The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.
problem Modeling chaotic positional dynamics of stars in celestial systems.
method Discrete dynamical systems, Ricci flow, Perelman entropy, Lyapunov exponents, bifurcation analysis.
result Entropy increases exponentially, indicating challenging long-term star position prediction.
Panda predicts chaotic systems without retraining, showing emergent properties.
problem Predicting chaotic systems with small errors.
method Trained on a synthetic dataset of chaotic dynamical systems using evolutionary algorithms.
result Panda predicts unseen chaotic systems with zero-shot learning.
ERDMD discovers sparse, nonuniformly timed DMD models from chaotic attractors.
problem Discovering high-fidelity, nonuniformly timed DMD models from chaotic data.
method Entropic regression for nonlinear information flow detection, combined with multi-step DMD.
result ERDMD produces highly efficient and robust models with minimal complexity.
New Finsler flow on 2-torus has chaotic dynamics.
problem Constructing chaotic dynamics on a 2-torus.
method Using Berger and Turaev's theorem, constructing a Finsler metric.
result Found a Finsler geodesic flow with positive metric entropy.
Cohesion uses deep Koopman operators to generate long-range forecasts of chaotic dynamics.
problem Challenges in data-driven emulation of chaotic dynamics, especially long-range skill decay.
method Generative modeling with coherent priors estimated using reduced-order models.
result Superior long-range forecasting skill on chaotic systems, including climate dynamics.
Unified theory of θ-expectations derived from chaotic dynamics.
problem Non-convex stochastic control problems outside G-expectations.
method Spectral theory of transfer operators for uniformly hyperbolic flows, viscosity solutions to HJB equations.
result Affine Hessian, non-convex gradient structure of θ-expectation. The chaotic geodesic flow on a jet space is non-integrable.
problem Non-integrability of the sub-Riemannian geodesic flow on J2(R2,R). method Analysis of the Hamiltonian geodesic flow on the metabelian Carnot group structure of J2(R2,R). result The reduced Hamiltonian Hμ is non-integrable by meromorphic functions for some values of μ. The reconstruction from observations of high-dimensional chaotic dynamics such as geophysical flows is hampered by (i) the partial and noisy observations that can realistically be obtained, (ii) the need to learn from long time series of data, and (iii) the unstable nature of the dynamics. To achieve such inference fro…
Analog forecasting uses local dynamics to predict chaotic systems.
problem Theoretical connections between analog forecasting and dynamical systems are overlooked.
method Local approximations of the system's dynamics, linear regression, and estimation of analog forecasting errors.
result Analog forecasting performances are highly linked to the local Jacobian matrix of the flow map.
Sparse Kernel Flows learns dynamical systems from data.
problem Learning dynamical systems from limited data.
method Sparse Kernel Flows: trains optimal kernel from a dictionary of kernels.
result Sparse Kernel Flows can learn from 132 chaotic systems.
Templates are branched 2-manifolds with semi-flows used to model `chaotic' hyperbolic invariant sets of flows on 3-manifolds. Knotted orbits on a template correspond to those in the original flow. Birman and Williams conjectured that for any given template the number of prime factors of the knots realized would be boun…
MDNs offer a data-efficient alternative to diffusion and flow models for multimodal scientific learning.
problem Capturing multimodal conditional uncertainty in scientific inverse problems.
method Mixture Density Networks (MDNs) as explicit parametric density estimators.
result MDNs achieve superior generalization, interpretability, and sample efficiency in scientific tasks.
Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. …
Researchers use quantum chaos and RMT to analyze turbulence, revealing unique scaling laws.
problem Understanding the statistical structure and scaling laws of turbulence.
method Applied tools from quantum chaos and Random Matrix Theory to analyze turbulence datasets.
result Turbulence Gram matrices exhibit power-law scalings distinct from classical chaos and random data.
Study chaotic behavior in homeomorphism groups of countable products of spaces.
problem Investigate chaotic behavior in homeomorphism groups of countable products of various metrizable topological spaces.
method Construct numerous examples of chaotic groups of homeomorphisms of countable products of spaces.
result New chaotic groups of homeomorphisms of countable products of various metrizable topological spaces are discovered.
