ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees
Andrea Goertzen, Sunbochen Tang, Navid Azizan
TL;DR
This paper introduces ECO, a novel energy-constrained operator learning framework that guarantees boundedness in predictions of chaotic dynamical systems, enabling stable long-term forecasting and invariant statistics estimation.
Contribution
ECO is the first data-driven chaotic dynamics model with formal boundedness guarantees using control theory and a quadratic projection layer.
Findings
ECO produces stable long-term forecasts for chaotic PDEs.
ECO accurately captures invariant statistical properties.
ECO provides a computationally efficient boundedness enforcement mechanism.
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…
Peer Reviews
Decision·ICLR 2026 Conference Withdrawn Submission
The paper has the following strengths worth credits: 1. Integrating control-theoretic Lyapunov energy functions to enforce dissipativity in learning chaotic systems. 2. The development of a convex quadratic projection layer to generate bounded predictions. 3. Appendix B provides meaningful metrics for evaluating long-term forecasting performance in chaotic systems.
The paper should be improved considering the following facts: 1. The novelty of ellipsoidal constraints is **incremental** compared with Markov neural operator (MNO). ECO replaces the spherical hard constraint with ellipsoidal constraints and introduces a differentiable projection layer. 2. Experiments and evaluations are limited to compare ECO with DeepONet. In Figure 4-5, DeepONet is not a strong baseline. Leading methods such as MNO (a constrained neural operator method), Poincaré Flow Neura
Good story on involving the control theory to help ML model
Despite good theoretic insights, the results doesn't look promising. Especially, I am not sure if the DeepOnet is tuned properly. No ML model parameters is provided. Also, wether the authors have checked MNO results. The PDF of the data looks shifted from the ground truth making the results less convincing.
Provides a clear and formal discrete-time boundedness guarantee. Integrates the constraint enforcement in a simple, computationally efficient closed-form layer. Demonstrates convincing empirical prevention of trajectory blow-up on canonical chaotic systems. Theoretical results are stated cleanly and tied directly to the implementation.
The novelty is limited—energy-based constraints, Lyapunov guarantees, and differentiable convex projections have been previously studied. Empirical baselines are thin (only DeepONet). Comparisons to FNO, reservoir computing, or dissipativity-aware models would greatly strengthen the manuscript. Overclaims originality (“first with guarantees for chaotic dynamics”)—this should be softened and scoped precisely. No ablations on key hyperparameters (k, alpha,Q) or quantification of how tightly th
## Strengths - The paper addresses a critical and well-known challenge in data-driven modeling of chaotic systems: the instability and divergence of long-term autoregressive predictions. The goal of enforcing stability by construction is highly relevant and important for the field. - The primary contribution is the theoretical, formal guarantees for the boundedness of the learned dynamics. Integrating Lyapunov stability theory into the neural operator architecture through a differentiable quadra
## Weaknesses While the core idea is promising, the paper has several notable weaknesses. - One weakness is the mismatch between the introduced Lyapunov energy-functional layer and the paper’s *operator-learning* framework. As stated in Theorem 1 (and its proof via bounds on the quadratic form), the energy layer and its guarantees are derived for a finite-dimensional system by spatial discretization. Specifically, the learned Lyapunov functional $V(w)=(w-w_c)^\top Q(w-w_c)$ acts on $w\in\mathbb{
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Taxonomy
TopicsModel Reduction and Neural Networks · Neural Networks and Reservoir Computing · Quantum many-body systems