Learning Dissipative Chaotic Dynamics with Boundedness Guarantees

TL;DR

The paper introduces a dissipative projection layer that enforces Lyapunov-based dissipativity in neural network models of chaotic dynamics, guaranteeing bounded long-horizon trajectories and an outer estimate of the strange attractor. By jointly learning a dynamics emulator and an energy function, the approach narrows the feasible parameter space and enables reliable long-term statistics in systems like Lorenz 63, Lorenz 96, and KS ROM. Empirical results show bounded trajectories and improved preservation of invariant statistics compared with unconstrained baselines, demonstrating practical significance for data-scarce, safety-critical chaotic systems. The framework offers a general, physics-informed inductive bias that can extend to more complex real-world dynamics and downstream control tasks. Overall, it provides a principled means to combine data-driven modeling with stability guarantees in chaotic regimes.

Abstract

Chaotic dynamics, commonly seen in weather systems and fluid turbulence, are characterized by their sensitivity to initial conditions, which makes accurate prediction challenging. Recent approaches have focused on developing data-driven models that attempt to preserve invariant statistics over long horizons since many chaotic systems exhibit dissipative behaviors and ergodicity. Despite the recent progress in such models, they are still often prone to generating unbounded trajectories, leading to invalid statistics evaluation. To address this fundamental challenge, we introduce a modular framework that provides formal guarantees of trajectory boundedness for neural network chaotic dynamics models. Our core contribution is a dissipative projection layer that leverages control-theoretic principles to ensure the learned system is dissipative. Specifically, our framework simultaneously learns a dynamics emulator and an energy-like function, where the latter is used to construct an algebraic dissipative constraint within the projection layer. Furthermore, the learned invariant level set provides an outer estimate for the system's strange attractor, which is known to be difficult to characterize due to its complex geometry. We demonstrate our model's ability to produce bounded long-horizon forecasts that preserve invariant statistics for chaotic dynamical systems, including Lorenz 96 and a reduced-order model of the Kuramoto-Sivashinsky equation.

Figures (9)

• Figure 1: Overview of our approach and main results. (A) Current autoregressive models without constraints suffer from accumulated error, leading to trajectories growing unbounded over extended rollout. Our approach overcomes this issue by building a dissipative projection layer that ensures the model is dissipative and guarantees bounded trajectories. (B) The dissipative constraint effectively narrows the search space of model parameters because the true dynamics of interest are dissipative. (C) Hovmöller diagrams of the Lorenz 96 system (with linear interpolation in the spatial dimension): Our approach ($f^*$ with projection, on the right) generates a bounded trajectory that reproduces flow characteristics seen in the ground truth trajectory (on the left), while the unconstrained neural network generates an unbounded trajectory failing to capture any meaningful statistics.
• Figure 2: Illustrations of theoretical results in Proposition \ref{['prop:invariance']} and \ref{['prop:attractivity']}: (A) The level set boundary serves as a barrier since the trajectory cannot gain energy outside. Once entering $M(c)$, the trajectory will be confined within. (B) The trajectory loses energy over time outside $M(c)$ because $\dot{V}(x) \leq 0$, resulting in convergence to the level set $M(c)$ eventually ($c_2 > c_1 > c > 0$).
• Figure 3: Lorenz 63 (A) Trajectories generated by the learned model ("fstar") and true dynamics ("GT") are visualized by their 2D projections, along with the learned invariant set (sampled with yellow points). (B) Comparison of the flow map projected onto $x_1-x_2$ plane. (C) Comparison of the learned energy function time history evaluated on the ground truth trajectory and the trajectory generated by our learned model.
• Figure 4: Principal component analysis (PCA) comparison between trajectories generated by unconstrained model ($\hat{f}$) and our proposed model with dissipative projection ($f^*$) for Lorenz 96 (first row, figures A and B) and a reduced-order KS model (second row, figures C and D). Unconstrained model (left figure in (A) and (C)) generates a trajectory that quickly deviates from the attractor and then grows unbounded. Our proposed model (right figure in (A) and (C)) provides boundedness guarantees, which guides the generated trajectory to enter and traverse the strange attractor. In addition, the learned invariant set (red ellipsoid point cloud) provides a tight outer-estimate of the attractor. The 2D histograms represent the probability density of trajectories in the PCA leading components for Lorenz 96 (B) and the reduced-order KS model (D). In both cases, the unconstrained model $\hat{f}$ (middle figure in (B) and (D)) produces an unstructured distribution that scatters across the PCA space, while our proposed model $f^*$ (right figure in (B) and (D)) is able to reproduce the shape and density of the ground truth distribution (left figure in (B) and (D)), which validates its capability to better preserve the system's invariant statistics.
• Figure 5: Reproduction of complex emergent dynamics in the Kuramoto-Sivashinsky system. The figure compares the spatiotemporal evolution of the ground truth (top) with predictions from the unconstrained ($\hat{f}$, middle) and our constrained ($f^*$, bottom) models. The unconstrained model exhibits finite-time blowup and grows unbounded. In contrast, our model's trajectory not only remains bounded but also faithfully captures the essential physical phenomena of the system, including the intricate patterns of coarsening events seen in the ground truth.

Theorems & Definitions (9)

• Definition 1
• Proposition 1: invariant level set
• Proposition 2: asymptotic stability
• Theorem 1
• proof
• Corollary 1
• proof
• proof : Proof for Proposition \ref{['prop:invariance']}
• proof : Proof for Proposition \ref{['prop:attractivity']}