Chaotic Time-Series Prediction

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Chaotic time-series prediction asks how to infer a system’s evolving state and useful long-run statistics from finite, noisy observations, in settings such as Lorenz dynamics, turbulence, and climate. Common approaches include reservoir computing and recurrent or transformer models, alongside Neural ODEs, Koopman operator learning, and physics-informed or structure-preserving surrogates; symbolic regression seeks interpretable equations. Because small errors grow quickly, evaluation distinguishes short-term trajectory skill from stable rollouts, long-run statistics, and rare-event forecasts, using benchmarks that vary noise, chaos level, and forecast horizon. The central tension is that exact trajectories become unreliable beyond Lyapunov-limited horizons, while models that match long-run statistics may miss event timing or physical mechanisms; robust uncertainty estimates and data-efficient generalization remain open challenges.

Latest papers

54 papers6 papers in the last four weeks, up 100% on the four weeks before. 0.0% of all new papers.

May 27, 2026

Adaptive Reservoir Computing for Multi-Scenario Chaotic System Forecasting

We present an adaptive reservoir computing framework for the CTF-4-Science Lorenz benchmark, which evaluates machine learning models across twelve distinct tasks spanning five qualitatively different scenarios: baseline forecasting, noisy signal reconstruction, forecasting under noise, few-shot learning, and parametric generalization. Rather than applying a uniform inference strategy, we tailor the training and prediction procedure of Echo State Networks (ESNs) to the specific demands of each evaluation scenario. Our key contributions are fourfold: (1) exact reservoir state synchronization that eliminates warmup approximation error in short-time prediction; (2) histogram-guided candidate selection that directly optimizes the long-time ergodic evaluation metric; (3) multi-seed reservoir search for few-shot regimes with severely limited training data; and (4) sequential multi-sequence training that resolves state-distribution mismatch in parametric generalization tasks. The proposed framework achieves a score of 74.91 on the public benchmark leaderboard, demonstrating that carefully adapted reservoir computing constitutes a competitive and computationally efficient approach for diverse chaotic system modeling challenges.

Shadmehr Zaregarizi, Khashayar Yavari

Politecnico di Torino Turin, Italy

Dynamic Neural Networks Forecasting Benchmarks Reservoir Computing Chaotic Time-Series Prediction

May 24, 2026

A comparative study of accuracy and rollout stability of temporal surrogate models

Temporal surrogate models are effective for predicting chaotic dynamical systems where computational cost can be prohibitive. Several deep neural network architectures can be used for such purposes. In this work, a few commonly used architectures are compared using a common training protocol. The objective is to fairly assess the impact of model architectures for long-horizon prediction stability. Experiments are carried out for three problems, the double pendulum, the Kuramoto-Sivashinsky equations, and the Kolmogorov flow. The experiments are carried out with matching model capacity. Analysis is also carried out for a scenario where each model is individually optimized. It is observed that in both scenarios, the models exhibit categorical differences in long-horizon rollouts. For a concrete quantification, stepwise error injections and perturbation amplifications are analyzed using metrics such as local jacobian, relative one-step bias, and finite-time Lyapunov growth. Additionally, an attractor analysis is also conducted to assess how well the learned models replicate the underlying system geometry. An ablation study to isolate the impact of each component of a continuous-update architecture is also carried out. It is concluded that models that having integrator-like updates show lower bias and perturbation amplification yielding stable long-horizon rollout and more accurate predictions.

Rajarshi Biswas

Cargill Inc. Wayzata, MN 55391, USA

Neural Surrogate Modeling Long-Term Time Series Forecasting PDE Surrogate Modeling Chaotic Time-Series Prediction

May 17, 2026

FEG-Pro: Forecast-Error Growth Profiling for Finite-Horizon Instability Analysis of Nonlinear Time Series

