Time Scaling Research Articles

Morphological evolution via surface diffusion learned by convolutional, recurrent neural networks: Extrapolation and prediction uncertainty

We use a convolutional, recurrent neural network approach to learn morphological evolution driven by surface diffusion. To this aim we first produce a training set using phase field simulations. Intentionally, we insert in such a set only relatively simple, isolated shapes. After proper data augmentation, training and validation, the model is shown to correctly predict also the evolution of previously unobserved morphologies and to have learned the correct scaling of the evolution time with size. Importantly, we quantify prediction uncertainties based on a bootstrap-aggregation procedure. The latter proved to be fundamental in pointing out high uncertainties when applying the model to more complex initial conditions (e.g., leading to the splitting of high aspect-ratio individual structures). The automatic smart augmentation of the training set and design of a hybrid simulation method are discussed.

Extreme value statistics of positive recurrent centrally biased random walks

We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.

Reduced-Scaling Double Hybrid Density Functional Theory with Rapid Basis Set Convergence through Localized Pair Natural Orbital F12.

Following earlier work [Mehta, N.; Martin, J. M. L. J. Chem. Theory Comput.2022, 10.1021/acs.jctc.2c00426] that showed how the slow basis set convergence of the double hybrid density functional theory can be obviated by the use of F12 explicit correlation in the GLPT2 step (second order Görling-Levy perturbation theory), we demonstrate here for the very large and chemically diverse GMTKN55 benchmark suite that the CPU time scaling of this step can be reduced (asymptotically linearized) using the localized pair natural orbital (PNO-L) approximation at negligible cost in accuracy.

Systematic description of COVID-19 pandemic using exact SIR solutions and Gumbel distributions

An epidemiological study is carried out in several countries analyzing the first wave of the COVID-19 pandemic using the SIR model and Gumbel distribution. The equations of the SIR model are solved exactly using the proper time as a parameter. The physical time is obtained by integration of the inverse of the infected function over proper time. Some properties of the solutions of the SIR model are studied such as time scaling and the asymmetry, which allows to obtain the basic reproduction number from the data. Approximations to the solutions of the SIR model are studied using Gumbel distributions by least squares fit or by adjusting the maximum of the infected function. Finally, the parameters of the SIR model and the Gumbel function are extracted from the death data and compared for the different countries. It is found that ten of the selected countries are very well described by the solutions of the SIR model, with a basic reproduction number between 3 and 8.

Wind farm array cable layout optimisation for complex offshore sites—A decomposition based heuristic approach

As the number of turbines in offshore wind farms increases, so does the complexity of cable routing optimisation problems. Optimising the collector network is a crucial task for developers contributing between 15% and 30% of initial investment costs. For some algorithms, increasing the number of turbines leads to unfavourable scaling of the computational time and memory required to reach optimal solutions. Heuristics offer an alternative but are likely to incur increases in total costs relative to the optimal solution since heuristic searching cannot guarantee optimality. This study proposes a novel optimisation algorithm based on the ant-colony heuristic by introducing decomposition techniques informed by the problem formulation to improve the computational performance. The new algorithm can reach near-optimal solutions and requires little computational resource. Three algorithms are compared on a set of six case studies, including mixed-integer linear programming (MILP), classical ant-colony optimisation (ACO) algorithm, and the proposed ACO with decomposition, ACOsp. Optimal solutions were found using the MILP algorithm. ACO algorithm solutions cost 0.4–7.6% more than optimal solutions, whereas ACOsp solutions cost only 0.0–1.4% more than optimal solutions. The proposed ACOsp algorithm has shown to be a robust approach for large-scale cable layout optimisation problems (>100 turbines) without requiring high-performance computing facilities.

Universal and Non-Universal Features in the Random Shear Model

The stochastic transport of particles in a disordered two-dimensional layered medium, driven by correlated y-dependent random velocity fields is usually referred to as random shear model. This model exhibits a superdiffusive behavior in the x direction ascribable to the statistical properties of the disorder advection field. By introducing layered random amplitude with a power-law discrete spectrum, the analytical expressions for the space and time velocity correlation functions, together with those of the position moments, are derived by means of two distinct averaging procedures. In the case of quenched disorder, the average is performed over an ensemble of uniformly spaced initial conditions: albeit the strong sample-to-sample fluctuations, and universality appears in the time scaling of the even moments. Such universality is exhibited in the scaling of the moments averaged over the disorder configurations. The non-universal scaling form of the no-disorder symmetric or asymmetric advection fields is also derived.

Scalable Spin Squeezing from Spontaneous Breaking of a Continuous Symmetry.

