Survey on computational models in dynamical systems, including new universality concepts.
problem Understanding the relationship between computational models and dynamical systems.
method Review of recent works on Turing universality, Topological Kleene Field Theories, and dynamical bordisms.
result Introduction of new perspectives on computability through dynamical systems.
Universal online optimization for dynamic environments using uniclass prediction.
problem Online optimization in changing environments with dynamic regret.
method Reduces dynamic online optimization to uniclass prediction problem, allowing control over dynamic regret bounds.
result First paper with state-of-the-art dynamic regret guarantees for general convex cost functions.
Modified dynamical systems retain Turing universality.
problem Embedding Turing machines into dynamical systems.
method Exploring flows with adapted 1-forms and homogeneity.
result Even slight modifications can lead to Turing universality.
New maps connect universal circles to ideal sphere for hyperbolic manifolds.
problem Understanding universal circles for Anosov foliations with branching.
method Introduced a new type of Cannon--Thurston map for leftmost universal circles.
result Fundamental group acts on leftmost universal circle with pseudo-Anosov dynamics.
A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.
problem Improving accuracy of neural networks in molecular dynamics.
method Developed a universal collection of 4 smooth scalar invariants on M(3) x M(3) and evaluated their effectiveness in a PONITA neural network architecture.
result Using a universal collection of invariants significantly improves neural network accuracy.
Deep learning models converge to Gaussian dynamics with mixed structured inputs.
problem Understanding neural network dynamics with complex input distributions.
method Extended hidden manifold model to Gaussian mixtures, analyzed via SGD.
result Learning dynamics with mixed inputs converge to Gaussian behavior.
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.
We investigated the critical dynamics on the daily Taiwan stock exchange index (TSE) from 1971 to 2005, and the 5-min intraday data from 1996 to 2005. A global persistence exponent θp was defined for non-equilibrium critical phenomena \cite{Janssen,Majumdar}, and describing dynamic behavior in an economic index \c…
Paper models Bitcoin market dynamics using 1+1D field theory.
problem Understanding stylized facts in Bitcoin markets.
method Collects order-book datasets, applies KPZ-like stochastic equations.
result Predicts order book dynamics with high precision.
The Basilica Julia set is universally equivalent to other complex dynamics sets.
problem Establishing the universality of the Basilica Julia set.
method Quasiconformal equivalence and geometric finiteness.
result The Basilica Julia set is quasiconformally equivalent to other complex dynamics sets.
Universal approximation for ODENet and ResNet with a single activation function.
problem Approximating complex dynamical systems with limited vector fields.
method Examined ODENet and ResNet with vector fields composed of a single activation function and affine mapping.
result ODENet and ResNet with restricted vector fields can uniformly approximate those with general vector fields.
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.
New model analyzes dynamic correlations in stock returns.
problem Analyzing time-varying correlations in high-dimensional data.
method Dynamic factor correlation model with novel parametrization.
result Model accurately captures heterogeneous heavy-tailed distributions and dependent shocks.
Study proves deep narrow RNNs can approximate any function, with minimum width independent of data length.
problem Proving universality of deep narrow RNNs with bounded widths.
method Analyzing RNNs as dynamical systems, proving universality for deep narrow structures with specific widths.
result Minimum width for universality of deep narrow RNNs is independent of data length.
We present a relatively detailed analysis of the persistence probability distributions in financial dynamics. Compared with the auto-correlation function, the persistence probability distributions describe dynamic correlations non-local in time. Universal and non-universal behaviors of the German DAX and Shanghai Index…
Collective phenomena with universal properties have been observed in many complex systems with a large number of components. Here we present a microscopic model of the emergence of scaling behavior in such systems, where the interaction dynamics between individual components is mediated by a global variable making the …
Study confirms financial bubbles' common patterns in isolated markets.
problem Testing universal dynamics of financial bubbles in isolated markets.
method Log-Periodic Power Law Singularity (LPPLS) model analysis of two major bubble episodes.
result Tehran Stock Exchange shows clear LPPLS hallmarks, supporting bubble universality.
A new model uses Toeplitz matrices to analyze time-series data transitions.
problem Analyzing transitions in time-series data from nonautonomous systems.
method Deep Koopman-layered models with learnable Toeplitz matrices, leveraging Toeplitz matrices' universal property.
result The model demonstrates universality and generalization, outperforming existing methods.
Deep neural networks near edge of chaos show universal scaling laws.
problem Understanding the behavior of deep neural networks near critical points.
method Analogy to absorbing phase transitions in statistical mechanics, deterministic propagation dynamics, mean-field and directed percolation universality classes.
result Deep neural networks exhibit universal scaling laws near the edge of chaos.
Volatility dynamics of wavelet - filtered stock price time series is studied. Using the universal thresholding method of wavelet filtering and a principle of minimal linear autocorrelation of noise component we find that the quantitative characteristics of volatility dynamics of denoised series are noticeably different…
Gradient descent dynamics in nonconvex models explained with universality.
problem Understanding long-time behavior of nonconvex gradient descent.
method Developed a state evolution system for tracking gradient descent iterates.
result Gradient descent iterates are approximately independent of data and strongly incoherent with feature vectors.