Estimating the largest Lyapunov exponent from a scalar time series is difficult when the governing equations, tangent dynamics, and full state vector are unavailable. We propose FEG-Pro, a forecast-error growth profiling framework for nonlinear scalar time series. The method constructs autocorrelation-guided sparse histories, performs distance-weighted k-nearest-neighbor multi-horizon forecasting, and analyzes the logarithmic growth of geometrically averaged forecast errors. Its primary output is the finite-horizon forecast-error growth slope, lambda_FEG. When the error-growth curve supports a quasi-linear regime, this slope can be compared with reference largest Lyapunov exponents as an estimate of the dominant instability rate. The same pipeline also extracts the formal fit-selection regime, curvature, residual roughness after quadratic detrending, monotonicity, and forecast-error distribution entropy (FEDE) from signed multi-horizon errors. These secondary descriptors are intended not only as diagnostic controls for the slope, but also as candidate machine-learning features for nonlinear signal analysis, because they encode profile geometry and distributional uncertainty not captured by lambda_FEG alone. We evaluate the method on chaotic maps, Mackey-Glass delay dynamics, and scalar Lorenz-63 observables with known or reference exponents. Full-record experiments show good agreement in quasi-linear cases and meaningful curve-shape information in curved or weak profiles. A dyadic length-halving experiment on representative logistic, Mackey-Glass, and Lorenz records shows that residual roughness and mean FEDE often change monotonically and remain interpretable as record length decreases, even when the slope becomes biased or highly variable. The results support treating forecast-error growth as a structured profile and feature-generation framework rather than a single-number estimator.

Andrei Velichko, N'Gbo N'Gbo, Bruno Carpentieri +1

Institute of Physics and Technology, Petrozavodsk State University, Petrozavodsk, Russia · School of Science and Engineering, International University of Grand Bassam, Grand Bassam, Cote d’Ivoire · Faculty of Engineering, Free University of Bozen-Bolzano, Bolzano, Italy +2

Time Series Forecasting Chaotic Dynamical Systems Chaotic Time-Series Prediction Lyapunov Exponent Estimation

May 14, 2026

QuChaTeR: A Hybrid Quantum-Chaotic Temporal Framework for Earthquake Prediction

Seismic prediction remains challenging due to the highly nonlinear and chaotic dynamics of earthquake signals. While classical deep learning models such as LSTMs and CNNs capture local temporal features, and quantum models offer richer state representations, their integration with chaos-driven mechanisms is underexplored. We introduce QuChaTeR, a hybrid architecture that combines wavelet-based preprocessing, chaotic maps, and variational quantum circuits with recurrent structures to enhance temporal feature extraction. Implemented in PyTorch and PennyLane, QuChaTeR is benchmarked against classical (LSTM, GRU, RNN, 1D-CNN, Reservoir Computing) and quantum-inspired (Quantum LSTM) baselines. On real-world seismic datasets, QuChaTeR consistently converges faster and achieves superior performance across multiple evaluation criteria. Despite promising results, scalability and quantum hardware limitations remain challenges. Overall, this work demonstrates how quantum-chaotic hybridization provides a practical pathway toward more accurate and robust earthquake prediction.

Emir Kaan Özdemir

Istanbul Erkek High School Ministry of National Education Istanbul, Türkiye · Ministry of National Education

Hybrid Quantum-Classical ML Spatiotemporal Forecasting Chaotic Time-Series Prediction

May 10, 2026

ChaosNetBench: Benchmarking Spatio-Temporal Graph Neural Networks on Chaotic Lattice Dynamics

Spatio-temporal graph neural networks (STGNNs) are widely used for short-term forecasting in dynamic physical systems such as traffic and weather. However, the prevailing evaluation practice uses real world benchmark data sets in a single domain with a single fixed holdout splits, making it difficult to compare architectures across different dynamical regimes. We introduce ChaosNetBench (CNB), a synthetic benchmark dataset and evaluation framework for studying STGNN performance under controlled multidimensional chaotic dynamics. CNB is built on a lattice of coupled standard maps with independently tunable local chaos (

K

), coupling strength (

\varepsilon

), and system size (

N

), providing known topology and known dynamics across 96 system instances and 9{,}600 trajectories. We introduce chaos indicators, evaluation metrics and a protocol to analyze and compare the capacity of STGNN architectures to deal with different levels of local and global chaos. We illustrate the usage of the framework by analyzing 13 architectures (5 STGNNs and 8 non-graph baselines). The results reveal a regime dependent transition in which non-graph baselines (TCN, N-BEATS, iTransformer) remain competitive when there is low local chaos, while STGNNs (e.g., Graph WaveNet, D2STGNN, STAEformer) are generally more resilient to higher levels of local and global chaos. CNB provides a practical, reusable testbed for systematically comparing and analyzing the capacity of STGNN architectures to handle different levels of local and global chaos.