Spontaneous symmetry breaking is a property of Hamiltonian equilibrium states which, in the thermodynamic limit, retain a finite average value of an order parameter even after a field coupled to it is adiabatically turned off. In the case of quantum spin models with continuous symmetry, we show that this adiabatic process is also accompanied by the suppression of the fluctuations of the symmetry generator-namely, the collective spin component along an axis of symmetry. In systems of S=1/2 spins or qubits, the combination of the suppression of fluctuations along one direction and of the persistence of transverse magnetization leads to spin squeezing-a much sought-after property of quantum states, both for the purpose of entanglement detection as well as for metrological uses. Focusing on the case of XXZ models spontaneously breaking a U(1) [or even SU(2)] symmetry, we show that the adiabatically prepared states have nearly minimal spin uncertainty; that the minimum phase uncertainty that one can achieve with these states scales as N^{-3/4} with the number of spins N; and that this scaling is attained after an adiabatic preparation time scaling linearly with N. Our findings open the door to the adiabatic preparation of strongly spin-squeezed states in a large variety of quantum many-body devices including, e.g., optical-lattice clocks.

Persistent homology analysis of a generalized Aubry-André-Harper model

Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize phases of a generalized Aubry-Andr\'e-Harper model. The persistent entropy and mean squared lifetime of features obtained using persistent homology behave similarly to conventional measures (Shannon entropy and inverse participation ratio) and can distinguish localized, extended, and critical phases. However, we find that the persistent entropy also clearly distinguishes ordered from disordered regimes of the model. The persistent homology approach can be applied to both the energy eigenstates and the wave packet propagation dynamics.

Highly scalable maximum likelihood and conjugate Bayesian inference for ERGMs on graph sets with equivalent vertices.

The exponential family random graph modeling (ERGM) framework provides a highly flexible approach for the statistical analysis of networks (i.e., graphs). As ERGMs with dyadic dependence involve normalizing factors that are extremely costly to compute, practical strategies for ERGMs inference generally employ a variety of approximations or other workarounds. Markov Chain Monte Carlo maximum likelihood (MCMC MLE) provides a powerful tool to approximate the maximum likelihood estimator (MLE) of ERGM parameters, and is generally feasible for typical models on single networks with as many as a few thousand nodes. MCMC-based algorithms for Bayesian analysis are more expensive, and high-quality answers are challenging to obtain on large graphs. For both strategies, extension to the pooled case—in which we observe multiple networks from a common generative process—adds further computational cost, with both time and memory scaling linearly in the number of graphs. This becomes prohibitive for large networks, or cases in which large numbers of graph observations are available. Here, we exploit some basic properties of the discrete exponential families to develop an approach for ERGM inference in the pooled case that (where applicable) allows an arbitrarily large number of graph observations to be fit at no additional computational cost beyond preprocessing the data itself. Moreover, a variant of our approach can also be used to perform Bayesian inference under conjugate priors, again with no additional computational cost in the estimation phase. The latter can be employed either for single graph observations, or for observations from graph sets. As we show, the conjugate prior is easily specified, and is well-suited to applications such as regularization. Simulation studies show that the pooled method leads to estimates with good frequentist properties, and posterior estimates under the conjugate prior are well-behaved. We demonstrate the usefulness of our approach with applications to pooled analysis of brain functional connectivity networks and to replicated x-ray crystal structures of hen egg-white lysozyme.

Invasiveness of a Growth-Migration System in a Two-dimensional Percolation cluster: A Stochastic Mathematical Approach.

We developed a framework based on the software Unstructured Reaction-Diffusion Master Equation (URDME) to address tumor cells' proliferation and migration in a heterogeneous space, herein a 2D percolation cluster. A mitogenic paracrine signaling pathway is utilized phenomenologically to reveal how cells cooperate with one another. We modeled the emerging Allee effect using low seeding density culture (LSDC) assays to fit the model parameters. A Finite time scaling (FTS) function has been formulated to quantitatively analyze invasiveness of a virtual Growth-Migration (GM) system in mimicking the cancer cell growth. Through such simulation, we analyzed the GM dynamics of virtual model in mimicking the growth of BT-474 cancer cell populations in vitro in a 2D percolation cluster and calculated the successful penetration rate (SPR). By analyzing the temporal trajectories of the SPR, we could determine the critical exponents of the critical SPR scaling relation. The SPR transition point ([Formula: see text]), which is a fundamentally different from a conventional percolation transition point, is found to be negatively correlated with the invasiveness of this cancer cell. The [Formula: see text] of the three variations of the virtual GM system distinctly designated by varying paracrine-regulated Allee (PAllee) model phenotypes is 0.3408, 0.3675, and 0.4454, respectively. FTS algorithm thereon may serve as an approach to quantify invasiveness of tumor cells. Through a phenomenological paracrine model, inter-cell cooperation and mutual mitogenic boosting are enabled to elicit the Allee effect in the GM systems. The rationale behind such computationally tunable virtual mechanism can be applied to other circumstances concerning emerging processes.

