mlsanityData to 14 Oct 2026

FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities

arXiv 2210.00272 ↗First posted 1 Oct 2022cs.LG cs.NA math.DScited by 12

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Abstract (arXiv)

Many real-world dynamical systems are associated with first integrals (a.k.a. invariant quantities), which are quantities that remain unchanged over time. The discovery and understanding of first integrals are fundamental and important topics both in the natural sciences and in industrial applications. First integrals arise from the conservation laws of system energy, momentum, and mass, and from constraints on states; these are typically related to specific geometric structures of the governing equations. Existing neural networks designed to ensure such first integrals have shown excellent accuracy in modeling from data. However, these models incorporate the underlying structures, and in most situations where neural networks learn unknown systems, these structures are also unknown. This limitation needs to be overcome for scientific discovery and modeling of unknown systems. To this end, we propose first integral-preserving neural differential equation (FINDE). By leveraging the projection method and the discrete gradient method, FINDE finds and preserves first integrals from data, even in the absence of prior knowledge about underlying structures. Experimental results demonstrate that FINDE can predict future states of target systems much longer and find various quantities consistent with well-known first integrals in a unified manner.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Builds on 35 of its 68 references that are in the map. First posted on arXiv 1 Oct 2022.

Builds on

35 of 68 references.

References · 35

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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 3; 2024: 3; 2025: 3; 2026: 3.

12 papers.

  1. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 52
  2. 2023Parallelizing non-linear sequential models over the sequence lengthcited by 45
  3. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 9
  4. 2024Structure-Preserving Physics-Informed Neural Networks With Energy or Lyapunov Structurecited by 6
  5. 2024Projected Neural Differential Equations for Learning Constrained Dynamicscited by 5
  6. 2023Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integratorscited by 4
  7. 2025Discovering Phase Space Structure in Learned Hamiltonian Systemscited by 2
  8. 2023Pseudo-Hamiltonian system identificationcited by 1
  9. 2024Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domainscited by 1
  10. 2026An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulationcited by 0
  11. 2026Learning partially observed systems with neural Hamiltonian ordinary differential equationscited by 0
  12. 2026OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systemscited by 0