Henok Tenaw Moges, Charalampos Skokos, Deshendran Moodley

Centre for Artificial Intelligence Research (CAIR) University of Cape Town, Cape Town, South Africa · Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics University of Cape Town, Cape Town, South Africa

Forecasting Benchmarks Synthetic Benchmark Generation Chaotic Dynamical Systems Chaotic Time-Series Prediction Spatiotemporal GNNs

Apr 28, 2026

Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

Andre Herz, Daniel Durstewitz, Georgia Koppe

Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany · Dept. of Theoretical Neuroscience, Central Institute of Mental Health (CIMH), Mannheim, Germany · Hector Institute for AI in Psychiatry and Dept. of Psychiatry and Psychotherapy, CIMH, Mannheim, Germany +1

Neural Surrogate Modeling Generalized Bayesian Inference Dynamical Systems Bayesian Inference Recurrent Neural Networks Chaotic Time-Series Prediction

Apr 26, 2026

An architectural capacity ceiling, not a barren plateau: why a fixed-encoding variational quantum circuit cannot fit the Lorenz-63 attractor

Variational quantum circuits train poorly on chaotic forecasting, usually blamed on barren plateaus (exponentially vanishing gradients). Using an exactly simulable four-qubit variational quantum physics-informed circuit fit to Lorenz-63, we show the barren-plateau explanation fails: the failure is an architectural capacity ceiling fixed by the circuit time-encoding, not its trainable depth. Four measurements support this. (i) A McClean-comparable gradient-variance estimator sits at the local-cost Haar/2-design scale 2^(-2n)=3.9e-3 at n=4; on structurally live parameters it decays about ninefold with depth then saturates there, large enough to train, not an exponential collapse. (ii) At a common budget of 200 optimiser iterations (600, in three stages, for layer-wise), gradient descent, layer-wise, and SPSA reach the same order of magnitude of loss, so no optimiser unlocks a better basin. (iii) The output-Jacobian rank saturates at 33 from five layers on, so depth buys no new output directions. (iv) A Fourier analysis explains why: the qubit-1 phase encoding acts on the initial |0> and is inert, so the maximum accessible frequency is 2.5/t_max=0.83 Hz, identical at every depth and about 4.4x below the narrowest Lorenz component bandwidth. The corrected band has dimension 1+2x5=11 per observable, and 3x11=33 equals the measured rank ceiling exactly, unifying the two diagnostics. A trained depth sweep agrees: mean loss improves with depth then flattens once the rank saturates. We correct our earlier preprint diagnosis, which compared unnormalised gradient norms to the McClean threshold, and place the advantage of fixed reservoirs and classical echo-state networks in architecture, not quantum mechanics.

Tushar Pandey

Texas A&M University, College Station, TX 77843, USA

Quantum Circuit Expressibility Variational Quantum Circuits Physics-Informed ML Quantum Circuit Trainability Chaotic Time-Series Prediction

Apr 23, 2026

Neuromorphic Computing Based on Parametrically-Driven Oscillators and Frequency Combs

Parametrically driven oscillators exhibit distinct dynamical regimes that strongly influence their computational capability as physical reservoirs. Here, we demonstrate reservoir computing in a two-mode system exhibiting 2:1 parametric resonance using Mackey--Glass, Rössler, and Lorenz chaotic time-series prediction tasks. Mapping the normalized mean squared error over parameter space reveals that the lowest prediction errors consistently occur within the parametric resonance regime, where nonlinear interactions are activated while temporal coherence is preserved. In contrast, although frequency-comb states increase spectral dimensionality, prediction performance degrades in the chaotic comb regime due to loss of phase coherence. We further show that the modulation depth

Δ\text{F}

, detuning

κ

, damping ratio

γ_{21}

, and input data rate systematically reshape the accessible dynamical regimes and thereby control the reservoir performance. These results identify parametric resonance as a robust operating regime for oscillator-based neuromorphic computing and establish dynamical regime engineering as a route toward physical reservoir optimization.

