Topic Editors

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA
ATOC/CIRES, University of Colorado, Boulder, CO 80303, USA
Department of Hydrology and Atmospheric Sciences, 1133 E. James E. Rogers Way, The University of Arizona, Tucson, AZ 85721, USA

Revisiting Butterfly Effect, Multiscale Dynamics, and Predictability Using Ai-Enhanced Modeling Framework (AEMF) and Chaos Theory

Abstract submission deadline
31 December 2024
Manuscript submission deadline
31 July 2025
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Topic Information

Dear Colleagues,

Strides in understanding the butterfly effect, chaos theory, and multiscale dynamics using theoretical chaotic models, empirical analysis, and numerical weather prediction models necessitate reevaluation of the conventionally accepted two-week weather forecast horizon (Shen et al., 2023a). To date, the capabilities of AI-driven models in improving both short-term and extended-range weather forecasts have also been impressive. This Topic invites papers that re-explore the butterfly effect and the limit of predictability, with the goal of revisiting predictability thresholds at various scales using differential equation-driven and AI-powered systems.

While recent reassessments confirm the finite predictability of chaotic systems, they also highlight factors such as multiscale interactions that can enhance forecast accuracy.

Butterfly Effect, Chaos, and Sensitivity: Lorenz's rediscovery of the butterfly effect in 1963 brought attention to the sensitive dependence on initial conditions (SDIC), a principle central to defining Chaos. Contemporary research expands on SDIC by examining various facets of the butterfly effect, including its metaphorical portrayal of a butterfly's wings creating a tornado, as well as the significance of small-scale processes in contributing to finite predictability (Shen et al., 2022a; Pielke et al. 2024).

Lorenz Models and the Scope of Predictability: Lorenz's work from the 1960s until 2008 exposed the inherent finite predictability of the atmosphere and laid the groundwork for the exploration of chaos, multistability, and (almost) intransitivity in meteorology and climate (Shen et al., 2023b). Although Lorenz's models suggest finite predictability, they do not establish an upper limit.

Multistability and Multiscale Predictability: Advancements from Lorenz's original research illustrate the coexistence of chaotic and orderly states, and the transitions between them. These findings challenge the notion of absolute atmospheric chaos by uncovering patterns and cycles, thus shedding light on the multifaceted nature of predictability and the critical role of feedback mechanisms within multiscale dynamics (Shen et al., 2022b).

AI Innovations and Their Forecasting Impact: The success of vision transformer technology in numerical weather prediction (Bi et al. 2023) has underlined the transformative potential of such technologies, also exemplified by comprehensive models such as ChatGPT and Gemini.

Integrating Equation and AI-Based Multiscale Systems: A synergistic approach is paramount. Construction of the similarity matrix using the self-attention mechanism (Vaswani et al., 2017) echoes traditional methods such as Singular Value Decomposition (SVD) and Principal Component Analysis (PCA; Cui and Shen 2021), which are employed in creating covariance matrices for discerning key scale interactions for enhanced predictability. Recognizing Lorenz's seminal contributions to SVD and PCA, as well as the integration of mode-mode interactions in his 1969 model (Shen et al. 2022c), we emphasized the impact of multiscale dynamics in both differential equation-driven and AI-powered systems.

Objectives and Hypotheses

Our goal is to deepen insights into the butterfly effect and multiscale interactions, unraveling the ways in which these interactions affect predictability in both traditional equation-based and modern AI-infused models. Suggested topics include, but are not limited to:

