A Response Theory Probe for Learned Stochastic AI Simulators, Tested on Lorenz-63

2026-10-05

AI Summary

This study addresses the lack of validated forced-response evaluation in machine learning (ML) chaos simulators. Grounded in Koopman theory, this work introduces a calibrated, mode-resolved response testing framework that integrates linear response theory with Ruelle–Pollicott resonances to systematically assess SINDy, multilayer perceptrons, reservoir computing, and neural ODEs/SDEs on the Lorenz-63 system. The results reveal a decoupling between statistical and response fidelity, demonstrating that training formulations fundamentally determine model properties: only sparse regression passes all tests, while other architectures exhibit either response or statistical biases. Furthermore, the static susceptibility χ(0) proves insufficient for identifying these critical discrepancies.

Abstract

Machine-learning emulators of chaotic and stochastic systems are usually validated on forecast skill and long-run statistics. Neither certifies that an emulator responds correctly to forcing, the property that projection and attribution studies rely on. Linear response theory makes this testable: the forced response follows from unperturbed correlations through a generalized fluctuation-dissipation relation, and decomposes over the stochastic Ruelle-Pollicott resonances of the Koopman generator. Building on the Koopmanism Response framework, we turn this into a calibrated, mode-resolved test for learned surrogates: each surrogate rollout passes or fails each check, and failure rates are compared with those of independent realizations of the true system. On stochastic Lorenz-63, a three-variable toy model, we evaluate SINDy, an MLP, a reservoir computer, a neural ODE and a neural SDE with learned diffusion, over up to 80 rollouts each. A sparse-regression model with the correct library passes every check at rates consistent with the true system. Invariant-statistics fidelity and response fidelity dissociate in both directions: a quarter of reservoir-computer rollouts pass every invariant-statistics check and match the static susceptibility $\chi(0)$, yet misrepresent the slow relaxation modes, while the neural ODE and SDE rarely meet the invariant-statistics floor but recover those modes in three quarters of rollouts. As expected of a time-integrated quantity dominated here by fast relaxation, $\chi(0)$ does not separate these cases. For a fixed network, the training formulation (one-step drift, flow map, or multi-step through the integrator) decides which of these properties it gets right.

Problem

• machine learning emulators
• linear response theory
• chaotic and stochastic systems
• surrogate validation
• forced response

Innovation

• Linear response theory
• Koopman generator
• Stochastic AI simulators
• Ruelle-Pollicott resonances
• Surrogate validation