Mahadev Sunil Kumar, Adarsh Ganesan

Amrita School of Computing, Amrita Vishwa Vidyapeetham, Kollam, India 690525 · Department of Electrical and Electronics Engineering, Birla Institute of Technology and Science, Pilani – Dubai Campus, Dubai, United Arab Emirates 345055

Dynamical Systems Reservoir Computing Neuromorphic Computing Chaotic Time-Series Prediction

Apr 22, 2026

A Hybridizable Neural Time Integrator for Stable Autoregressive Forecasting

For autoregressive modeling of chaotic dynamical systems over long time horizons, the stability of both training and inference is a major challenge in building scientific foundation models. We present a hybrid technique in which an autoregressive transformer is embedded within a novel shooting-based mixed finite element scheme, exposing topological structure that enables provable stability. For forward problems, we prove preservation of discrete energies, while for training we prove uniform bounds on gradients, provably avoiding the exploding gradient problem. Combined with a vision transformer, this yields latent tokens admitting structure-preserving dynamics. We outperform modern foundation models with a

65\times

reduction in model parameters and long-horizon forecasting of chaotic systems. A "mini-foundation" model of a fusion component shows that 12 simulations suffice to train a real-time surrogate, achieving a

9{,}000\times

speedup over particle-in-cell simulation.

Brooks Kinch, Xiaozhe Hu, Yilong Huang +6

Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA · Department of Mathematics, Tufts University, Medford, MA, USA · Department of Mathematics and Cybernetics, SINTEF Digital, Oslo, Norway +3

Neural Surrogate Modeling Time Series Transformers Chaotic Time-Series Prediction Autoregressive Modeling Neural Network Stability Autoregressive Transformer

Apr 22, 2026

Learning to Emulate Chaos: Adversarial Optimal Transport Regularization

Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model with data-driven methods such as machine learning emulators. While emulators are promising tools for accelerating simulations and solving inverse problems, they still struggle to learn chaotic dynamics, where sensitivity to initial conditions renders exact long-term forecasts infeasible, especially given noisy data. Recent work instead trains emulators to match the statistical properties of chaotic attractors, but these approaches often rely on handcrafted summary statistics or large, diverse multi-environment datasets. In this work, we propose a family of adversarial optimal transport objectives that can jointly learn high-quality summary statistics and a physically consistent emulator from a single noisy trajectory. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein) of our approach. Numerical experiments across a variety of chaotic systems, including ones with high-dimensional spatiotemporal chaos, show that emulators trained using our proposed objectives have significantly improved long-term statistical fidelity.

Gabriel Melo, Leonardo Santiago, Peter Y. Lu

LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau · Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC · University of Campinas +1

Neural Surrogate Modeling Chaotic Dynamical Systems Optimal Transport Chaotic Time-Series Prediction

Apr 17, 2026

Horizon-Constrained Rashomon Sets for Chaotic Forecasting

Predictive multiplicity and chaotic dynamics represent two fundamental challenges in machine learning that have evolved independently despite their conceptual connections. We bridge this gap by introducing horizon-constrained Rashomon sets, a theoretical framework that characterizes how model multiplicity evolves with prediction horizon in chaotic systems. Unlike static prediction tasks where the Rashomon set remains fixed, chaos induces exponential divergence among initially similar models, fundamentally transforming the nature of predictive equivalence. We prove that the effective Rashomon set contracts exponentially with lead time at a rate determined by the maximum Lyapunov exponent and introduce Lyapunov-weighted metrics that provide tighter bounds on predictive disagreement. Leveraging these insights, we develop decision-aligned selection algorithms that choose among near-optimal models based on downstream utility rather than forecast accuracy alone. Extensive experiments on synthetic chaotic systems (Lorenz-96, Kuramoto-Sivashinsky) and real-world applications (wind power, traffic, weather) demonstrate that our framework improves decision quality by 18-34% while maintaining competitive predictive performance. This work establishes the first rigorous connection between chaos theory and predictive multiplicity, providing principled guidance for deploying machine learning in safety-critical chaotic domains.

