Related papers: Chaos Meets Attention: Transformers for Large-Scal…
Forecasting chaotic systems is a cornerstone challenge in many scientific fields, complicated by the exponential amplification of even infinitesimal prediction errors. Modern machine learning approaches often falter due to two opposing…
The applicability of machine learning for predicting chaotic dynamics relies heavily upon the data used in the training stage. Chaotic time series obtained by numerically solving ordinary differential equations embed a complicated noise of…
We develop a powerful and general method to provide rigorous and accurate upper and lower bounds for Lyapunov exponents of stochastic flows. Our approach is based on computer-assisted tools, the adjoint method and established results on the…
Multi-agent learning is intrinsically harder, more unstable and unpredictable than single agent optimization. For this reason, numerous specialized heuristics and techniques have been designed towards the goal of achieving convergence to…
Brains process information through the collective dynamics of large neural networks. Collective chaos was suggested to underlie the complex ongoing dynamics observed in cerebral cortical circuits and determine the impact and processing of…
Reliable prediction of large chaotic sytems in the short to middle time range is of interest in a number of fields, including climate, ecology, seismology, and economics. In this paper, results from chaos theory, and statistical theory are…
Deep neural network models have shown a great potential in accelerating the simulation of fluid dynamic systems. Once trained, these models can make inference within seconds, thus can be extremely efficient. However, they suffer from a…
Chaos is an inherently dynamical phenomenon traditionally studied for trajectories that are either permanently erratic or transiently influenced by permanently erratic ones lying on a set of measure zero. The latter gives rise to the final…
A deterministic multi-scale dynamical system is introduced and discussed as prototype model for relative dispersion in stationary, homogeneous and isotropic turbulence. Unlike stochastic diffusion models, here trajectory transport and…
Large-scale foundation models for scientific machine learning adapt to physical settings unseen during training, such as zero-shot transfer between turbulent scales. This phenomenon, in-context learning, challenges conventional…
An extreme event is a sudden and violent change in the state of a nonlinear system. In fluid dynamics, extreme events can have adverse effects on the system's optimal design and operability, which calls for accurate methods for their…
The predictability of weather and climate is strongly state-dependent: special and extremely relevant atmospheric states like blockings are associated with anomalous instability. Indeed, typically, the instability of a chaotic dynamical…
We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in…
Designing continuous trajectories whose time-averaged occupancy provably matches a prescribed spatial density (the \emph{ergodic coverage} problem) is central to UAV-assisted data collection and sensing, robotic exploration, and mobile…
In many real world chaotic systems, the interest is typically in determining when the system will behave in an extreme manner. Flooding and drought, extreme heatwaves, large earthquakes, and large drops in the stock market are examples of…
We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can…
Transformers are widely used deep learning architectures. Existing transformers are mostly designed for sequences (texts or time series), images or videos, and graphs. This paper proposes a novel transformer model for massive (up to a…
We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite- time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered…
With the attention mechanism, transformers achieve significant empirical successes. Despite the intuitive understanding that transformers perform relational inference over long sequences to produce desirable representations, we lack a…
It is frequently asserted that in a chaotic system two initially close points will separate at an exponential rate governed by the largest global Lyapunov exponent. Local Lyapunov exponents, however, are more directly relevant to…