Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

TL;DR

This paper introduces a global structural constraint based on the semi-group property to recover continuous dynamics from discrete data, improving accuracy and efficiency over traditional local supervision methods.

Contribution

It proposes a novel approach using symmetry rupture as a regularizer and inference tool, enabling adaptive step size selection and better long-term predictions in dynamical systems.

Findings

Reduces rollout RMSE by 87% on diffusion-reaction benchmarks.

Uses 5x fewer function evaluations than Neural ODE baseline.

Maintains low RMSE in auto-regressive PDE benchmarks while adapting compute based on complexity.

Abstract

Recovering continuous-time dynamics from discrete observations is difficult because local supervision (e.g., pointwise regression targets, derivative approximations, or equation residuals) 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 purposes. As a training regularizer, it confines the hypothesis space to flows that compose consistently across temporal scales. As an inference oracle, it lets the solver select the largest step size that preserves internal consistency, replacing the local truncation error that conventional adaptive solvers depend on. On the diffusion-reaction benchmark under time-informed inference, our method reduces rollout RMSE by 87\% while using 5x fewer function evaluations than a Neural ODE baseline. In the more demanding direct auto-regressive setting, where the model must predict distant future frames without intermediate temporal cues, our adaptive solver allocates compute based on local geometric complexity -- maintaining the lowest rollout RMSE on two of three PDE benchmarks while baselines either diverge or require up to an order of magnitude more function evaluations to remain stable.