Gauri Kale, Rahul Vishwakarma, Holly Diamond +2

Dept. of Electrical Engineering, California State University Long Beach, USA · WorkOnward Inc., USA · Dept. of Computer Engineering & Computer Science, California State University Long Beach, USA

Chaotic Dynamical Systems Chaotic Time-Series Prediction

Apr 3, 2026

Recurrent Quantum Feature Maps for Reservoir Computing

Reservoir computing promises a fast method for handling large amounts of temporal data. This hinges on constructing a good reservoir--a dynamical system capable of transforming inputs into a high-dimensional representation while remembering properties of earlier data. In this work, we introduce a reservoir based on recurrent quantum feature maps where a fixed quantum circuit is reused to encode both current inputs and a classical feedback signal derived from previous outputs. We evaluate the model on the Mackey-Glass time-series prediction task using our recently introduced CP feature map, and find that it achieves lower mean squared error than standard classical baselines, including echo state networks and multilayer perceptrons, while maintaining compact circuit depth and qubit requirements. We further analyze memory capacity and show that the model effectively retains temporal information, consistent with its forecasting accuracy. Finally, we study the impact of realistic noise and find that performance is robust to several noise channels but remains sensitive to two-qubit gate errors, identifying a key limitation for near-term implementations.

Utkarsh Singh, Aaron Z. Goldberg, Christoph Simon +1

Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario, K1N 6N5 Canada · National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1N 5A2, Canada · Institute for Quantum Science and Technology, Department of Physics and Astronomy, University of Calgary, Alberta T2N 1N4, Canada +1

Quantum Reservoir Computing Quantum Feature Maps Chaotic Time-Series Prediction

Mar 23, 2026

Noise Titration: Exact Distributional Benchmarking for Probabilistic Time Series Forecasting

Modern time series forecasting is evaluated almost entirely through passive observation of single historical trajectories, rendering claims about a model's robustness to non-stationarity fundamentally unfalsifiable. We propose a paradigm shift toward interventionist, exact-statistical benchmarking. By systematically titrating calibrated Gaussian observation noise into known chaotic and stochastic dynamical systems, we transform forecasting from a black-box sequence matching game into an exact distributional inference task. Because the underlying data-generating process and noise variance are mathematically explicit, evaluation can rely on exact negative log-likelihoods and calibrated distributional tests rather than heuristic approximations. To fully leverage this framework, we extend the Fern architecture into a probabilistic generative model that natively parameterizes the Symmetric Positive Definite (SPD) cone, outputting calibrated joint covariance structures without the computational bottleneck of generic Jacobian modeling. Under this rigorous evaluation, we find that state-of-the-art zero-shot foundation models behave consistently with the context-parroting mechanism, failing systematically under non-stationary regime shifts and elevated noise. In contrast, Fern explicitly captures the invariant measure and multivariate geometry of the underlying dynamics, maintaining structural fidelity and statistically sharp calibration precisely where massive sequence-matching models collapse.

Qilin Wang

Independent Researcher

Probabilistic Time Series Forecasting Forecasting Benchmarks Non-Stationary Time Series Forecasting Chaotic Time-Series Prediction

Mar 16, 2026

Storage and selection of multiple chaotic attractors in minimal reservoir computers

Modern predictive modeling increasingly calls for a single learned dynamical substrate to operate across multiple regimes. From a dynamical-systems viewpoint, this capability decomposes into the storage of multiple attractors and the selection of the appropriate attractor in response to contextual cues. In reservoir computing (RC), multi-attractor learning has largely been pursued using large, randomly wired reservoirs, on the assumption that stochastic connectivity is required to generate sufficiently rich internal dynamics. At the same time, recent work shows that minimal deterministic reservoirs can match random designs for single-system chaotic forecasting. Under which conditions can minimal topologies learn multiple chaotic attractors? In this paper, we find that minimal architectures can successfully store multiple chaotic attractors. However, these same architectures struggle with task switching, in which the system must transition between attractors in response to external cues. We test storage and selection on all 28 unordered system pairs formed from eight three-dimensional chaotic systems. We do not observe a robust dependence of multi-attractor performance on reservoir topology. Over the ten topologies investigated, we find that no single one consistently outperforms the others for either storage or cue-dependent selection. Our results suggest that while minimal substrates possess the representational capacity to model coexisting attractors, they may lack the robust temporal memory required for cued transitions.

