While deep generative models (DGMs) have demonstrated remarkable success in capturing complex data distributions, they consistently fail to learn strict symbolic constraints that encode domain knowledge, thus requiring a neurosymbolic approach to constraint integration. Existing solutions to this challenge have primarily relied on projection-based methods and often ignore the underlying data distribution, harming generative performance. In this work, we propose a probabilistically sound neurosymbolic framework for enforcing hard linear equality constraints into DGMs to generate constraint-compliant and realistic data. This is achieved by our proposed gradient estimators that allow the constrained distribution, the data distribution conditioned on constraints, to be differentiably learned. We carry out extensive experiments with various DGM model architectures over five image datasets and three scientific applications in which domain knowledge is governed by linear equality constraints. Among all the constraint integration strategies, ours not only guarantees the satisfaction of constraints in generation but also archives superior generative performance across every benchmark.

@inproceedings{LiNeSy26,
  title     = {Learning Deep Generative Models under Hard Linear Equality Constraints},
  author    = {Li, Ruoyan and Sahu, Dipti Ranjan and Van den Broeck, Guy and Zeng, Zhe},
  booktitle = {Proceedings of the 20th International Conference on Neurosymbolic Learning and Reasoning (NeSy)},
  series    = {Proceedings of Machine Learning Research},
  volume    = {284},
  pages     = {1--37},
  url       = "https://starai.cs.ucla.edu/papers/LiNeSy26.pdf",
  month     = 9,
  year      = {2026},
  keywords  = {workshop}
}