We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of…
New method models complex dynamics using a base variable.
problem Modeling complex high-frequency dynamics from time series.
method Constructing a joint model with a base variable and a target variable.
result Successfully models chaotic behavior and reconstructs statistical properties.
TSSC images enhance chaotic signal classification using ConvNets.
problem Classifying chaotic signals accurately and robustly.
method Triad State Space Construction (TSSC) for image encoding, Convolutional Neural Network (ConvNet) for classification.
result TSSC-ConvNet achieves high accuracy and robustness in chaotic signal classification.
The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first …
The use of artificial neural networks as models of chaotic dynamics has been rapidly expanding. Still, a theoretical understanding of how neural networks learn chaos is lacking. Here, we employ a geometric perspective to show that neural networks can efficiently model chaotic dynamics by becoming structurally chaotic t…
Reservoir computing predicts chaotic systems for long horizons with sparse updates.
problem Predicting chaotic systems with long horizons using limited data.
method Sparse, time-dependent data inputs into reservoir computing.
result Achieves arbitrarily long prediction horizons for chaotic systems.
Proposes NSSNNs to predict nonseparable Hamiltonian systems.
problem Predicting nonseparable Hamiltonian systems with coupled dynamics.
method Augmented symplectic time integrator to decouple position and momentum.
result Long-term, accurate, and robust predictions for large-scale Hamiltonian systems.
In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic…
Bayesian ANN method predicts chaotic systems with uncertainty.
problem Estimating chaotic dynamical systems from noisy data.
method Bayesian Artificial Neural Networks for ODE inverse problems.
result Accurate time predictions and uncertainty bounds.
The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete" k-surface, i.e. a surface of constant (extrinsic) curvature k, where k belongs to ]0…
New risk models use chaotic attractors to predict extreme events.
problem Predicting Black Swan events in financial markets.
method Combining heavy-tailed priors with chaotic dynamics (Lorenz and Rossler systems).
result Models generate volatility clustering, fat tails, and extreme events.
Deep learning models learn chaotic system dynamics from real and simulated data.
problem Training deep learning models for chaotic systems requires big data.
method Jointly train deep neural networks on real and simulated data, enforcing physical laws.
result Proposes knowledge-based deep learning (KDL) for accurate forecasting of chaotic systems.
This paper considers the ideal gas-like model of trading markets, where each individual is identified as a gas molecule that interacts with others trading in elastic or money-conservative collisions. Traditionally this model introduces different rules of random selection and exchange between pair agents. Real economic …
We propose a physics-informed Echo State Network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional lo…
Method infers causal structure from system behaviors using RKHS and kernel ε-machines.
problem Discovering causal structure in systems with varying external and measurement noise.
method Combines causal states and RKHS for efficient representation and inference of causal structure.
result Robustly estimates causal structure in high-dimensional data with varying noise.
Non-vanishing steady Euler flows and Beltrami fields found in high dimensions.
problem Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
method Using open books, proved existence of non-vanishing steady solutions to the Euler equations for vector fields in odd dimensions.
result Existence of non-vanishing steady Euler flows and Beltrami fields in high dimensions.
Due to the dynamic nature, chaotic time series are difficult predict. In conventional signal processing approaches signals are treated either in time or in space domain only. Spatio-temporal analysis of signal provides more advantages over conventional uni-dimensional approaches by harnessing the information from both …
Gradient descent with chaotic perturbations improves generalization.
problem Improving generalization of gradient descent.
method Introducing chaotic perturbations to gradient descent to achieve improved generalization.
result Gradient descent with chaotic perturbations converges to a heavy-tailed SDE, leading to improved generalization.
The geometry of oscillatory integrals on manifolds with intermediate symmetry.
problem Classification of curvature conditions in Sogge's program.
method Proposing a classification of curvature conditions.
result No manifolds satisfy the chaotic curvature condition of order 1.
