This work is devoted to data-driven identification of discrete breathers in numerical simulations of a two-dimensional crystal lattice using locally sampled wave data. Different lattice wave datasets are considered, with data collected from regions of different shapes and sizes defined by the lattice particles in mechanical equilibrium. Specifically, in addition to regions with a regular hexagonal shape, one- and quasi-one-dimensional regions reflecting the quasi-one-dimensionality of discrete breathers in two-dimensional hexagonal crystal lattices are proposed. To improve numerical efficiency, dataset dimensionality is reduced using Principal Component Analysis, and highly accurate Support Vector Machine classifiers are trained to distinguish between linear and nonlinear wave data. The obtained classifiers, together with the sliding window method, are applied to detect localization regions in two-dimensional hexagonal crystal lattice numerical simulations. High-precision algorithms for detected localization region segmentation and localized wave directionality estimation within the detected regions are further proposed, and their performance is evaluated. The presented methods are successfully applied to detect localized waves and their collision regions, as well as their directionality, performing a numerical study of stationary and traveling two-dimensional discrete breather interactions. Qualitatively better results are obtained when considering wave-data collection regions respecting the quasi-one-dimensional nature of two-dimensional discrete breathers in the hexagonal crystal lattice model.

In this article, we explore the lifetime of localized excitations in nonlinear lattices, called breathers, when a thermalized lattice is perturbed with localized energy delivered to a single site. We develop a method to measure the time it takes for the system to approach equilibrium based on a single scalar quantity, the participation number, and deduce the value corresponding to thermal equilibrium. We observe the time to achieve thermalization as a function of the energy of the excited site. We explore a variety of different physical system models. The result is that the lifetime of breathers increases exponentially with the breather energy for all the systems. This increase becomes observable when this energy is larger than approximately ten times the local average thermal energy. These results may provide a method to detect the existence of breathers in real systems.

Structure-preserving deep learning has recently received high attention, e.g., the development of the symplecticity-preserving neural networks SympNets for learning the flow of a Hamiltonian system. The incorporation of structural properties in neural networks has been shown to produce qualitatively better long-time predictions. It is still a great challenge to obtain computationally efficient learning algorithms for high-dimensional problems. In this work, we propose dimensionality reduction with the proper symplectic decomposition (PSD) of the time-series training datasets in conjunction with SympNets . PSD was originally proposed to obtain symplectic reduced-order models of Hamiltonian systems. We demonstrate the proposed purely data-driven approach by learning nonlinear localized discrete breather (DB) solutions in a one-dimensional crystal lattice model. We find that learning the SPD-reduced Hamiltonian dynamics is not only more computationally efficient compared to learning the whole high-dimensional model, but we also recover accurate spectral results from the neural network predictions in contrast to predictions by learned non-symplectic proper orthogonal decomposition (POD) dimensionality-reduced dynamics.

Structure-preserving data-driven learning algorithms have recently received high attention, e.g., the development of the symplecticity-preserving neural networks SympNets for learning the flow of a Hamiltonian system. The preservation of structural properties by neural networks has been shown to produce qualitatively better long-time predictions. Learning the flow of high- dimensional Hamiltonian dynamics still poses a great challenge due to the increase in neural network model complexity and, thus, the significant increase in training time. In this work, we investigate dimensionality reduction techniques of training datasets of solutions to Hamiltonian dynamics, which can be well modeled in a lower-dimensional subspace. For learning the flow of such Hamiltonian dynamics with symplecticity-preserving neural networks SympNets, we propose dimensionality reduction with the proper symplectic decomposition (PSD). PSD was originally proposed to obtain symplectic reduced-order models of Hamiltonian systems. We demonstrate the proposed purely data-driven approach by learning the nonlinear localized discrete breather solutions in a one-dimensional crystal lattice model. Considering three near-optimal PSD solutions, i.e., cotangent lift, complex SVD, and dimension-reduced nonlinear programming solutions, we find that learning the SPD-reduced Hamiltonian dynamics is not only more computationally efficient compared to learning the whole high-dimensional model, but we can also obtain comparably qualitatively good long-time predictions. Specifically, the cotangent lift and nonlinear programming PSD solutions demonstrate significantly enhanced long-term prediction capabilities, outperforming the approach of learning Hamiltonian dynamics with non-symplectic proper orthogonal decomposition (POD) dimensionality reduction.

