ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees
This addresses the issue of unbounded predictions in chaotic systems modeling for applications like weather and fluid turbulence, offering the first formal guarantees for data-driven models, though it is incremental in improving existing methods.
The paper tackles the problem of data-driven models for chaotic dynamical systems generating unbounded predictions, which prevents meaningful statistics evaluation, by introducing the Energy-Constrained Operator (ECO) that learns system dynamics while enforcing boundedness, resulting in stable long-horizon forecasts that capture invariant statistics on systems like the Kuramoto-Sivashinsky and Navier-Stokes equations.
Chaos is a fundamental feature of many complex dynamical systems, including weather systems and fluid turbulence. These systems are inherently difficult to predict due to their extreme sensitivity to initial conditions. Many chaotic systems are dissipative and ergodic, motivating data-driven models that aim to learn invariant statistical properties over long time horizons. While recent models have shown empirical success in preserving invariant statistics, they are prone to generating unbounded predictions, which prevent meaningful statistics evaluation. To overcome this, we introduce the Energy-Constrained Operator (ECO) that simultaneously learns the system dynamics while enforcing boundedness in predictions. We leverage concepts from control theory to develop algebraic conditions based on a learnable energy function, ensuring the learned dynamics is dissipative. ECO enforces these algebraic conditions through an efficient closed-form quadratic projection layer, which provides provable trajectory boundedness. To our knowledge, this is the first work establishing such formal guarantees for data-driven chaotic dynamics models. Additionally, the learned invariant level set provides an outer estimate for the strange attractor, a complex structure that is computationally intractable to characterize. We demonstrate empirical success in ECO's ability to generate stable long-horizon forecasts, capturing invariant statistics on systems governed by chaotic PDEs, including the Kuramoto--Sivashinsky and the Navier--Stokes equations.