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Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks.…
A general algebraic approach, incorporating both invariance groups and dynamic symmetry algebras, is developed to reveal hidden coherent structures (closed complexes and configurations) in quantum many-body physics models due to symmetries…
Electronic transitions involving core-level orbitals offer a localized, atomic-site and element specific peek window into statistical systems such as molecular liquids. Although formally understood, the complex relation between structure…
In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry…
We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are…
Model order reduction provides low-complexity high-fidelity surrogate models that allow rapid and accurate solutions of parametric differential equations. The development of reduced order models for parametric \emph{nonlinear} Hamiltonian…
The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability.…
For many-particle systems defined on lattices we investigate the global structure of effective Hamiltonians and observables obtained by means of a suitable basis transformation. We study transformations which lead to effective Hamiltonians…
The problem of detecting and quantifying the presence of symmetries in datasets is useful for model selection, generative modeling, and data analysis, amongst others. While existing methods for hard-coding transformations in neural networks…
In this paper, we introduces a Pseudo-Symplectic Neural Network (PSNN) for learning general Hamiltonian systems (both separable and non-separable) from data. To address the limitations of existing structure-preserving methods (e.g.,…
Measurement data is often sampled irregularly i.e. not on equidistant time grids. This is also true for Hamiltonian systems. However, existing machine learning methods, which learn symplectic integrators, such as SympNets [20] and…
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation…
Machine learning techniques are essential tools to compute efficient, yet accurate, force fields for atomistic simulations. This approach has recently been extended to incorporate quantum computational methods, making use of variational…
The dynamics of learning in modern large AI systems is hierarchical, often characterized by abrupt, qualitative shifts akin to phase transitions observed in physical systems. While these phenomena hold promise for uncovering the mechanisms…
The neural mechanisms supporting flexible relational inferences, especially in novel situations, are a major focus of current research. In the complementary learning systems framework, pattern separation in the hippocampus allows rapid…
In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. Basic idea is to associate with the incidence matrix of the graph a Dirac…
Biological intelligence is remarkable in its ability to produce complex behaviour in many diverse situations through data efficient, generalisable and transferable skill acquisition. It is believed that learning "good" sensory…
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path…
This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian $G$-actions. Within a framework of noncommutative integrability we study integrability of $G$-invariant systems, collective motions and reduced…
In the study of bi-Hamiltonian systems (both classical and quantum) one starts with a given dynamics and looks for all alternative Hamiltonian descriptions it admits.In this paper we start with two compatible Hermitian structures (the…