Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

Researchers present a novel method for reconstructing continuous-time physical dynamics from discrete observations by enforcing the semi-group property of autonomous flows, using a metric called Symmetry Rupture to regularize training and guide adaptive step selection. The approach significantly outperforms Neural ODE baselines on diffusion-reaction and PDE benchmarks, reducing errors by 87% while requiring 5x fewer function evaluations.

This work addresses a fundamental challenge in scientific machine learning: inferring smooth continuous dynamics from sparse, discrete measurements. Traditional approaches rely on local supervision signals—pointwise regression targets or derivative approximations—that degrade in accuracy as observation intervals expand. The researchers replace this local paradigm with a global structural constraint rooted in dynamical systems theory: any autonomous flow must satisfy the semi-group property, meaning time translations compose consistently.

The innovation centers on a time-conditioned secant velocity field whose deviation from this property—termed Symmetry Rupture—serves dual purposes. During training, it acts as a regularizer that restricts the hypothesis space to physically plausible flows. During inference, it functions as an adaptive solver oracle, enabling the model to select step sizes based on internal geometric consistency rather than conventional local truncation error estimates.

The method demonstrates substantial practical advantages across benchmarks. On diffusion-reaction problems with time-informed inference, it achieves 87% error reduction using 5x fewer evaluations than Neural ODE baselines. In the more challenging auto-regressive setting—where the model must predict distant frames without intermediate guidance—the method maintains stability and low rollout error while allocating computation proportionally to local geometric complexity.

This represents progress toward more reliable neural surrogate models for physical systems, particularly valuable for scientific computing, materials simulation, and climate modeling where computational efficiency and accuracy both matter. The approach bridges classical dynamical systems theory with modern deep learning, suggesting that explicit structural constraints can substitute for massive data requirements.

• →Symmetry Rupture measures deviation from semi-group property to regularize training and guide adaptive step selection
• →Method reduces rollout error by 87% on diffusion-reaction benchmarks while using 5x fewer function evaluations than Neural ODE
• →Approach maintains lowest rollout error on two of three PDE benchmarks in direct auto-regressive prediction without intermediate cues
• →Global structural constraints from dynamical systems theory improve generalization beyond local supervision signals
• →Adaptive solver allocates compute based on local geometric complexity rather than conventional truncation error estimates