  • Optimizing Numerical Methods Using Multiple Time Steps: In equation-based frameworks, multiple time-step methods such as Adams-Bashforth have been used to solve initial value problems. Promising outcomes based on AI models containing extended sequences suggest that adding temporal steps could potentially enhance predictability within these equation-based models;
  • Evaluating a Reanalysis Datasets' Impact on AI Model Development and Predictive Precision: Lorenz's investigations indicated that forecast accuracy hinges on both mathematical formulations and initial input data. This principle can be empirically assessed in AI models by evaluating how forecast accuracy varies when a specific AI model is initialized using different reanalysis datasets. Comparative analyses of forecasts from AI models, each trained and then tested using separate datasets, could shed light on the inherent predictability of real-world weather phenomena;
  • Integrating AI With Ensemble Forecasting: Ensemble forecasting, which runs multiple simulations containing slight variations, could be significantly improved by employing AI algorithms that better manage and interpret the vast amounts of generated data, leading to more accurate predictions;
  • Quantifying the Butterfly Effect and Chaos Using Lyapunov Exponents: Computing Lyapunov Exponents and evaluating ensemble model runs, such as analyzing variances, can help assess the degree of sensitivity to initial conditions within models;
  • Enhancing Detail Based on AI-Enhanced Downscaling: While current AI models show promise, their resolution is often limited by the coarse resolution of global model data, thereby missing finer meteorological details. Utilizing AI for downscaling, in conjunction with high-resolution regional models, may offer a cost-effective method for overcoming this limitation;
  • Integrating Non-Forecast Variables to Broaden the Scope of AI Models: Many AI models are designed using a limited set of forecast variables, often omitting significant variables such as precipitation. By developing diagnostic algorithms that leverage existing forecast data in order to compute these missing variables, the variables can be incorporated into AI models, thus improving their comprehensiveness;
  • Uncovering Multiscale Processes Using PCA and SVD: Performing singular value analyses of the query, key, value matrices, and eigenvalue analyses on similarity matrices within AI models can reveal critical multiscale interactions;
  • Conceptual Models for Understanding Predictability: Classical and generalized Lorenz models offer insights that a nonlinear feedback loop and its extensions, akin to a complex network of interconnected highways and byways, have the capacity to enhance or suppress the flow and impact of information and noise at various scales. This conceptual model could illustrate factors that strengthen or undermine the predictability of weather and climate;
  • Revisiting the Bounds of Predictability: As scientists leverage the growing power of computational resources to develop more advanced AI-driven models, the expectation is that these models will continually improve their efficacy, thus advancing their ability to accurately forecast meteorological and climatic events. This raises a profound and complicated question: Is the predictability of weather and climate inherently limited, despite the escalating proficiency of AI-enabled tools? A more challenging question is the following: Will a Laplace Demon eventually be created?
  • Investigating AI's Role in Extreme Weather Event Prediction: Exploring how AI models can be trained to recognize and predict extreme weather patterns, which are often outliers in data sets and which present significant challenges to traditional prediction models, is important;
  • Exploring the Dependence of Climate Projection Capability on the Extent of Reanalysis Data: Present AI-based models utilize a comprehensive set of global reanalysis data, which for ECMWF Reanalysis v5 (ERA5) dates back to January 1940. While equation-based models can formulate fundamental dynamics across both weather and climate timescales, it is important to determine the maximum timescale represented by AI-driven models;
  • Exploring Hallucination and Its Connection to the Sensitive Dependence on Initial Conditions (SDIC): In the realm of AI tools, hallucination presents a major challenge, bearing resemblance to the sudden emergence of SDIC, out of what appears to be a regular solution within a period of continuous dependence on initial conditions (CDIC). The use of ensemble methods, with minor initial differences, can be instrumental in assessing the possible duration for which information remains reliable before the onset of divergence. Contemporary developments in Chaos theory indicate that the duration of intervals is contingent upon initial conditions (Shen et al., 2021). Using a variety of model ensembles may serve as an alternative method;
  • Cross-Disciplinary Approaches to Forecasting: This multidisciplinary topic invites contributions that bring together meteorology, computer science, mathematics & statistics, and other disciplines to develop comprehensive models. These models should more effectively navigate the intricacies of weather and climate systems, as well as enhance the processing of model-based digital data, imagery, and video content using generative technologies. The consortium of AEMF journals includes titles such as Atmosphere, Encyclopedia, Fractal and Fractional, Machine Learning and Knowledge Extraction, and Meteorology. The first letters of these journals’ titles create the acronym AE3M2F2, aligning with the multidisciplinary topic title: Ai-Extended Modeling Framework.