Francesco Martinuzzi, Holger Kantz

Max Planck Institute for the Physics of Complex Systems, Dresden, Germany

Reservoir Computing Dynamical Systems Chaotic Dynamical Systems Chaotic Time-Series Prediction

Mar 16, 2026

A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systems

We introduce a score-filter-enhanced data assimilation framework designed to reduce predictive uncertainty in machine learning (ML) models for data-driven dynamical system forecasting. Machine learning serves as an efficient numerical model for predicting dynamical systems. However, even with sufficient data, model uncertainty remains and accumulates over time, causing the long-term performance of ML models to deteriorate. To overcome this difficulty, we integrate data assimilation techniques into the training process to iteratively refine the model predictions by incorporating observational information. Specifically, we apply the Ensemble Score Filter (EnSF), a generative AI-based training-free diffusion model approach, for solving the data assimilation problem in high-dimensional nonlinear complex systems. This leads to a hybrid data assimilation-training framework that combines ML with EnSF to improve long-term predictive performance. We shall demonstrate that EnSF-enhanced ML can effectively reduce predictive uncertainty in ML-based Lorenz-96 system prediction and the Korteweg-De Vries (KdV) equation prediction.

Jingqiao Tang, Ryan Bausback, Feng Bao +2

Data Assimilation Scientific ML Chaotic Time-Series Prediction

Mar 14, 2026

Hierarchy of extreme-event predictability in turbulence revealed by machine learning

Extreme-event predictability in turbulence is strongly state dependent, yet event-by-event predictability horizons are difficult to quantify without access to governing equations or costly perturbation ensembles. Here we train an autoregressive conditional diffusion model on direct numerical simulations of the two-dimensional Kolmogorov flow and use a CRPS-based skill score to define an event-wise predictability horizon. Enstrophy extremes exhibit a pronounced hierarchy: forecast skill persists from

\approx 1

to

> 4

Lyapunov times across events. Spectral filtering shows that these horizons are controlled predominantly by large-scale structures. Extremes are preceded by intense strain cores organizing quadrupolar vortex packets, whose lifetime sharply separates long- from short-horizon events. These results identify coherent-structure persistence as a governing mechanism for the predictability of turbulence extremes and provide a data-driven route to diagnose predictability limits from observations.

Yuxuan Yang, Chenyu Dong, Gianmarco Mengaldo

National University of Singapore Singapore

Chaotic Time-Series Prediction Computational Fluid Dynamics Probabilistic Forecasting

Mar 13, 2026

Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems

High-fidelity numerical simulations of chaotic, high dimensional nonlinear dynamical systems are computationally expensive, necessitating the development of efficient surrogate models. Most surrogate models for such systems are deterministic, for example when neural operators are involved. However, deterministic models often fail to capture the intrinsic distributional uncertainty of chaotic systems. This work presents a surrogate modeling formulation that leverages generative machine learning, where a deep learning diffusion model is used to probabilistically forecast turbulent flows over long horizons. We introduce a multi-step autoregressive diffusion objective that significantly enhances long-rollout stability compared to standard single-step training. To handle complex, unstructured geometries, we utilize a multi-scale graph transformer architecture incorporating diffusion preconditioning and voxel-grid pooling. More importantly, our modeling framework provides a unified platform that also predicts spatiotemporally important locations for sensor placement, either via uncertainty estimates or through an error-estimation module. Finally, the observations of the ground truth state at these dynamically varying sensor locations are assimilated using diffusion posterior sampling requiring no retraining of the surrogate model. We present our methodology on two-dimensional homogeneous and isotropic turbulence and for a flow over a backwards-facing step, demonstrating its utility in forecasting, adaptive sensor placement, and data assimilation for high dimensional chaotic systems.

Dibyajyoti Chakraborty, Hojin Kim, Romit Maulik

College of Information Sciences and Technology, Pennsylvania State University, University Park, 16802, PA, USA · School of Mechanical Engineering, Purdue University, West Lafayette, 47907, IN, USA

PDE Surrogate Modeling Chaotic Time-Series Prediction Autoregressive Diffusion Active Sensing Data Assimilation Diffusion Posterior Sampling

