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1. Tunneling dynamics of an oscillating universe model

   https://doi.org/10.1088/1475-7516/2022/05/007  

   Bojowald, Martin; Petersen, Pip (May 2022, Journal of Cosmology and Astroparticle Physics)

   Abstract Quasiclassical methods for non-adiabatic quantum dynamics can reveal new features of quantum effects, such as tunneling evolution, that are harder to analyze in standard treatments based on wave functions of stationary states. Here, these methods are applied to an oscillating universe model introduced recently. Our quasiclassical treatment correctly describes several expected features of tunneling states, in particular just before and after tunneling into a trapped region where a model universe may oscillate through many cycles of collapse and expansion. As a new result, the oscillating dynamics is found to be much less regular than in the classical description, revealing a succession of cycles with varying maximal volume even when the matter ingredients and their parameters do not change. 

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2. Canonical description of cosmological backreaction

   https://doi.org/10.1088/1475-7516/2021/03/083  

   Bojowald, Martin; Ding, Ding (March 2021, Journal of Cosmology and Astroparticle Physics)

   Abstract Canonical methods of quasiclassical dynamics make it possible to go beyond a strict background approximation for cosmological perturbations by including independent fields such as correlation degrees of freedom. New models are introduced and analyzed here for cosmological dynamics in the presence of quantum correlations between background and perturbations, as well as cross-correlations between different modes of a quantum field. Evolution equations for moments of a perturbation state reveal conditions required for inhomogeneity to build up out of an initial vacuum. A crucial role is played by quantum non-locality, formulated by canonical methods as an equivalent local theory with non-classical degrees of freedom given by moments of a quantum state. 

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3. The branch‐cut quantum gravity with a self‐coupling inflation scalar field: Dynamical equations

   https://doi.org/10.1002/asna.20230152  

   Weber, Fridolin; Hess, Peter O.; Bodmann, Benno; de Freitas Pacheco, José; Hadjimichef, Dimiter; Marzola, Marcelo; Naysinger, Geovane; Razeira, Moisés; Zen Vasconcellos, César A. (December 2023, Astronomische Nachrichten)

   Abstract This article focuses on the implications of the recently developed commutative formulation based on branch‐cutting cosmology, the Wheeler–DeWitt equation, and Hořava–Lifshitz quantum gravity. Assuming a mini‐superspace of variables, we explore the impact of an inflaton‐type scalar field on the dynamical equations that describe the trajectories evolution of the scale factor of the Universe, characterized by the dimensionless helix‐like function . This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the big bang and big crunch singularities. Taking the Hořava–Lifshitz action as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives, with running coupling constants denoted as , the commutative quantum gravity approach preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt–Deser–Misner formalism. We investigate both chaotic and nonchaotic inflationary scenarios, demonstrating the sensitivity of the branch‐cut Universe's dynamics to initial conditions and parameterizations of primordial matter content. The results suggest a continuous connection of Riemann surfaces, overcoming primordial singularities and exhibiting diverse evolutionary behaviors, from big crunch to moderate acceleration. 

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4. Horizon Forcing: Improving the Recurrent Forecasting of Chaotic Systems

   https://doi.org/10.1145/3718090  

   Zhuang, Yong; Almeida, Matthew; Flynn, Patrick; Ding, Wei; Islam, Shaqiful; Li, Zihan; Chen, Ping (June 2025, ACM Transactions on Intelligent Systems and Technology)

   Chaotic dynamics are ubiquitous in many real-world systems, ranging from biological and industrial processes to climate dynamics and the spread of viruses. These systems are characterized by high sensitivity to initial conditions, making it challenging to predict their future behavior confidently. In this study, we propose a novel deep-learning framework that addresses this challenge by directly exploiting the long-term compounding of local prediction errors during model training, aiming to extend the time horizon for reliable predictions of chaotic systems. Our approach observes the future trajectories of initial errors at a time horizon, modeling the evolution of the loss to that point through the use of two major components: 1) a recurrent architecture (Error Trajectory Tracing) designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on three classic chaotic systems and six real-world time series prediction tasks with chaotic characteristics. The results show that our approach outperforms the state-of-the-art methods. 

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5. Widening the Time Horizon: Predicting the Long-Term Behavior of Chaotic Systems

   https://doi.org/10.1109/ICDM54844.2022.00094  

   Zhuang, Yong; Almeida, Matthew; Ding, Wei; Flynn, Patrick D; Islam, Shafiqul; Chen, Ping (November 2022, IEEE ICDM)

   The understanding of chaotic systems is challenging not only for theoretical research but also for many important applications. Chaotic behavior is found in many nonlinear dynamical systems, such as those found in climate dynamics, weather, the stock market, and the space-time dynamics of virus spread. A reliable solution for these systems must handle their complex space-time dynamics and sensitive dependence on initial conditions. We develop a deep learning framework to push the time horizon at which reliable predictions can be made further into the future by better evaluating the consequences of local errors when modeling nonlinear systems. Our approach observes the future trajectories of initial errors at a time horizon to model the evolution of the loss to that point with two major components: 1) a recurrent architecture, Error Trajectory Tracing, that is designed to trace the trajectories of predictive errors through phase space, and 2) a training regime, Horizon Forcing, that pushes the model’s focus out to a predetermined time horizon. We validate our method on classic chaotic systems and real-world time series prediction tasks with chaotic characteristics, and show that our approach outperforms the current state-of-the-art methods. 

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