mlsanityData to 7 Oct 2026

Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators

arXiv 2306.03548 ↗First posted 6 Jun 2023cs.LG cs.NA math.NAcited by 4

Not on the map. Why?

Abstract (arXiv)

We introduce the mean inverse integrator (MII), a novel approach to increase the accuracy when training neural networks to approximate vector fields of dynamical systems from noisy data. This method can be used to average multiple trajectories obtained by numerical integrators such as Runge-Kutta methods. We show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular advantages when used in connection with MII. When training vector field approximations, explicit expressions for the loss functions are obtained when inserting the training data in the MIRK formulae, unlocking symmetric and high-order integrators that would otherwise be implicit for initial value problems. The combined approach of applying MIRK within MII yields a significantly lower error compared to the plain use of the numerical integrator without averaging the trajectories. This is demonstrated with experiments using data from several (chaotic) Hamiltonian systems. Additionally, we perform a sensitivity analysis of the loss functions under normally distributed perturbations, supporting the favorable performance of MII.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 16 of its 52 references that are in the map. First posted on arXiv 6 Jun 2023.

Builds on

16 of 52 references.

References · 16

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2019Hamiltonian Neural Networkscited by 702
  5. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  6. 2019Symplectic Recurrent Neural Networkscited by 148
  7. 2019Hamiltonian Graph Networks with ODE Integratorscited by 107
  8. 2020Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraintscited by 102
  9. 2021Symplectic Learning for Hamiltonian Neural Networkscited by 35
  10. 2020Sparse Symplectically Integrated Neural Networkscited by 20
  11. 2022Pseudo-Hamiltonian Neural Networks with State-Dependent External Forcescited by 17
  12. 2022Learning Hamiltonians of constrained mechanical systemscited by 14
4 more
  1. 2022FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantitiescited by 12
  2. 2020Variational Integrator Graph Networks for Learning Energy Conserving Dynamical Systemscited by 7
  3. 2022Bayesian Identification of Nonseparable Hamiltonian Systems Using Stochastic Dynamic Modelscited by 3
  4. 2023Learning Hamiltonian Systems with Mono-Implicit Runge-Kutta Methodscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 1; 2026: 2.

4 papers.

  1. 2025Learning mechanical systems from real-world data using discrete forced Lagrangian dynamicscited by 6
  2. 2024Symplectic Neural Networks Based on Dynamical Systemscited by 4
  3. 2026Controlled oscillation modeling using port-Hamiltonian neural networkscited by 1
  4. 2026Learning partially observed systems with neural Hamiltonian ordinary differential equationscited by 0