Symplecticity-preserving neural networks such as SympNets have been proposed to learn the flow of symplectic Hamiltonian dynamics and to obtain qualitatively better long-term predictions. Computationally, learning high-dimensional problems still poses a great challenge. Structure-preserving dimensionality reduction methods have been developed to improve computational efficiency, such as proper symplectic decomposition (PSD), to preserve the inherent geometric properties of the system when learning Hamiltonian dynamics. Several near-optimal PSD solutions, such as a cotangent lift solution, have also been constructed. In this work, we propose a symplecticity-preserving unconstrained parametrization of the symplectic lift matrices, such that the dimensionality reduction can be learned simultaneously with learning Hamiltonian dynamics in the dimension-reduced phase space. With this approach, we obtain more accurate numerical results, especially longterm predictions, compared to learning dimension-reduced dynamics with the previously introduced constant PSD cotangent lift solution.

In this work we propose new structure-preserving multiple time stepping methods for numerical simulation of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Proposed multiple time stepping schemes are based on symplecticity-preserving symmetric splitting methods recently developed by the authors. Originally developed explicit splitting methods do not exactly conserve total charge probability, thus, to improve charge probability conservation and to better resolve high frequency oscillations of the charge in numerical simulations with large time steps we incorporate multiple time stepping approach when solving split charge equations. Improved numerical results with multiple time stepping methods of charge transfer by mobile discrete breathers are demonstrated in a crystal lattice model example.

It has been observed in fossil tracks and experiments in the layered silicate mica muscovite the transport of charge through the cation layers sandwiched between the layers of tetrahedra-octahedra-tetrahedra. A classical model for the propagation of anharmonic vibrations along the cation chains has been proposed based on first principles and empirical functions. In that model, several propagating entities have been found as kinks or crowdions and breathers, both with or without wings, the latter for specific velocities and energies. Crowdions are equivalent to moving interstitials and transport electric charge if the moving particle is an ion, but they also imply the movement of mass, which was not observed in the experiments. Breathers, being just vibrational entities, do not transport charge. In this work, we present a semiclassical model obtained by adding a quantum particle, electron or hole to the previous model. We present the construction of the model based on the physics of the system. In particular, the strongly nonlinear vibronic interaction between the nuclei and the extra electron or hole is essential to explain the localized charge transport, which is not compatible with the adiabatic approximation. The formation of vibrational localized charge carriers breaks the lattice symmetry group in a similar fashion to the Jahn-Teller Effect, providing a new stable dynamical state. We study the properties and the coherence of the model through numerical simulations from initial conditions obtained by tail analysis and other means. We observe that although the charge spreads from an initial localization in a lattice at equilibrium, it can be confined basically to a single particle when coupled to a chaotic quasiperiodic breather. This is coherent with the observation that experiments imply that a population of charge is formed due to the decay of potassium unstable isotopes.

In this paper, we study the spectral properties of polarobreathers, that is, breathers carrying charge in a one-dimensional semiclassical model. We adapt recently developed numerical methods that preserve the charge probability at every step of time integration without using the Born–Oppenheimer approximation, which is the assumption that the electron is not at equilibrium with the atoms or ions. We develop an algorithm to obtain exact polarobreather solutions. The properties of polarobreathers, both stationary and moving ones, are deduced from the lattice and charge variable spectra in the frequency–momentum space. We consider an efficient approach to produce approximate polarobreathers with long lifespans. Their spectrum allows for the determination of the initial conditions and the necessary parameters to obtain numerically exact polarobreathers. The spectra of exact polarobreathers become extremely simple and easy to interpret. We also solve the problem that the charge frequency is not an observable, but the frequency of the charge probability certainly is an observable.

We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector field and the splitting methods based on symplectic integrators. Symplectic gradient modules of the recently proposed symplecticity-preserving neural networks SympNets are used to construct invertible locally-symplectic modules, which compositions result in volume-preserving neural networks LocSympNets. To further preserve properties of the flow of a dynamical system LocSympNets are extended to symmetric locally-symplectic neural networks SymLocSympNets, such that the inverse of SymLocSympNets is equal to the feed-forward propagation of SymLocSympNets with the negative time step, which is a general property of the flow of a dynamical system. LocSympNets and SymLocSympNets are studied numerically considering learning linear and nonlinear volume-preserving dynamics. In particular, we demonstrate learning of linear traveling wave solutions to the semi-discretized advection equation, periodic trajectories of the Euler equations of the motion of a free rigid body, and quasi-periodic solutions of the charged particle motion in an electromagnetic field. LocSympNets and SymLocSympNets can learn linear and nonlinear dynamics to a high degree of accuracy even when random noise is added to the training data. In all numerical experiments, SymLocSympNets have produced smaller errors in long-time predictions compared to the LocSympNets. When learning a single trajectory of the rigid body dynamics locally-symplectic neural networks can learn both quadratic invariants of the system with absolute relative errors below 1%. In addition, SymLocSympNets produce qualitatively good long-time predictions, when the learning of the whole system from randomly sampled data is considered. LocSympNets and SymLocSympNets can produce accurate short-time predictions of quasi-periodic solutions, which is illustrated in the example of the charged particle motion in an electromagnetic field.

