Trajectory-Optimized Time Reparameterization for Learning-Compatible Reduced-Order Modeling of Stiff Dynamical Systems

TL;DR

This paper introduces a trajectory-optimized time reparameterization method to improve the learnability and stability of reduced-order models for stiff dynamical systems, enabling better explicit integration and prediction accuracy.

Contribution

It proposes an optimization-based time reparameterization approach that enhances the conditioning and smoothness of trajectories in neural ODE reduced-order models for stiff systems.

Findings

Significantly reduces training loss compared to existing methods.

Produces smoother, more learnable trajectories in stiff systems.

Achieves improved physical-time predictions across multiple test cases.

Abstract

Stiff dynamical systems present a challenge for machine-learning reduced-order models (ML-ROMs), as explicit time integration becomes unstable in stiff regimes while implicit integration within learning loops is computationally expensive and often degrades training efficiency. Time reparameterization (TR) offers an alternative by transforming the independent variable so that rapid physical-time transients are spread over a stretched-time coordinate, enabling stable explicit integration on uniformly sampled grids. Although several TR strategies have been proposed, their effect on learnability in ML-ROMs remains incompletely understood. This work investigates time reparameterization as a stiffness-mitigation mechanism for neural ODE reduced-order modeling and introduces a trajectory-optimized TR (TOTR) formulation. The proposed approach casts time reparameterization as an optimization problem in arc-length coordinates, in which a traversal-speed profile is selected to penalize acceleration in stretched time. By targeting the smoothness of the training dynamics, this formulation produces reparameterized trajectories that are better conditioned and easier to learn than existing TR methods. TOTR is evaluated on three stiff problems: a parameterized stiff linear system, the van der Pol oscillator, and the HIRES chemical kinetics model. Across all cases, the proposed approach yields smoother reparameterizations and improved physical-time predictions under identical training regimens than other TR approaches. Quantitative results demonstrate loss reductions of one to two orders of magnitude compared to benchmark algorithms. These results highlight that effective stiffness mitigation in ML-ROMs depends critically on the regularity and learnability of the time map itself, and that optimization-based TR provides a robust framework for explicit reduced-order modeling of multiscale dynamical systems.