Although extensivework hasrecently been done inthisfield, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable.

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.

A Novel Reservoir Computing Framework for Chaotic Time Series Prediction Using Time Delay Embedding and Random Fourier Features S. K. Laha Advanced Design and Analysis Group CSIR - Central Mechanical Engineering Research Institute MG Avenue, Durgapur, West Bengal, PIN - 713209, India Abstract: Forecasting chaotic time series requires models that can capture the intrinsic geometry of the underlying attractor while remaining computationally efficient. We introduce a novel reservoir computing (RC) framework that integrates time - delay embedding with Random Fourier Feature (RFF) mappings to construct a dynamical reservoir without the need for traditional recurrent architectures. Unlike standard RC, which relies on high - dimensional recurrent connectivity, the proposed RFF - RC explicitly approximates non linear kernel transformations that uncover latent dynamical relations in the reconstructed phase space. This hybrid formulation offers two key advantages: (i) it provides a principled way to approximate complex nonlinear interactions among delayed coordina tes, thereby enriching the effective dynamical representation of the reservoir, and (ii) it reduces reliance on manual reservoir hyperparameters such as spectral radius and leaking rate. We evaluate the framework on canonical chaotic systems - the Mackey - Gla ss equation, the Lorenz system, and the Kuramoto - Sivashinsky equation. This novel formulation demonstrates that RFF - RC not only achieves superior prediction accuracy but also yields robust attractor reconstructions and long - horizon forecasts.

Long-term forecasting of chaotic systems remains a fundamental challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Conventional approaches, such as reservoir computing, typically require training data that incorporates long-term continuous dynamical behavior to comprehensively capture system dynamics. While advanced deep sequence models can capture transient dynamics within the training data, they often struggle to maintain predictive stability and dynamical coherence over extended horizons. Here, we propose PhyxMamba, a framework that integrates a Mamba-based state-space model with physics-informed principles to forecast long-term behavior of chaotic systems given short-term historical observations on their state evolution. We first reconstruct the attractor manifold with time-delay embeddings to extract global dynamical features. After that, we introduce a generative training scheme that enables Mamba to replicate the physical process. It is further augmented by multi-patch prediction and attractor geometry regularization for physical constraints, enhancing predictive accuracy and preserving key statistical properties of systems. Extensive experiments on simulated and real-world chaotic systems demonstrate that PhyxMamba delivers superior forecasting accuracy and faithfully captures essential statistics from short-term historical observations.

Accurately forecasting chaotic systems, prevalent in domains such as weather prediction and fluid dynamics, remains a significant scientific challenge. The inherent sensitivity of these systems to initial conditions, coupled with a scarcity of observational data, severely constrains traditional modeling approaches. Since these models are typically trained for a specific system, they lack the generalization capacity necessary for real-world applications, which demand robust zero-shot or few-shot forecasting on novel or data-limited scenarios. To overcome this generalization barrier, we propose ChaosNexus, a foundation model pre-trained on a diverse corpus of chaotic dynamics. ChaosNexus employs a novel multi-scale architecture named ScaleFormer augmented with Mixture-of-Experts layers, to capture both universal patterns and system-specific behaviors. The model demonstrates state-of-the-art zero-shot generalization across both synthetic and real-world benchmarks. On a large-scale testbed comprising over 9,000 synthetic chaotic systems, it improves the fidelity of long-term attractor statistics by more than 40% compared to the leading baseline. This robust performance extends to real-world applications with exceptional data efficiency. For instance, in 5-day global weather forecasting, ChaosNexus achieves a competitive zero-shot mean error below 1 degree, a result that further improves with few-shot fine-tuning. Moreover, experiments on the scaling behavior of ChaosNexus provide a guiding principle for scientific foundation models: cross-system generalization stems from the diversity of training systems, rather than sheer data volume.

Recent time-series foundation models exhibit strong abilities to predict physical systems. These abilities include zero-shot forecasting, in which a model forecasts future states of a system given only a short trajectory as context, without knowledge of the underlying physics. Here, we show that foundation models often forecast through a simple parroting strategy, and when they are not parroting they exhibit some shared failure modes such as converging to the mean. As a result, a naive context parroting model that copies directly from the context scores higher than leading time-series foundation models on predicting a diverse range of dynamical systems, including low-dimensional chaos, turbulence, coupled oscillators, and electrocardiograms -- and at a tiny fraction of the computational cost. We draw a parallel between context parroting and induction heads, which explains recent works showing that large language models can often be repurposed for time series forecasting. Our dynamical systems perspective also ties the scaling between forecast accuracy and context length to the fractal dimension of the underlying chaotic attractor, providing insight into previously observed in-context neural scaling laws. By revealing the performance gaps and failure modes of current time-series foundation models, context parroting can guide the design of future foundation models and help identify in-context learning strategies beyond parroting.