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal.

In this work we propose Support Vector Machine classification algorithms to classify one-dimensional crystal lattice waves from locally sampled data. Different learning datasets of particle displacements, momenta and energy density values are considered. Efficiency of the classification algorithms is further improved by two dimensionality reduction techniques: Principal Component Analysis and Locally Linear Embedding. Robustness of classifiers is investigated and demonstrated. Developed algorithms are successfully applied to detect localized intrinsic modes in three numerical simulations considering a case of two localized stationary breather solutions, a single stationary breather solution in noisy background and two mobile breather collision.

We study nonlinear excitations propagating in a hexagonal layer which is a model for the cation layer of silicates. We consider their properties in the frequency–momentum or ω−k representation, extending the theory on pterobreathers in their moving frame for the first time to two dimensions. It can also be easily extended to three dimensions. Exact traveling waves in the ω−k representation are within resonant planes, each plane corresponding in the moving frame to a single frequency. These frequencies are integer multiples of a frequency called the fundamental frequency. A breather is within a resonant plane called the breather plane and has a single frequency in the moving frame. The intersection of the resonant planes with the phonon surfaces produce co-traveling wings with a small set of frequencies. The traveling waves obtained by perturbing the system consist of a breather and a soliton traveling together and are quasi-exact. These traveling waves can be used as seeds to obtain exact traveling waves, also formed by a breather and a soliton. The wings do exist but they are usually very small.

Temperatūras un dislokāciju blīvuma sadalījumu modelēšana Si

We describe the full two-dimensional scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated with the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two-dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^{∘} scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

A recently proposed phase-space boundary integral model for the stochastic propagation of ray densities is presented and, for the first time, explicit connections between this model and parametric uncertainties arising in the underlying physical model are derived. In particular, an asymptotic analysis for a weak noise perturbation of the propagation speed is used to derive expressions for the probability distribution of the phase-space boundary coordinates after transport along uncertain, and in general curved, ray trajectories. Furthermore, models are presented for incorporating geometric uncertainties in terms of both the location of an edge within a polygonal domain, as well as small scale geometric fluctuations giving rise to rough boundary reflections. Uncertain source terms are also considered in the form of stochastically distributed point sources and uncertain boundary data. A series of numerical experiments is then performed to illustrate these uncertainty models in two-dimensional convex polygonal domains. (c) 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)

We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^∘ scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry.

Discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures. Wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator. In this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces. Many applications require the resolution of wave energy distributions in three-dimensional domains, such as in room acoustics, underwater acoustics and for electromagnetic cavity problems. In this work we extend discrete flow mapping to three-dimensional domains by propagating wave energy densities through tetrahedral meshes. The geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration. The important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space.

We consider the approximation of the phase-space flow of a dynamical system on a triangulated surface using an approach known as Discrete Flow Mapping. Such flows are of interest throughout statistical mechanics, but the focus here is on flows arising from ray tracing approximations of linear wave equations. An orthogonal polynomial basis approximation of the phase-space density is applied in both the position and direction coordinates, in contrast with previous studies where piecewise constant functions have typically been applied for the spatial approximation. In order to improve the tractability of an orthogonal polynomial approximation in both phase-space coordinates, we propose a careful strategy for computing the propagation operator. For the favourable case of a Legendre polynomial basis we show that the integrals in the definition of the propagation operator may be evaluated analytically with respect to position and via a spectrally convergent quadrature rule for the direction coordinate. A generally applicable spectral quadrature scheme for integration with respect to both coordinates is also detailed for completeness. Finally, we provide numerical results that motivate the use of p-refinement in the orthogonal polynomial basis.

A boundary integral operator method for stochastic ray tracing in billiards was recently proposed in [1]. In particular, a phase-space boundary integral model for propagating uncertain ray or particle flows was described and shown to interpolate between deterministic and random models of the flow propagation. In this work we describe discretisation schemes for this class of boundary integral operators using piecewise constant collocation in the spatial variable and either the Nystrom method or the collocation method in the momentum variable. The simplicity of the spatial basis means that the corresponding spatial integration can be performed analytically. Convergence properties of the discretisation schemes and strategies for numerical implementation are presented and discussed.