mlsanityData to 7 Oct 2026

Neural Conservation Laws: A Divergence-Free Perspective

arXiv 2210.01741 ↗First posted 4 Oct 2022cs.LGcited by 67

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Abstract (arXiv)

We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that any solution of the continuity equation can be represented as a divergence-free vector field. We hence propose building divergence-free neural networks through the concept of differential forms, and with the aid of automatic differentiation, realize two practical constructions. As a result, we can parameterize pairs of densities and vector fields that always exactly satisfy the continuity equation, foregoing the need for extra penalty methods or expensive numerical simulation. Furthermore, we prove these models are universal and so can be used to represent any divergence-free vector field. Finally, we experimentally validate our approaches by computing neural network-based solutions to fluid equations, solving for the Hodge decomposition, and learning dynamical optimal transport maps.

Key facts. Cited by 67 papers in the map (data through 7 Oct 2026). Builds on 30 of its 68 references that are in the map. First posted on arXiv 4 Oct 2022.

Builds on

30 of 68 references.

References · 30

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18 more
  1. 2019Residual Flows for Invertible Generative Modelingcited by 256
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 15; 2024: 12; 2025: 21; 2026: 18.

67 papers.

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  1. 2023A Mathematical Guide to Operator Learningcited by 74
  2. 2023Learning Physical Models that Can Respect Conservation Lawscited by 58
  3. 2025AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformerscited by 49
  4. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 46
  5. 2023Self-Supervised Learning with Lie Symmetries for Partial Differential Equationscited by 29
  6. 2023Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputationcited by 26
  7. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
  8. 2023Harnessing the Power of Neural Operators with Automatically Encoded Conservation Lawscited by 22
  9. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  10. 2023Latent Traversals in Generative Models as Potential Flowscited by 17
  11. 2024Graph Neural PDE Solvers with Conservation and Similarity-Equivariancecited by 15
  12. 2023Flow Factorized Representation Learningcited by 11
38 more
  1. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  2. 2023Fast Dynamic 1D Simulation of Divertor Plasmas with Neural PDE Surrogatescited by 9
  3. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  4. 2023p-Poisson surface reconstruction in curl-free flow from point cloudscited by 8
  5. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 8
  6. 2023Invariant preservation in machine learned PDE solvers via error correctioncited by 7
  7. 2025Approximation theory for 1-Lipschitz ResNetscited by 7
  8. 2025Enabling Automatic Differentiation with Mollified Graph Neural Operatorscited by 6
  9. 2025High precision PINNs in unbounded domains: application to singularity formulation in PDEscited by 6
  10. 2023Lagrangian Flow Networks for Conservation Lawscited by 5
  11. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  12. 2024A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solidscited by 4
  13. 2024Differentiable programming across the PDE and Machine Learning barriercited by 4
  14. 2025Representing Flow Fields with Divergence-Free Kernels for Reconstructioncited by 4
  15. 2025Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step casecited by 3
  16. 2026Project and Generate: Divergence-Free Neural Operators for Incompressible Flowscited by 3
  17. 2024Stable Autonomous Flow Matchingcited by 2
  18. 2024Unsupervised Representation Learning from Sparse Transformation Analysiscited by 2
  19. 2025Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networkscited by 2
  20. 2025Simulation-Free Differential Dynamics through Neural Conservation Lawscited by 2
  21. 2025Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticitycited by 2
  22. 2025Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networkscited by 2
  23. 2026Divergence-Free Diffusion Models for Incompressible Fluid Flowscited by 2
  24. 2026Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flowscited by 2
  25. 2022Harmonic (Quantum) Neural Networkscited by 1
  26. 2023Constrained Empirical Risk Minimization: Theory and Practicecited by 1
  27. 2023Learning Latent Dynamics via Invariant Decomposition and (Spatio-)Temporal Transformerscited by 1
  28. 2025Adaptive Correction for Ensuring Conservation Laws in Neural Operatorscited by 1
  29. 2025KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planningcited by 1
  30. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  31. 2026LagrangianSplats: Divergence-Free Transport of Gaussian Primitives for Fluid Reconstructioncited by 1
  32. 2026Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errorscited by 1
  33. 2024Scalable physical source-to-field inference with hypernetworkscited by 0
  34. 2024Control, Transport and Sampling: Towards Better Loss Designcited by 0
  35. 2024A nudge to the truth: atom conservation as a hard constraint in models of atmospheric composition using an uncertainty-weighted correctioncited by 0
  36. 2024Transport-Embedded Neural Architecture: Redefining the Landscape of physics aware neural models in fluid mechanicscited by 0
  37. 2025Learning and Transferring Physical Models through Derivativescited by 0
  38. 2025Identifying Memorization of Diffusion Models through p-Laplace Analysis: Estimators, Bounds and Applicationscited by 0

50 most cited of 67 All 67 (JSON)