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Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural…
Reward and representation learning are two long-standing challenges for learning an expanding set of robot manipulation skills from sensory observations. Given the inherent cost and scarcity of in-domain, task-specific robot data, learning…
Vision Transformers (ViTs) have shown promising performance compared with Convolutional Neural Networks (CNNs), but the training of ViTs is much harder than CNNs. In this paper, we define several metrics, including Dynamic Data Proportion…
Recent work in financial machine learning has shown the virtue of complexity: the phenomenon by which deep learning methods capable of learning highly nonlinear relationships outperform simpler approaches in financial forecasting. While…
A time-reversal invariant topological insulator can be generally defined by the effective topological field theory with a quantized \theta coefficient, which can only take values of 0 or \pi. This theory is generally valid for an…
This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of…
For a field $\ef$, the discrete topological vector spaces over $\ \ef$ are essentially of the form $\ef^{\alpha}$ where $\alpha$ is an ordinal. With additional appropriate properties, they are isomorphic to $\ef^{(\beta)}$ where $\beta$ is…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Dynamical phase transitions are defined as non-analytic points of the large deviation function of current fluctuations. We show that for boundary driven systems, many dynamical phase transitions can be identified using the geometrical…
Learning-based methods have effectively promoted the community of image compression. Meanwhile, variational autoencoder (VAE) based variable-rate approaches have recently gained much attention to avoid the usage of a set of different…
In this paper, the intuitive idea of tilt is formalised into the rigorous concept of tilt rotations. This is motivated by the high relevance that pure tilt rotations have in the analysis of balancing bodies in 3D, and their applicability to…
This article reviews the concepts and methods of variational path sampling. These methods allow computational studies of rare events in systems driven arbitrarily far from equilibrium. Based upon a statistical mechanics of trajectory space…
We show here that matrix Darboux-Toda transformation can be written as a product of a number of mappings. Each of these mappings is a symmetry of the matrix nonlinear Shrodinger system of integro-differential equations. We thus introduce a…
Transformers have revolutionized performance in Natural Language Processing and Vision, paving the way for their integration with Graph Neural Networks (GNNs). One key challenge in enhancing graph transformers is strengthening the…
We consider dynamical systems on the space of functions taking values in a free associative algebra. The system is said to be integrable if it possesses an infinite dimensional Lie algebra of commuting symmetries. In this paper we propose a…
For discrete classical Molecular dynamics (MD) obtained by the "Verlet" algorithm (VA) with the time increment $h$ there exists a shadow Hamiltonian $\tilde{H}$ with energy $\tilde{E}(h)$, for which the discrete particle positions lie on…
Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
This is a survey on natural local torus actions which arise in integrable dynamical systems, and their relations with other subjects, including: reduced integrability, local normal forms, affine structures, monodromy, global invariants,…
We study shadowing-type properties for set-valued dynamical systems. In particular, we investigate the periodic shadowing property and its relationship with expansivity and chain transitivity. We establish that for positively expansive…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…