mlsanityData to 14 Oct 2026

Imposing Hard Constraints on Deep Networks: Promises and Limitations

arXiv 1706.02025 ↗First posted 7 Jun 2017cs.CVcited by 43

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Abstract (arXiv)

Imposing constraints on the output of a Deep Neural Net is one way to improve the quality of its predictions while loosening the requirements for labeled training data. Such constraints are usually imposed as soft constraints by adding new terms to the loss function that is minimized during training. An alternative is to impose them as hard constraints, which has a number of theoretical benefits but has not been explored so far due to the perceived intractability of the problem. In this paper, we show that imposing hard constraints can in fact be done in a computationally feasible way and delivers reasonable results. However, the theoretical benefits do not materialize and the resulting technique is no better than existing ones relying on soft constraints. We analyze the reasons for this and hope to spur other researchers into proposing better solutions.

Key facts. Cited by 43 papers in the map (data through 14 Oct 2026). First posted on arXiv 7 Jun 2017.

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Citing papers per year: 2021: 8; 2022: 5; 2023: 7; 2024: 8; 2025: 7; 2026: 8.

43 papers.

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  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  2. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 41
  3. 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
  4. 2024HardNet: Hard-Constrained Neural Networks with Universal Approximation Guaranteescited by 31
  5. 2023RAYEN: Imposition of Hard Convex Constraints on Neural Networkscited by 30
  6. 2024Robust Calibration of Large Vision-Language Adapterscited by 28
  7. 2023Random-Set Neural Networks (RS-NN)cited by 19
  8. 2025Pinet: Optimizing hard-constrained neural networks with orthogonal projection layerscited by 19
  9. 2024HANNA: Hard-constraint Neural Network for Consistent Activity Coefficient Predictioncited by 17
  10. 2025Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraintscited by 14
  11. 2023A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraintscited by 12
  12. 2021Deep Learning Explicit Differentiable Predictive Control Laws for Buildingscited by 11
31 more
  1. 2022Neural network training under semidefinite constraintscited by 11
  2. 2021Physics-informed linear regression is competitive with two Machine Learning methods in residential building MPCcited by 9
  3. 2024Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guaranteescited by 8
  4. 2022Guaranteed Conformance of Neurosymbolic Models to Natural Constraintscited by 6
  5. 2021Lagrangian dual framework for conservative neural network solutions of kinetic equationscited by 5
  6. 2021HybridSDF: Combining Deep Implicit Shapes and Geometric Primitives for 3D Shape Representation and Manipulationcited by 5
  7. 2021Fast Jacobian-Vector Product for Deep Networkscited by 4
  8. 2024Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithmcited by 4
  9. 2025DAE-HardNet: A Physics Constrained Neural Network Enforcing Differential-Algebraic Hard Constraintscited by 3
  10. 2026Soft-Radial Projection for Constrained End-to-End Learningcited by 3
  11. 2021Leveraging Unlabeled Data for Entity-Relation Extraction through Probabilistic Constraint Satisfactioncited by 2
  12. 2023Hierarchical learning, forecasting coherent spatio-temporal individual and aggregated building loadscited by 2
  13. 2024Class and Region-Adaptive Constraints for Network Calibrationcited by 2
  14. 2026SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraintscited by 2
  15. 2022Towards Positive Jacobian: Learn to Postprocess Diffeomorphic Image Registration with Matrix Exponentialcited by 1
  16. 2023Constrained Empirical Risk Minimization: Theory and Practicecited by 1
  17. 2025Implicit Bias in Matrix Factorization and its Explicit Realization in a New Architecturecited by 1
  18. 2025BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problemscited by 1
  19. 2026The Standard Interpretable Model: A general theory of interpretable machine learning to deductively design interpretable methods using Lagrangian mechanicscited by 1
  20. 2022Neuro-physical dynamic load modeling using differentiable parametric optimizationcited by 0
  21. 2023CQnet: convex-geometric interpretation and constraining neural-network trajectoriescited by 0
  22. 2023Differentially Private Distributed Convex Optimizationcited by 0
  23. 2024A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correctioncited by 0
  24. 2024Unsupervised Domain Adaptation Approaches for Chessboard Recognitioncited by 0
  25. 2025Heterogeneous Causal Learning for Optimizing Aggregated Functions in User Growthcited by 0
  26. 2025Homotopy-Guided Self-Supervised Learning of Parametric Solutions for AC Optimal Power Flowcited by 0
  27. 2026Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNscited by 0
  28. 2026Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networkscited by 0
  29. 2026NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guaranteescited by 0
  30. 2026CAffNet: Hard Constraint-Affine Neural Networkscited by 0
  31. 2026A Multi-stage Constrained Optimization Framework for Data-driven Problemscited by 0