Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model using data-driven emulators, including neural operator architectures. For chaotic systems, the inherent sensitivity to initial conditions makes exact long-term forecasts theoretically infeasible, meaning that traditional squared-error losses often fail when trained on noisy data. Recent work has focused on training emulators to match the statistical properties of chaotic attractors by introducing regularization based on handcrafted local features and summary statistics, as well as learned statistics extracted from a diverse dataset of trajectories. In this work, we propose a family of adversarial optimal transport objectives that jointly learn high-quality summary statistics and a physically consistent emulator. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein). Our experiments across a variety of chaotic systems, including systems with high-dimensional chaotic attractors, show that emulators trained with our approach exhibit significantly improved long-term statistical fidelity.

In this setting, neural operators trained to minimize squared error losses, while capable of accurate short-term forecasts, often fail to reproduce statistical or structural properties of the dynamics over longer time horizons and can yield degenerate results.

Chaotic systems make long-horizon forecasts difficult because small perturbations in initial conditions cause trajectories to diverge at an exponential rate. In this setting, neural operators trained to minimize squared error losses, while capable of accurate short-term forecasts, often fail to reproduce statistical or structural properties of the dynamics over longer time horizons and can yield degenerate results. In this paper, we propose an alternative framework designed to preserve invariant measures of chaotic attractors that characterize the time-invariant statistical properties of the dynamics. Specifically, in the multi-environment setting (where each sample trajectory is governed by slightly different dynamics), we consider two novel approaches to training with noisy data. First, we propose a loss based on the optimal transport distance between the observed dynamics and the neural operator outputs. This approach requires expert knowledge of the underlying physics to determine what statistical features should be included in the optimal transport loss. Second, we show that a contrastive learning framework, which does not require any specialized prior knowledge, can preserve statistical properties of the dynamics nearly as well as the optimal transport approach. On a variety of chaotic systems, our method is shown empirically to preserve invariant measures of chaotic attractors.

Fundamental limits to predictability are central to our understanding of many physical and computational systems. Here we show that, despite its remarkable capabilities, deep learning exhibits such fundamental limits rooted in the fractal, riddled geometry of its basins of attraction: any initialization that leads to one solution lies arbitrarily close to another that leads to a different one. We derive sufficient conditions for the emergence of riddled basins by analytically linking features widely observed in deep learning, including chaotic learning dynamics and symmetry-induced invariant subspaces, to reveal a general route to riddling in realistic deep networks. The resulting basins of attraction possess an infinitely fine-scale fractal structure characterized by an uncertainty exponent near zero, so that even large increases in the precision of initial conditions yield only marginal gains in outcome predictability. Riddling thus imposes a fundamental limit on the predictability and hence reproducibility of neural network training, providing a unified account of many empirical observations. These results reveal a general organizing principle of deep learning with important implications for optimization and the safe deployment of artificial intelligence.

Reservoir computing can embed attractors into random neural networks (RNNs), generating a ``mirror'' of a target attractor because of its inherent symmetrical constraints. In these RNNs, we report that an attractor-merging crisis accompanied by intermittency emerges simply by adjusting the global parameter. We further reveal its underlying mechanism through a detailed analysis of the phase-space structure and demonstrate that this bifurcation scenario is intrinsic to a general class of RNNs, independent of training data.

Department of Physics, Arizona State University, Tempe, Arizona 85287, USA (Dated: September 5, 2025) Finding the governing equations from data by sparse optimization has become a popular approach to deterministic modeling of dynamical systems. Considering the physical situations where the data can be imperfect due to disturbances and measurement errors, we show that for many chaotic systems, widely used sparse-optimization methods for discovering governing equations produce models that depend sensitively on the measurement procedure, yet all such models generate virtually identical chaotic attractors, leading to a striking limitation that challenges the conventional notion of equation-based modeling in complex dynamical systems. Calculating the Koopman spectra, we find that the different sets of equations agree in their large eigenvalues and the differences begin to appear when the eigenvalues are smaller than an equation-dependent threshold. The results suggest that finding the governing equations of the system and attempting to interpret them physically may lead to misleading conclusions. It would be more useful to work directly with the available data using, e.g., machine-learning methods. In physical science, a methodology of biblical importance is developing a quantitative model by extracting a set of governing equations from experimental data.

Handling regime shifts and non-stationary time series in deep learning systems presents a significant challenge. In the case of online learning, when new information is introduced, it can disrupt previously stored data and alter the model's overall paradigm, especially with non-stationary data sources. Therefore, it is crucial for neural systems to quickly adapt to new paradigms while preserving essential past knowledge relevant to the overall problem. In this paper, we propose a novel training algorithm for neural networks called \textit{Lyapunov Learning}. This approach leverages the properties of nonlinear chaotic dynamical systems to prepare the model for potential regime shifts. Drawing inspiration from Stuart Kauffman's Adjacent Possible theory, we leverage local unexplored regions of the solution space to enable flexible adaptation. The neural network is designed to operate at the edge of chaos, where the maximum Lyapunov exponent, indicative of a system's sensitivity to small perturbations, evolves around zero over time. Our approach demonstrates effective and significant improvements in experiments involving regime shifts in non-stationary systems. In particular, we train a neural network to deal with an abrupt change in Lorenz's chaotic system parameters. The neural network equipped with Lyapunov learning significantly outperforms the regular training, increasing the loss ratio by about $96\%$.

Machine learning has emerged as an alternative approach for solving partial differential equations, reproducing trajectories of dynamical systems, emulating statistical properties of chaotic systems, etc. Neural networks and deep learning play a particularly important role in developing new techniques for understanding and solving various dynamical systems. Reservoir computing [15, 30] is a particular class of machine learning models; it utilizes a large recurrent network (reservoir), and only a linear output layer is trained to match the trajectory. Echo-State Networks refer to reservoirs that have the Echo-State Property (see e.g.

International Research Center for Neurointelligence, University of Tokyo Institutes for Advanced Study, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan (Dated: July 16, 2024) Chaotic dynamics are ubiquitous in nature and useful in engineering, but their geometric design can be challenging. Here, we propose a method using reservoir computing to generate chaos with a desired shape by providing a periodic orbit as a template, called a skeleton. We exploit a bifurcation of the reservoir to intentionally induce unsuccessful training of the skeleton, revealing inherent chaos. The emergence of this untrained attractor, resulting from the interaction between the skeleton and the reservoir's intrinsic dynamics, offers a novel semi-supervised framework for designing chaos. Chaotic dynamics are prevalent in nature, including biological neural systems [1-3], and are applied in engineering, such as for random number generation [4, 5], communication systems [6, 7], optimization [8, 9], deep learning [10, 11], and robot control [12-14]. A notable challenge is designing the geometric shapes of chaotic attractors, for which no practical methods have been proposed so far, to the best of our knowledge.