Mar 11, 2026

Dynamics-Informed Deep Learning for Predicting Extreme Events

Predicting extreme events in high-dimensional chaotic dynamical systems remains a fundamental challenge, as such events are rare, intermittent, and arise from transient dynamical mechanisms that are difficult to infer from limited observations. Accordingly, real-time forecasting calls for precursors that encode the mechanisms driving extremes, rather than relying solely on statistical associations. We propose a fully data-driven framework for long-lead prediction of extreme events that constructs interpretable, mechanism-aware precursors by explicitly tracking transient instabilities preceding event onset. The approach leverages a reduced-order formulation to compute finite-time Lyapunov exponent (FTLE)-like precursors directly from state snapshots, without requiring knowledge of the governing equations. To avoid the prohibitive computational cost of classical FTLE computation, instability growth is evaluated in an adaptively evolving low-dimensional subspace spanned by Optimal Time-Dependent (OTD) modes, enabling efficient identification of transiently amplifying directions. These precursors are then provided as input to a Transformer-based model, enabling forecast of extreme event observables. We demonstrate the framework on Kolmogorov flow, a canonical model of intermittent turbulence. The results show that explicitly encoding transient instability mechanisms substantially extends practical prediction horizons compared to baseline observable-based approaches.

Eirini Katsidoniotaki, Themistoklis P. Sapsis

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139

Chaotic Time-Series Prediction Chaotic Dynamical Systems

Mar 7, 2026

Turning Time Series into Algebraic Equations: Symbolic Machine Learning for Interpretable Modeling of Chaotic Time Series

Chaotic time series are notoriously difficult to forecast. Small uncertainties in initial conditions amplify rapidly, while strong nonlinearities and regime dependent variability constrain predictability. Although modern deep learning often delivers strong short horizon accuracy, its black box nature limits scientific insight and practical trust in settings where understanding the underlying dynamics matters. To address this gap, we propose two complementary symbolic forecasters that learn explicit, interpretable algebraic equations from chaotic time series data. Symbolic Neural Forecaster (SyNF) adapts a neural network based equation learning architecture to the forecasting setting, enabling fully differentiable discovery of compact and interpretable algebraic relations. The Symbolic Tree Forecaster (SyTF) builds on evolutionary symbolic regression to search directly over equation structures under a principled accuracy complexity trade off. We evaluate both approaches in a rolling window nowcasting setting with one step ahead forecasting using several accuracy metrics and compare against a broad suite of baselines spanning classical statistical models, tree ensembles, and modern deep learning architectures. Numerical experiments cover a benchmark of 132 low dimensional chaotic attractors and two real world chaotic time series, namely weekly dengue incidence in San Juan and the Nino 3.4 sea surface temperature index. Across datasets, symbolic forecasters achieve competitive one step ahead accuracy while providing transparent equations that reveal salient aspects of the underlying dynamics.

Madhurima Panja, Grace Younes, Tanujit Chakraborty

SAFIR, Sorbonne University Abu Dhabi, UAE · Sorbonne Center for Artificial Intelligence, Sorbonne University, Paris, France

Time Series Forecasting Chaotic Time-Series Prediction Explainable Time Series Forecasting Symbolic Regression

Mar 4, 2026

Adaptive Sensing of Continuous Physical Systems for Machine Learning

Physical dynamical systems can be viewed as natural information processors: their systems preserve, transform, and disperse input information. This perspective motivates learning not only from data generated by such systems, but also how to measure them in a way that extracts the most useful information for a given task. We propose a general computing framework for adaptive information extraction from dynamical systems, in which a trainable attention module learns both where to probe the system state and how to combine these measurements to optimize prediction performance. As a concrete instantiation, we implement this idea using a spatiotemporal field governed by a partial differential equation as the underlying dynamics, though the framework applies equally to any system whose state can be sampled. Our results show that adaptive spatial sensing significantly improves prediction accuracy on canonical chaotic benchmarks. This work provides a perspective on attention-enhanced reservoir computing as a special case of a broader paradigm: neural networks as trainable measurement devices for extracting information from physical dynamical systems.

Felix Köster, Atsushi Uchida

Department of Information and Computer Sciences, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama City, Saitama, 338–8570, Japan

Adaptive Sensing Active Perception Scientific ML Chaotic Time-Series Prediction

Nov 10, 2025

A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series

The accurate forecasting of complex, high-dimensional dynamical systems from observational data is a fundamental task across numerous scientific and engineering disciplines. A significant challenge arises from noise-corrupted measurements, which severely degrade the performance of data-driven models. In chaotic dynamical systems, where small initial errors amplify exponentially, it is particularly difficult to develop a model from noisy data that achieves short-term accuracy while preserving long-term invariant properties. To overcome this, we consider the weak formulation as a complementary approach to the classical

