Journal Article

Cross‐Attractor Transforms: Improving Forecasts by Learning Optimal Maps Between Dynamical Systems and Imperfect Models

Amrhein, D. E., Agarwal, N., Grooms, I.

2025

Overview

Biased, incomplete numerical models are often used for forecasting states of complex dynamical systems by mapping an estimate of a “true” initial state into model phase space, making a forecast, and then mapping back to the “true” space. While advances have been made to reduce errors associated with model initialization and model forecasts, we lack a general framework for discovering optimal mappings between “true” dynamical systems and model phase spaces. Here, we propose using a data‐driven approach to infer these maps. Our approach consistently reduces errors in the Lorenz‐96 system with an imperfect model constructed to produce significant model errors compared to a reference configuration. Optimal pre‐ and post‐processing transforms leverage “shocks” and “drifts” in the imperfect model to make more skillful forecasts of the reference system. The implemented machine learning architecture using neural networks constructed with a custom analog‐adjoint layer makes the approach generalizable across applications. Plain Language Summary Modeling and forecasting natural systems, such as Earth's oceans and atmosphere, is difficult due to their inherent unpredictability, our incomplete understanding of their dynamics, and their vastness and complexity. One way to improve forecasts is by improving physical representations within numerical models. However, models will always have shortcomings. The alternative approach explored here is to maximize the utility of available imperfect or incomplete models by revising how the model is used and how its forecast is interpreted. Here, we employ machine learning to learn pre‐ and post‐processing operators, called cross‐attractor transforms (CATs), which reduce the overall forecast errors from imperfect models. We demonstrate the framework's efficacy by using a simplified dynamical model as an imperfect representation of a higher‐dimensional chaotic dynamical system, analogous to using a simple pendulum to forecast the behavior of a double pendulum. In addition to improving forecasts, CATs offer insights into how the two systems evolve in time. The approach is generalizable across dynamical systems and disciplines. Key Points Forecasts from imperfect models are improved using optimized pre‐ and post‐processing operators Neural networks trained on imperfect models and reference truth forecasts efficiently derive these operators Forecasts from this hybrid machine‐learning approach are more accurate than purely data‐driven methods when applied to an idealized system

• Publisher: John Wiley & Sons, Inc,Wiley
• Subject: Attractors (mathematics) / Chaos theory / Complexity / Dynamic models / Dynamical systems / Earth system prediction / Error reduction / forecast error / Forecast errors / Forecast improvement / Forecasting / Learning algorithms / Machine learning / Mapping / Mathematical models / Neural networks / nonlinear processes / Numerical models / Oceans / Pendulums / Reference systems / Representations