mlsanityData to 7 Oct 2026

Projected Neural Differential Equations for Learning Constrained Dynamics

arXiv 2410.23667 ↗First posted 31 Oct 2024cs.LG physics.comp-ph stat.MLcited by 5

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Abstract (arXiv)

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 56 of its 110 references that are in the map. First posted on arXiv 31 Oct 2024.

Builds on

56 of 110 references.

References · 56

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  3. 2019Hamiltonian Neural Networkscited by 702
  4. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  5. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  8. 2021Machine learning accelerated computational fluid dynamicscited by 461
  9. 2020Lagrangian Neural Networkscited by 388
  10. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  11. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  12. 2022On Neural Differential Equationscited by 264
38 more
  1. 2023Neural General Circulation Models for Weather and Climatecited by 232
  2. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  3. 2019Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learningcited by 205
  4. 2019Symplectic ODE-Net: Learning Hamiltonian Dynamics with Controlcited by 183
  5. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  6. 2020Riemannian Continuous Normalizing Flowscited by 143
  7. 2020Normalizing Flows on Tori and Spherescited by 117
  8. 2020Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraintscited by 102
  9. 2016First-order Methods for Geodesically Convex Optimizationcited by 92
  10. 2020Neural Manifold Ordinary Differential Equationscited by 81
  11. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 67
  12. 2022Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systemscited by 50
  13. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  14. 2021LEADS: Learning Dynamical Systems that Generalize Across Environmentscited by 39
  15. 2021Adaptive-Control-Oriented Meta-Learning for Nonlinear Systemscited by 38
  16. 2019Variational Integrator Networks for Physically Structured Embeddingscited by 36
  17. 2022Learning differentiable solvers for systems with hard constraintscited by 34
  18. 2022Hard-Constrained Deep Learning for Climate Downscalingcited by 29
  19. 2019A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimizationcited by 25
  20. 2024Scaling physics-informed hard constraints with mixture-of-expertscited by 24
  21. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  22. 2022Differentiable Programming for Earth System Modelingcited by 21
  23. 2020Learning Physical Constraints with Neural Projectionscited by 19
  24. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  25. 2021How to Avoid Trivial Solutions in Physics-Informed Neural Networkscited by 14
  26. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  27. 2020Variational Optimization on Lie Groups, with Examples of Leading (Generalized) Eigenvalue Problemscited by 12
  28. 2022FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantitiescited by 12
  29. 2022FC-PINO: High Precision Physics-Informed Neural Operators via Fourier Continuationcited by 12
  30. 2022Learning reversible symplectic dynamicscited by 11
  31. 2022Constants of motion networkcited by 10
  32. 2023ClimSim-Online: A Large Multi-scale Dataset and Framework for Hybrid ML-physics Climate Emulationcited by 10
  33. 2024Feedback Favors the Generalization of Neural ODEscited by 8
  34. 2022Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systemscited by 7
  35. 2023HelmFluid: Learning Helmholtz Dynamics for Interpretable Fluid Predictioncited by 7
  36. 2021PowerDynamics.jl -- An experimentally validated open-source package for the dynamical analysis of power gridscited by 6
  37. 2022Approximation of nearly-periodic symplectic maps via structure-preserving neural networkscited by 6
  38. 2021A Variational Formulation of Accelerated Optimization on Riemannian Manifoldscited by 5

50 most cited of 56 All 56 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 4; 2026: 1.

5 papers.

  1. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 8
  2. 2025Causal Climate Emulation with Bayesian Filteringcited by 5
  3. 2025A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equationscited by 4
  4. 2025ArchesClimate: Probabilistic Decadal Ensemble Generation With Flow Matchingcited by 3
  5. 2026Exact Fixed-Point Constraints in Neural-ODEs with Provable Universalitycited by 1