A ML model accurately replicates chaotic dynamics across various parameters.
problem Replicating chaotic characteristics of non-linear dynamics using machine learning.
method A ML model trained to predict one-step-ahead states from historic states captures bifurcation diagrams and Lyapunov exponents universally.
result Variational quantum circuit outperforms classical models in reproducing long-term chaotic characteristics.
Investigates chaotic financial time series with monthly contributions and devaluation.
problem Analyzing chaotic behavior in financial processes with piecewise contributions and negative interest rates.
method Examines a financial process with monthly contributions and devaluation, showing dichotomy in behavior.
result Financial time series exhibit either periodic sequences or Cantor set of ω-limit points, with chaotic behavior at points of a Cantor attractor.
We use standard deep neural networks to classify univariate time series generated by discrete and continuous dynamical systems based on their chaotic or non-chaotic behaviour. Our approach to circumvent the lack of precise models for some of the most challenging real-life applications is to train different neural netwo…
RNNs struggle with chaotic dynamics due to exploding gradients, but we found a way to optimize training.
problem Challenging training of RNNs with chaotic dynamics due to exploding gradients.
method Relating loss gradients to Lyapunov spectrum to optimize training on chaotic data.
result RNNs with chaotic dynamics always have diverging gradients, while stable ones have bounded gradients.
This paper proves long-time accuracy of ensemble Kalman filters for chaotic and machine-learned systems.
problem Ensuring long-term accuracy of ensemble Kalman filters for complex dynamical systems.
method Established conditions for long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems.
result Ensemble Kalman filters maintain small estimation error over long time horizons for chaotic and machine-learned systems.
Apparently random financial fluctuations often exhibit varying levels of complexity, chaos. Given limited data, predictability of such time series becomes hard to infer. While efficient methods of Lyapunov exponent computation are devised, knowledge about the process driving the dynamics greatly facilitates the complex…
The integrability of the geodesic flow on the three-folds M3 admitting SL(2,R)-geometry in Thurston's sense is investigated. The main examples are the quotients MΓ3=Γ\PSL(2,R), where Γ⊂PSL(2,R) is a cofinite Fuchsian group. We show that the correspon…
FCOC framework improves financial volatility forecasting.
problem Tackles dual challenges of feature fidelity and model responsiveness in financial volatility forecasting.
method Synergizes fractal feature extraction and dynamic chaotic oscillation processing.
result Demonstrates profound and generalizable impact on S\&P 500 and DJI datasets.
A new framework reduces inconsistencies in chaotic surrogate modeling.
problem Consistency issues between probabilistic objectives and dynamical system dynamics.
method KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter.
result KAFFEE mitigates the dynamic-probabilistic consistency gap, improving reconstruction and predictive scores.
A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time…
A pairwise clustering approach is applied to the analysis of the Dow Jones index companies, in order to identify similar temporal behavior of the traded stock prices. To this end, the chaotic map clustering algorithm is used, where a map is associated to each company and the correlation coefficients of the financial ti…
ESNs with transfer learning predict long-term chaotic patterns in spatiotemporal dynamical systems.
problem Predicting long-term statistical patterns of spatiotemporally chaotic dynamical systems.
method Echo state networks (ESNs) with transfer learning.
result ESNs with transfer learning accurately predict long-term statistical properties of spatiotemporally chaotic PDEs.
Combining LETKF and RC improves chaotic system prediction from noisy, sparse data.
problem Improving chaotic system prediction from imperfect observations and models.
method Combining LETKF and RC to predict spatio-temporal chaotic systems from noisy and sparsely distributed observations.
result The proposed method using LETKF and RC outperforms LETKF in predicting chaotic systems from noisy and sparse observations.
Transfer learning improves chaotic dynamics predictions with less data.
problem Efficiently predicting chaotic dynamics with limited data.
method Transfer learning for nonlinear dynamics, optimizing transfer rate and leveraging small-scale turbulence universality.
result Significantly more accurate inference of chaotic dynamics achieved.