mlsanityData to 14 Oct 2026

Learning differentiable solvers for systems with hard constraints

arXiv 2207.08675 ↗First posted 18 Jul 2022cs.LGcited by 34

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Abstract (arXiv)

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable PDE-constrained layer that can be incorporated into any NN architecture. Our method leverages differentiable optimization and the implicit function theorem to effectively enforce physical constraints. Inspired by dictionary learning, our model learns a family of functions, each of which defines a mapping from PDE parameters to PDE solutions. At inference time, the model finds an optimal linear combination of the functions in the learned family by solving a PDE-constrained optimization problem. Our method provides continuous solutions over the domain of interest that accurately satisfy desired physical constraints. Our results show that incorporating hard constraints directly into the NN architecture achieves much lower test error when compared to training on an unconstrained objective.

Key facts. Cited by 34 papers in the map (data through 14 Oct 2026). Builds on 23 of its 42 references that are in the map. First posted on arXiv 18 Jul 2022.

Builds on

23 of 42 references.

References · 23

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2016Group Equivariant Convolutional Networkscited by 1,165
  5. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  6. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  7. 2017OptNet: Differentiable Optimization as a Layer in Neural Networkscited by 599
  8. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  9. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  10. 2019Differentiable Convex Optimization Layerscited by 465
  11. 2016Input Convex Neural Networkscited by 414
  12. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
11 more
  1. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
  2. 2021Efficient and Modular Implicit Differentiationcited by 203
  3. 2021DC3: A learning method for optimization with hard constraintscited by 184
  4. 2019Implicit Deep Learningcited by 124
  5. 2016A Differentiable Physics Engine for Deep Learning in Roboticscited by 108
  6. 2017Linearly constrained Gaussian processescited by 42
  7. 2022Learning continuous models for continuous physicscited by 28
  8. 2018On the Differentiability of the Solution to Convex Optimization Problemscited by 23
  9. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
  10. 2020Linearly Constrained Neural Networkscited by 18
  11. 2019ANODEV2: A Coupled Neural ODE Evolution Frameworkcited by 12

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 9; 2024: 8; 2025: 13; 2026: 3.

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  1. 2023Learning Physical Models that Can Respect Conservation Lawscited by 59
  2. 2025AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformerscited by 53
  3. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 52
  4. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 38
  5. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 24
  6. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
  7. 2023SuperBench: A Super-Resolution Benchmark Dataset for Scientific Machine Learningcited by 18
  8. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 15
  9. 2024AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamicscited by 15
  10. 2023A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraintscited by 12
  11. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  12. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 9
22 more
  1. 2024Stability-Aware Training of Machine Learning Force Fields with Differentiable Boltzmann Estimatorscited by 8
  2. 2024Learning Physics for Unveiling Hidden Earthquake Ground Motions via Conditional Generative Modelingcited by 7
  3. 2025Enabling Automatic Differentiation with Mollified Graph Neural Operatorscited by 6
  4. 2025Strictly Constrained Generative Modeling via Split Augmented Langevin Samplingcited by 6
  5. 2025EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scalecited by 6
  6. 2023Neural Fields with Hard Constraints of Arbitrary Differential Ordercited by 5
  7. 2023Gradient-free online learning of subgrid-scale dynamics with neural emulatorscited by 5
  8. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  9. 2025Adaptive Physics-informed Neural Networks: A Surveycited by 5
  10. 2025Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Modelscited by 4
  11. 2024Guaranteeing Conservation Laws with Projection in Physics-Informed Neural Networkscited by 3
  12. 2023CONFIDE: Contextual Finite Differences Modelling of PDEscited by 2
  13. 2023Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equationcited by 2
  14. 2025Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modelingcited by 2
  15. 2022Neural DAEs: Constrained neural networkscited by 1
  16. 2025HoP: Homeomorphic Polar Learning for Hard Constrained Optimizationcited by 1
  17. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  18. 2026Constraint-Aware Neurosymbolic Uncertainty Quantification with Bayesian Deep Learning for Scientific Discoverycited by 1
  19. 2026Unveiling Multi-regime Patterns in SciML: Distinct Failure Modes and Regime-specific Optimizationcited by 1
  20. 2025Spatio-temporal, multi-field deep learning of shock propagation in meso-structured mediacited by 0
  21. 2025Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networkscited by 0
  22. 2026PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generationcited by 0