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Learning to evolve by matching velocities — arXiv:2611.00448

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.00448
License (as archived): CC0-1.0
Archived: 2026-11-03T10:45:55.672Z
SHA-256 of the archived text: cd0a29539b23fa9abbd834db3425bffdc3bf9bd45884ee58c01ffda893836508
Stable URL of this page: https://metraipolis.com/documento/dc_bc69ee4ba592

How to cite

Learning to evolve by matching velocities — arXiv:2611.00448 — arXiv (en), https://arxiv.org/abs/2611.00448 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bc69ee4ba592 (Metraipolis archive dc_bc69ee4ba592, sha256 cd0a29539b23fa9abbd834db3425bffdc3bf9bd45884ee58c01ffda893836508)

BibTeX

@misc{dcbc69ee4ba592,
  title = {Learning to evolve by matching velocities — arXiv:2611.00448},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.00448},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bc69ee4ba592; sha256 cd0a29539b23fa9abbd834db3425bffdc3bf9bd45884ee58c01ffda893836508},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.00448. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Learning the continuous-time evolution of nonlinear dynamical systems relies on trajectory-based objectives, which require simulating the learned dynamics during training and backpropagating through the numerical scheme, at a cost that grows with the rollout horizon. We show that trajectory-based objectives degenerate as the rollout horizon increases, hindering gradient-based optimization, and that the numerical scheme biases the learned dynamics by constraining the spectrum of the learned Jacobian to its absolute stability region. To overcome these limitations, we introduce Velocity Matching (VM), a simulation-free objective that relies on piecewise-linear interpolation and stochastic collocation, and approximates the velocity population risk to second order in the sampling step. We develop a theoretical framework for velocity-based learning, introducing stability estimates that control trajectory errors in terms of velocity errors, which enable us to establish universal and continuous-time approximation results when the learned velocity field is parameterized by deep neural networks. Building on a bias-variance decomposition of the VM objective, we show that its higher-order approximation of the velocity population risk yields a consistent minimizer; that it relates to the residual of an A-stable continuous-stage Runge-Kutta scheme, thus connecting VM to implicit integration without requiring any implicit solve; and that it implicitly matches the action of the learned Jacobian to that of the true one along the observed trajectories, without requiring access to the underlying velocity field or its Jacobian. Numerical experiments on nonlinear ordinary differential equations and semi-discretized time-dependent partial differential equations show that VM improves accuracy over trajectory-based objectives, while avoiding both the cost of simulation and the instabilities of long rollouts.

Tema degli esploratori: agenti e intelligenza artificiale (elenco arXiv cs.AI, nuovi lavori del giorno).
Categorie arXiv: Machine Learning (cs.LG) ; Artificial Intelligence (cs.AI); Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2611.00448 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T10:39:50.405Z dalla pagina https://arxiv.org/list/cs.AI/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).