Spectral form factor in a minimal bosonic model of many-body quantum chaos.

We study spectral form factor in periodically kicked bosonic chains. We consider a family of models where a Hamiltonian with the terms diagonal in the Fock space basis, including random chemical potentials and pairwise interactions, is kicked periodically by another Hamiltonian with nearest-neighbor hopping and pairing terms. We show that, for intermediate-range interactions, the random phase approximation can be used to rewrite the spectral form factor in terms of a bistochastic many-body process generated by an effective bosonic Hamiltonian. In the particle-number conserving case, i.e., when pairing terms are absent, the effective Hamiltonian has a non-Abelian SU(1,1) symmetry, resulting in universal quadratic scaling of the Thouless time with the system size, irrespective of the particle number. This is a consequence of degenerate symmetry multiplets of the subleading eigenvalue of the effective Hamiltonian and is broken by the pairing terms. In such a case, we numerically find a nontrivial systematic system-size dependence of the Thouless time, in contrast to a related recent study for kicked fermionic chains.

Random Generation of Lattice Structures with Short-Range Order

Thermodynamically driven short-range order (SRO) is known to exist in multi-component alloys and can influence their mechanical properties. Atomistic and mesoscale simulations are frequently used to study the effects of SRO, but introducing SRO into the lattice usually requires costly Monte Carlo simulations. Here, we develop a novel method and open-source code to create lattices with specified SRO parameters that can be used as inputs to simulations to study the interaction of SRO with dislocations or other deformation mechanisms. The method, order through informed swapping (OTIS), uses a statistical smart swapping procedure to create lattices with known SRO parameters from an initially random lattice. We illustrate the use of OTIS on ternary and quinary equimolar alloys to create both BCC and FCC lattices with hundreds of thousands of atoms. The computation time and size scaling are discussed. As an example application of the method, the generated SRO lattices are incorporated into a phase-field dislocation dynamics to study dislocation loop growth.

Linking the mixing times of random walks on static and dynamic random graphs

In this paper, which is a culmination of our previous research efforts, we provide a general framework for studying mixing profiles of non-backtracking random walks on dynamic random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph tends to infinity. Subject to mild general conditions, we link two mixing times: one for a static version of the random graph, the other for a class of dynamic versions of the random graph in which the edges are randomly rewired but the degrees are preserved. With the help of coupling arguments we show that the link is provided by the probability that the random walk has not yet stepped along a previously rewired edge.To demonstrate the utility of our framework, we rederive our earlier results on mixing profiles for global edge rewiring under weaker assumptions, and extend these results to an entire class of rewiring dynamics parametrised by the range of the rewiring relative to the position of the random walk. Along the way we establish that all the graph dynamics in this class exhibit the trichotomy we found earlier, namely, no cut-off, one-sided cut-off or two-sided cut-off.For interpolations between global edge rewiring, the only Markovian graph dynamics considered here, and local edge rewiring (i.e., only those edges that are incident to the random walk can be rewired), we show that the trichotomy splits further into a hexachotomy, namely, three different mixing profiles with no cut-off, two with one-sided cutoff, and one with two-sided cut-off. Proofs are built on a new and flexible coupling scheme, in combination with sharp estimates on the degrees encountered by the random walk in the static and the dynamic version of the random graph. Some of these estimates require sharp control on possible short-cuts in the graph between the edges that are traversed by the random walk.

“Second-Order Primal” + “First-Order Dual” Dynamical Systems With Time Scaling for Linear Equality Constrained Convex Optimization Problems

Second-order dynamical systems are important tools for solving optimization problems, and most of the existing works in this field have focused on unconstrained optimization problems. In this article, we propose an inertial primal–dual dynamical system with constant viscous damping and time scaling for the linear equality constrained convex optimization problem, which consists of a second-order ordinary differential equation (ODE) for the primal variable and a first-order ODE for the dual variable. When the scaling satisfies certain conditions, we prove its convergence property without assuming strong convexity. Even the convergence rate can become exponential when the scaling grows exponentially. We also show that the obtained convergence property of the dynamical system is preserved under a small perturbation.

Numerical modeling and optimization of hundred-watt-level 2.8 μm and 1.6 μm cascaded heavily-erbium-doped fluoride fiber amplifiers

We propose a continuous-wave dual-seed cascaded heavily erbium-doped fluoride fiber amplifier scheme with a 981 nm bi-directional pump configuration for hundred-watt-level power scaling for the first time. When the powers of the 2.8 μm and 1.6 μm seed lasers and the total pump light are 5 W, 20 W, and 300 W, respectively, the 2.8 μm laser theoretically achieves an output power of ∼100 W. The amplifier scheme can also reduce the fiber temperature. Additionally, the results show that there is no excited state absorption at ∼1.675 μm, and thus injecting a 1.675 μm laser to the amplifier is not conducive to the power scaling.