For many externally driven complex systems neither the noisy driving force, nor the internal dynamics are a priori known. Here we focus on systems for which the time dependent activity of a large number of components can be monitored, allowing us to separate each signal into a component attributed to the external drivi…
Universal preconditioning reduces sequential prediction regret.
problem Improving sequential prediction performance.
method Convolve target sequence with orthogonal polynomial coefficients.
result First sublinear and hidden-dimension-independent regret bounds.
We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespectively of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in…
Proposes a new neural network architecture combining MLP and basis functions.
problem Function approximation and operator learning in scientific machine learning.
method Combines robust MLP inner functions with flexible basis functions outer functions.
result KKAN outperforms MLPs and KANs in function approximation and operator learning tasks.
In this paper we investigate some connections between Topological Dynamics, the theory of G-Principal Bundles, and the theory of Locally Trivial Groupoids.
Develops theory for data-driven methods in dynamical systems.
problem Lack of analysis for data-driven methods in dynamical systems.
method Establishes existence of mapping and properties of operator learning architecture.
result Novel universal approximation theorems for smoothing and forecasting.
New proof shows incremental flow models are essential for universal generation.
problem Understanding the universality of flow-based models in generating natural maps.
method Topological-dynamical argument and algebraic properties of flows.
result Incremental generation is necessary and sufficient for universal flow-based generation.
Enhances RSCNs with hybrid regularization for nonlinear dynamics.
problem Modeling nonlinear dynamic systems with uncertainties.
method Recurrent stochastic configuration networks with hybrid regularization.
result The method outperforms other models in nonlinear system identification and industrial tasks.
URT layer improves few-shot image classification across diverse domains.
problem Few-shot image classification in multi-domain settings.
method Meta-learns to dynamically re-weight and compose domain-specific representations.
result Sets new state-of-the-art on Meta-Dataset.
A new machine-learned CG model predicts protein structures efficiently.
problem Developing a universal, computationally efficient protein simulation model.
method Combining deep learning with all-atom protein simulations to create a transferable CG force field.
result The model predicts protein structures, intermediates, and fluctuations efficiently.
Improves bandit convex optimization with gradient variations.
problem Bandit Convex Optimization with Gradient Variations.
method Refined analysis of non-consecutive gradient variation.
result Improved dimension dependence for convex and strongly convex functions.
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a s…
DeepRSCN models nonlinear systems using stochastic configurations.
problem Modeling nonlinear dynamic systems efficiently.
method Incrementally constructed deep reservoir computing framework with random parameters and online weight updates.
result DeepRSCN outperforms single-layer networks in efficiency, learning, and generalization.
A phenomenological investigation of the endogenous and exogenous dynamics in the fluctuations of capital fluxes is investigated on the Chinese stock market using mean-variance analysis, fluctuation analysis and their generalizations to higher orders. Non-universal dynamics have been found not only in α exponents diff…
nUDEs use neural networks to model biology without negative values.
problem Unrealistic negative values in hybrid models of biology.
method Developed non-negative UDEs (nUDEs) with regularization techniques.
result nUDEs provide realistic solutions for biological models.
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.
New method controls linear systems with partial info and disturbances.
problem Controlling linear dynamical systems under partial observation and adversarial disturbances.
method Double Spectral Control (DSC) using two-level spectral approximation strategy.
result Matches best known regret guarantees with exponential runtime improvement.
The objective of transfer reinforcement learning is to generalize from a set of previous tasks to unseen new tasks. In this work, we focus on the transfer scenario where the dynamics among tasks are the same, but their goals differ. Although general value function (Sutton et al., 2011) has been shown to be useful for k…
Echo state networks with random weights can approximate any continuous system.
problem Approximating continuous dynamical systems using echo state networks.
method Randomly generated internal weights and a sampling procedure for activation functions.
result Echo state networks with random weights can approximate any continuous casual time-invariant operators with high probability.
Neural networks with learned biases can approximate any function.
problem Whether neural networks with only learned biases can approximate any continuous function.
method Theoretical and numerical analysis of random weights and learned biases in neural networks.
result Feedforward and recurrent neural networks with random weights can approximate any continuous function and dynamical systems.
We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuous-time deep residual networks, which can …
Machine learning (ML) and artificial intelligence (AI) algorithms are now being used to automate the discovery of physics principles and governing equations from measurement data alone. However, positing a universal physical law from data is challenging without simultaneously proposing an accompanying discrepancy model…
For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…
We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends work of Bonatti, Gogolev, Hammerlindl and Potrie to the whole isotopy class. We relate the techniques with the study of certain p…
Study bounds topological entropy of toroidal attractors.
problem Bounding entropy of toroidal attractors.
method Analyzing topological properties of toroidal sets to bound entropy.
result Entropy of toroidal attractors is bounded from below.
Numerous networks in the real world change over time, in the sense that nodes and edges enter and leave the networks. Various dynamic random graph models have been proposed to explain the macroscopic properties of these systems and to provide a foundation for statistical inferences and predictions. It is of interest to…