  • Bi, K.; Xie, L.; Zhang, H.; et al. Accurate medium-range global weather forecasting with 3D neural networks. Nature 2023, 619, 533–538.
  • Cui, J.; Shen, B.-W. A Kernel Principal Component Analysis of Coexisting Attractors within a Generalized Lorenz Model. Chaos Solitons Fractals 2021, 146.
  • Pielke Sr, R.; Shen, B.-W.; Zeng, X. The Butterfly Effect: Can a butterfly in Brazil cause a tornado in Texas? Weatherwise 2024, in press.
  • Shen, B.-W. A Review of Lorenz's Models from 1960 to 2008. Int. J. Bifurc. Chaos 2023, 33, 2330024.
  • Shen, B.-W., R. A. Pielke Sr., X. Zeng, and X. Zeng, 2023a: Lorenz’s View on the Predictability Limit. Encyclopedia 2023, 3, 887-899.
  • Shen, B.-W.; Pielke Sr., R. A.; Zeng, X. 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Special Issue on Multistability, Multiscale Predictability, and Sensitivity in Numerical Models. Atmosphere 2023, 14, 1279. (22 journal pages)
  • Shen, B.-W.; Pielke Sr., R. A.; Zeng, X. One Saddle Point and Two Types of Sensitivities Within the Lorenz 1963 and 1969 Models. Atmosphere 2022, 13, 753.
  • Shen, B.-W.; Pielke Sr., R. A.; Zeng, X.; Cui, J.; Faghih-Naini, S.; Paxson, W.; Kesarkar, A.; Zeng, X.; Atlas, R. The Dual Nature of Chaos and Order in the Atmosphere. Atmosphere 2022, 13, 1892. (Editor's Choice) (a long list)
  • Shen, B.-W.; Pielke Sr., R. A.; Zeng, X.; Cui, J.; Faghih-Naini, S.; Paxson, W.; Atlas, R. Three Kinds of Butterfly Effects Within Lorenz Models. Encyclopedia 2022, 2, 1250-1259.
  • Shen, B.; Pielke, R.A.; Zeng, X.; Baik, J.; Faghih-Naini, S.; Cui, J.; Atlas, R. Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model. Bull. Amer. Meteor. Soc. 2021, 102, E148–E158,
  • Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A.N.; Kaiser, Ł.; Polosukhin, I. "Attention is All you Need". Advances in Neural Information Processing Systems. 30. Curran Associates, Inc. 2017. Paper.pdf

Dr. Bo-Wen Shen
Prof. Dr. Roger A. Pielke Sr.
Prof. Dr. Xubin Zeng
Topic Editors


  • artificial intelligence (AI)
  • butterfly effect
  • Chaos theory
  • AI-based downscaling
  • ensemble forecasting
  • hallucination
  • multiscale dynamics
  • neural networks
  • Lorenz model
  • predictability
  • sensitive dependence on initial conditions
  • self-attention
  • vision transformer

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
2.9 4.6 2010 17.7 Days CHF 2400 Submit
- 3.3 2020 17.6 Days CHF 1200 Submit
- - 2021 24.6 Days CHF 1000 Submit
2.7 4.9 1999 20.8 Days CHF 2600 Submit
Fractal and Fractional
5.4 4.6 2017 18.9 Days CHF 2700 Submit
Machine Learning and Knowledge Extraction
3.9 6.3 2019 19.9 Days CHF 1800 Submit
- - 2022 28.3 Days CHF 1000 Submit is a multidiscipline platform providing preprint service that is dedicated to sharing your research from the start and empowering your research journey.

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