The Hausdorff Dimension and Capillary Imbibition

The time scaling exponent for the analytical expression of capillary rise ℓ∼tδ for several theoretical fractal curves is derived. It is established that the actual distance of fluid travel in self-avoiding fractals at the first stage of imbibition is in the Washburn regime, whereas at the second stage it is associated with the Hausdorff dimension dH. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals F governed by the geodesic dimension dg=dH/dℓ, where dℓ is the chemical dimension of F. The imbibition measured by the chemical distance ℓg is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustrative examples to discuss some physical implications are presented.

New insights into spatio-temporal dynamics of irradiation defects rafting

The self-organization of irradiation defects has continuously attracted numerous experimental and theoretical attentions, due to its influence on the service performance of irradiated materials. This work developed the discrete dislocation dynamics method by considering the thermal fluctuations, in order to statistically clarify the contribution of elastic interactions and thermal fluctuations to the temporal and spatial dynamics of raft formation. On temporal scales, this work for the first time demonstrates the existence of power-law scaling of raft formation time, and its insensitivity to the thermal fluctuations, and loop size distribution. The power-law exponent is consistent with many other self-organized critical phenomena, such as earthquakes, sandpile sliding, and dislocation avalanches, etc. On spatial scales, the deviation between the normal direction of the raft habit plane and the Burgers vector of the irradiation-induced interstitial loops was carefully analyzed, which agrees well with the experimental results.

Finite Element Simplifications and Simulation Reliability in Single Point Incremental Forming.

Single point incremental forming (SPIF) is one of the most promising technologies for the manufacturing of sheet metal prototypes and parts in small quantities. Similar to other forming processes, the design of the SPIF process is a demanding task. Nowadays, the design process is usually performed using numerical simulations and virtual models. The modelling of the SPIF process faces several challenges, including extremely long computational times caused by long tool paths and the complexity of the problem. Path determination is also a demanding task. This paper presents a finite element (FE) analysis of an incrementally formed truncated pyramid compared to experimental validation. Focus was placed on a possible simplification of the FE process modelling and its impact on the reliability of the results obtained, especially on the geometric accuracy of the part and bottom pillowing effect. The FE modelling of SPIF process was performed with the software ABAQUS, while the experiment was performed on a conventional milling machine. Low-carbon steel DC04 was used. The results confirm that by implementing mass scaling and/or time scaling, the required calculation time can be significantly reduced without substantially affecting the pillowing accuracy. An innovative artificial neural network (ANN) approach was selected to find the optimal values of mesh size and mass scaling in term of minimal bottom pillowing error. However, care should be taken when increasing the element size, as it has a significant impact on the pillow effect at the bottom of the formed part. In the range of selected mass scaling and element size, the smallest geometrical error regarding the experimental part was obtained by mass scaling of 19.01 and tool velocity of 16.49 m/s at the mesh size of 1 × 1 mm. The obtained results enable significant reduction of the computational time and can be applied in the future for other incrementally formed shapes as well.

Subdiffusive Thouless time scaling in the Anderson model on random regular graphs

The scaling of the Thouless time with system size is of fundamental importance to characterize dynamical properties in quantum systems. In this work, we study the scaling of the Thouless time in the Anderson model on random regular graphs with on-site disorder. We determine the Thouless time from two main quantities: the spectral form factor and the power spectrum. Both quantities probe the long-range spectral correlations in the system and allow us to determine the Thouless time as the time scale after which the system is well described by random matrix theory. We find that the scaling of the Thouless time is consistent with the existence of a sub-diffusive regime anticipating the localized phase. Furthermore, to reduce finite-size effects, we break energy conservation by introducing a Floquet version of the model and show that it hosts a similar sub-diffusive regime.

On computational capabilities of Ising machines based on nonlinear oscillators

Dynamical Ising machines are actively investigated from the perspective of finding efficient heuristics for NP-hard optimization problems. However, the existing data demonstrate super-polynomial scaling of the running time with the system size, which is incompatible with large NP-hard problems. We show that oscillator networks implementing the Kuramoto model of synchronization are capable of demonstrating polynomial scaling. The dynamics of these networks is related to the semidefinite programming relaxation of the Ising model ground state problem. Consequently, such networks, as we numerically demonstrate, are capable of producing the best possible approximation in polynomial time. To reach such performance, however, the reconstruction of the binary Ising state (rounding) must be specially addressed. We demonstrate that commonly implemented forced collapse to a close-to-Ising state may diminish the computational capabilities up to their complete invalidation. Therefore, consistent treatment of rounding may cardinally improve various operation metrics of already existing and upcoming dynamical Ising machines.