Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency

2026-05-08

AI Summary

This work addresses the challenge of recovering continuous-time dynamics from discrete observations, where conventional locally supervised methods suffer from significant distortion over large time intervals. The authors propose a novel approach grounded in global structural constraints: they enforce the semigroup property under time translation for autonomous dynamical flows and introduce a time-conditioned secant vector field. A key innovation lies in leveraging “symmetry breaking” as a regularizer, which simultaneously constrains the hypothesis space to ensure cross-scale consistency and replaces local truncation error to guide adaptive step-size selection. On diffusion-reaction benchmarks, the method reduces rollback RMSE by 87% and requires five times fewer function evaluations. In an autoregressive setting without intermediate hints, it achieves the lowest RMSE on two of three PDE benchmarks, whereas baseline methods either diverge or demand an order-of-magnitude more computation to remain stable.

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.

Problem

• continuous-time dynamics
• discrete observations
• physical dynamics recovery
• time-series modeling
• differential equations

Innovation

• Symmetry Rupture
• semi-group property
• intrinsic differential consistency
• adaptive solver
• time-conditioned secant velocity field