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This research introduces a novel method for recovering continuous-time dynamics from discrete data by enforcing global structural constraints, specifically the semi-group property, to minimize computational cost. By treating deviations from this property as a regularizer, the approach achieves superior stability and efficiency compared to standard Neural ODEs and flow-matching baselines on complex physical benchmarks like Navier-Stokes equations. The technique effectively internalizes temporal consistency, allowing adaptive solvers to operate with significantly fewer function evaluations while maintaining accuracy over long horizons.
Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency
arXiv:2605.08454v2 Announce Type: replace Abstract: Recovering continuous-time dynamics from discrete observations is difficult because local supervision loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property---which we call Symmetry Rupture---serves two roles: as a training regularizer it confines the hypothesis space to flows that compose consistently across temporal scales; as an inference oracle it guides an adaptive solver to select the largest step size that preserves internal consistency, without relying on local truncation error estimates. On the diffusion-reaction and Navier-Stokes benchmarks, our method achieves the lowest rollout RMSE among all evaluated methods using $\leq 3$ function evaluations per step. On the near-conservative shallow water benchmark, our method matches or exceeds competing flow-based methods at far lower computational cost; the strongest Neural ODE baseline achieves lower absolute RMSE but at the expense of $>100$ function evaluations per step. In the direct auto-regressive setting, our solver maintains stable long-horizon rollouts where flow-matching baselines diverge and Neural ODE requires up to $12\times$ more function evaluations. The core contribution is a model that internalizes temporal consistency, removing the need for high-order external solvers to compensate for structural bias.
Abstract:Recovering continuous-time dynamics from discrete observations is difficult because local supervision loses fidelity as the observation interval grows. We replace local supervision with a global structural constraint: any flow representing autonomous dynamics must satisfy the semi-group property under time translation. We train a time-conditioned secant velocity field whose deviation from this property---which we call Symmetry Rupture---serves two roles: as a training regularizer it confines the hypothesis space to flows that compose consistently across temporal scales; as an inference oracle it guides an adaptive solver to select the largest step size that preserves internal consistency, without relying on local truncation error estimates. On the diffusion-reaction and Navier-Stokes benchmarks, our method achieves the lowest rollout RMSE among all evaluated methods using $\leq 3$ function evaluations per step. On the near-conservative shallow water benchmark, our method matches or exceeds competing flow-based methods at far lower computational cost; the strongest Neural ODE baseline achieves lower absolute RMSE but at the expense of $>100$ function evaluations per step. In the direct auto-regressive setting, our solver maintains stable long-horizon rollouts where flow-matching baselines diverge and Neural ODE requires up to $12\times$ more function evaluations. The core contribution is a model that internalizes temporal consistency, removing the need for high-order external solvers to compensate for structural bias.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2605.08454 [cs.LG]
(or arXiv:2605.08454v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.08454 Focus to learn more arXiv-issued DOI via DataCite
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