Learning Stiff Dynamical Operators: Scaling, Fast-Slow Excitation, and Eigen-Consistent Neural Models

TL;DR

This paper introduces methods to improve neural operator learning for stiff dynamical systems by addressing spectral fidelity, enabling accurate modeling of both slow and fast modes across different stiffness regimes.

Contribution

It proposes three novel techniques—stiffness-aware scaling, local trajectory excitation, and Jacobian diagnostics—to enhance spectral fidelity in neural models of stiff systems.

Findings

Reduced fast eigenvalue error by an order of magnitude

Improved recovery of slow and fast modes across regimes

Enhanced rollout fidelity in stiff dynamical systems

Abstract

Stiff dynamical systems represent a central challenge in multi scale modeling across combustion, chemical kinetics, and nonlinear dynamical systems. Neural operator learning has recently emerged as a promising approach to approximate dynamical generators from data, yet stiffness imposes severe obstacles: training errors concentrate on slow manifold states, collapse of fast dynamics occurs, and the learned operator may fail to reproduce the true eigenstructure. We demonstrate three key advances enabling accurate learning of stiff operators and preserving spectral fidelity: (i) stiffness aware scaling of time derivatives, (ii) fast direction excitation via local trajectory cloud bursts, and (iii) autograd-based Jacobian diagnostics ensuring eigenstructure fidelity. Applied to the Davis-Skodje system, the approach recovers both slow and fast modes across stiffness regimes, reducing fast eigenvalue error by an order of magnitude while improving rollout fidelity. These results argue that spectral fidelity - not trajectory accuracy alone - should be a first-class target in data driven learning of stiff operators.