cited first, 4 of 4)

- [A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction](https://mlsanity.com/paper/2503.19466/) (2025)
- [Constrained Machine Learning Through Hyperspherical Representation](https://mlsanity.com/paper/2504.08415/) (2025)
- [FourierCSP: Differentiable Constraint Satisfaction Problem Solving by Walsh-Fourier Expansion](https://mlsanity.com/paper/2510.04480/) (2025)
- [SMiLE: Provably Enforcing Global Relational Properties in Neural Networks](https://mlsanity.com/paper/2511.07208/) (2025)

## Abstract (arXiv)

As machine learning models, specifically neural networks, are becoming increasingly popular, there are concerns regarding their trustworthiness, specially in safety-critical applications, e.g. actions of an autonomous vehicle must be safe. There are approaches that can train neural networks where such domain requirements are enforced as constraints, but they either cannot guarantee that the constraint will be satisfied by all possible predictions (even on unseen data) or they are limited in the type of constraints that can be enforced. In this paper, we present an approach to train neural networks which can enforce a wide variety of constraints and guarantee that the constraint is satisfied by all possible predictions. The approach builds on earlier work where learning linear models is formulated as a constraint satisfaction problem (CSP). To make this idea applicable to neural networks, two crucial new elements are added: constraint propagation over the network layers, and weight updates based on a mix of gradient descent and CSP solving. Evaluation on various machine learning tasks demonstrates that our approach is flexible enough to enforce a wide variety of domain constraints and is able to guarantee them in neural networks.