ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees

2025-12-01

AI Summary

Long-term prediction of chaotic dynamical systems—such as the Kuramoto–Sivashinsky and Navier–Stokes equations—suffers from unbounded trajectory divergence and poor statistical consistency. Method: We propose an energy-constrained operator learning framework that, for the first time, provides provable trajectory boundedness for data-driven chaotic models. Our approach integrates a learnable energy function with a differentiable, control-theoretically designed algebraic dissipation constraint, enforced via an efficient closed-form quadratic projection layer embedded end-to-end during training. Contributions/Results: (1) Theoretical guarantee that predicted trajectories remain within an outer estimate of the singular attractor’s invariant level set; (2) accurate recovery of long-term invariant statistics; and (3) stable, long-horizon predictions across multiple chaotic PDE systems, significantly outperforming existing unconstrained models.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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.

Problem

• Ensuring bounded predictions in chaotic dynamics models
• Learning dissipative dynamics with provable trajectory boundedness
• Capturing invariant statistics for chaotic PDE systems

Innovation

• Energy-constrained operator learning for chaotic dynamics
• Algebraic conditions from control theory ensure dissipative dynamics
• Closed-form quadratic projection layer guarantees trajectory boundedness