L2

-loss function for training models of dynamical systems. We empirically verify that the weak formulation, with a proper choice of test function and integration domain, effectively filters noisy data. This insight explains why a weak form loss function is analogous to fitting a model to filtered data and provides a practical way to parameterize the weak form. Subsequently, we demonstrate how this approach overcomes the instability and inaccuracy of standard Neural ODE (NODE) in modeling chaotic systems. Through numerical examples, we show that our proposed training strategy, the Weak Penalty NODE, is computationally efficient, solver-agnostic, and yields accurate and robust forecasts across benchmark chaotic systems and a real-world climate dataset.

Xuyang Li, John Harlim, Dibyajyoti Chakraborty +1

aSchool of Construction, University of North Carolina at Charlotte, Charlotte, 28223, NC, USA · cDepartment of Mathematics, Institute for Computational and Data Sciences, The Pennsylvania State University, University Park, 16802, PA, USA · bCollege of Information Sciences and Technology, The Pennsylvania State University, University Park, 16802, PA, USA +1

Nonlinear System Identification Neural ODEs Chaotic Time-Series Prediction

Jul 21, 2025

Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

While data-driven techniques are powerful tools for reduced-order modeling of systems with chaotic dynamics, great potential remains for leveraging known physics (i.e. a full-order model (FOM)) to improve predictive capability. We develop a hybrid reduced order model (ROM), informed by both data and FOM, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found using an autoencoder. This approach projects the vector field of the FOM onto the invariant manifold; then, this physics-derived vector field is either corrected using dynamic data, or used as a Bayesian prior that is updated with data. In both cases, the neural ordinary differential equation approach is used. We consider simulated data from the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations. Relative to the data-only approach, for scenarios of abundant data, scarce data, and even an incorrect FOM (i.e. erroneous parameter values), the hybrid approach yields substantially improved time-series predictions.

Alex Guo, Michael D. Graham

Department of Chemical and Biological Engineering, University of Wisconsin-Madison, Madison WI 53706, USA

Reduced-Order Modeling Chaotic Time-Series Prediction PDE Surrogate Modeling Latent Dynamics Modeling

Apr 24, 2025

Tailored minimal reservoir computing: on the bidirectional connection between nonlinearities in the reservoir and in data

We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.

Davide Prosperino, Haochun Ma, Christoph Räth

Ludwig-Maximilians-Universität München, Faculty of Physics, Geschwister-Scholl-Platz 1, 80539 Munich, Germany · Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institute for Frontier Materials on Earth and in Space, Linder Höhe, 51147 Cologne, Germany

Nonlinear System Identification Reservoir Computing Chaotic Time-Series Prediction

Jul 8, 2024

Adversarial dynamical systems characterize when data-driven learning succeeds or fails

Many systems resist analytical modeling, making data-driven inference of dynamics important. Yet data-driven methods can fail to converge or generalize, leaving open a central question: When can system behavior be learned reliably from data, and when is such learning impossible? We answer this question using adversarial dynamical systems to identify the boundary between accessible and inaccessible regimes. In Koopman operator learning, a leading framework for representing nonlinear dynamics through linear spectral objects, we design optimal data-driven spectral algorithms with convergence and certification guarantees under conditions arising broadly in physical systems. This yields a convergence theory for Koopman-operator approximations and resolves a longstanding open problem in Koopman spectral analysis. Conversely, by constructing adversarial systems, we prove matching impossibility results: without these conditions, no single-sequence limiting procedure can guarantee learning, regardless of data quality. These results sharply characterize when data-driven spectral learning can succeed and when it must fail. We validate the framework on oscillators, chaotic fluid flows and Arctic sea ice concentration forecasting. In the latter, we uncover hidden modes of Arctic sea ice decline, deliver long-range forecasts with geographic error bounds, and outperform state-of-the-art dynamical and deep learning models at substantially lower computational cost, enabling real-time deployment on standard CPUs.

Matthew J. Colbrook, Igor Mezić, Alexei Stepanenko

DAMTP, University of Cambridge, Cambridge, CB3 0WA, UK. · University of California, Santa Barbara, CA 93106, USA.

Koopman Operator Learning Spectral Methods Statistical Learning Theory Chaotic Time-Series Prediction