Metastability and paramagnetism in superconducting mesoscopic disks

A projected order parameter is used to calculate not only local minima of the Ginzburg-Landau energy functional but also saddle points or energy barriers responsible for the metastabilities observed in superconducting mesoscopic disks [A. K. Geim et al., Nature (London) 396, 144 (1998)]. We calculate the local minima magnetization and find the energetic instability points between vortex configurations with different vorticity. We also find that, for any vorticity, the supercurrent can reverse its flow direction on decreasing the magnetic field before one vortex can escape.     

Organization and energy properties of metastable states for the random-field Ising model

Random-field Ising model (RFIM) systems are characterized by a large number of metastable states corresponding to local minima of the system energy with respect to single spin flip. We classified the minima in a hierarchical way based on the possibility of a given state to escape from a basin of mutually reachable states. We investigate the energy properties of the metastable states in relation to the basin they belong to: states of particularly high energy, obtained by fast-quenching randomly initial spin configurations, tend to have access to a complex structure of correlated basins, opposite to what is found for low-energy states. The purpose of this paper is to investigate the connection between the properties of the basin oriented graph and the energy of the corresponding states.     

An evolutional approach to the numerical construction of polycrystalline structures using the Voronoi tessellation

We developed a method to numerically construct a polycrystalline structure with a specified grain-size distribution by utilizing the genetic algorithm, which is suitable for global optimization of complex parametric space, and the Voronoi tessellation. The computational merits of our new method compared with the conventional optimization method are: (a) it is able to escape more easily from local minima of the penalty function; and (b) it is suitable for parallel processing. The method efficiently creates input data for numerical simulations of microstructural evolution such as recrystallization, grain growth, deformation, and fracture.     

Nonlinear Time Series Forecast Using Radial Basis Function Neural Networks

In the research of using Radial Basis Function Neural Network (RBF NN) forecasting nonlinear time series, we investigate how the different clusterings affect the process of learning and forecasting. We find that k-means clustering is very suitable. In order to increase the precision we introduce a nonlinear feedback term to escape from the local minima of energy, then we use the model to forecast the nonlinear time series which are produced by Mackey-Glass equation and stocks. By selecting the k-means clustering and the suitable feedback term, much better forecasting results are obtained.     

Speeding up Evolutionary Search by Small Fitness Fluctuations

We consider a fixed size population that undergoes an evolutionary adaptation in the weak mutation rate limit, which we model as a biased Langevin process in the genotype space. We show analytically and numerically that, if the fitness landscape has a small highly epistatic (rough) and time-varying component, then the population genotype exhibits a high effective diffusion in the genotype space and is able to escape local fitness minima with a large probability. We argue that our principal finding that even very small time-dependent fluctuations of fitness can substantially speed up evolution is valid for a wide class of models.     

Lens Design: Global Optimization with Escape Function

The technique of using the ‘escape function’ for global optimization in lens design is described. This includes how to identify two solutions as independent; the threshold value for this criterion can be chosen to determine how to explore local solutions—rough or fine. Choice of appropriate values for two parameters in the escape function is most important, since this will affect the efficiency of the automatic global optimization process. There are two problems, i.e. giving default values at the beginning of the design, and determining a default rule to change them when the escape is unsuccessful. The latter was solved by assuming a ‘saddle path’ as the best route to escape. An exact solution for the former was not found, but a hint for finding a second best solution is shown based on the statistical study of local minima.     

FAST EVOLVING MULTI-LAYER PERCEPTRONS FOR NOISY CHAOTIC TIME SERIES MODELING AND PREDICTIONS

A fast evolutionary programming (FEP) is proposed to train multi-layer perceptrons (MLP) for noisy chaotic time series modeling and predictions. This FEP, which uses a Cauchy mutation operator that results in a significantly faster convergence to the optimal solution, can help MLP to escape from local minima. A comparison against back-propagation-trained networks was performed. Numerical experimental results show that the FEP can help MLP better capturing dynamics from noisy chaotic time series than the back-propagation algorithm and produce a more consistently modeling and prediction.     

Chaotic Dynamics in Weight Space of Neural Networks

When a special nonlinear self-feedback term is introduced into the dynamical equation of the backpropagation training algorithm for networks, the dynamics in weight space of networks can become chaotic. Chaotic dynamics of the system can help it escape from the most commonplace local minima of the energy. Simulation on the XOR problem and the prediction of chaotic time series have shown that the proposed chaotic training algorithm can converge to the global minimum or its approximate solutions efficiently and dramatically faster than the original backpropagation training algorithm.     

Adaptive gradient-based source and mask co-optimization with process awareness

We develop a source and mask co-optimization framework incorporating the minimization of edge placement error(EPE) and process variability band(PV Band) into the cost function to compensate simultaneously for the image distortion and the increasingly pronounced lithographic process conditions. Explicit differentiable functions of the EPE and the PV Band are presented, and adaptive gradient methods are applied to break symmetry to escape suboptimal local minima. Dependence on the initial mask conditions is also investigated. Simulation results demonstrate the efficacy of the proposed source and mask optimization approach in pattern fidelity improvement, process robustness enhancement, and almost unaffected performance with random initial masks.     

Nonlinear Time Series Prediction Using Chaotic Neural Networks

A nonlinear feedback term is introduced into the evaluation equation of weights of the backpropagation algorithm for neural network,the network becomes a chaotic one.For the purpose of that we can investigate how the different feedback terms affect the process of learning and forecasting,we use the model to forecast the nonlinear time series which is produced by Makey-Glass equation.By selecting the suitable feedback term,the system can escape from the local minima and converge to the global minimum or its approximate solutions,and the forecasting results are better than those of backpropagation algorithm.     

Transport in randomly-fluctuating spatially-periodic potentials

The motion of overdamped particles in a one-dimensional spatially-periodic potential is considered. The potential is also randomly-fluctuating in time, due to multiplicative colored noise terms, and has a deterministic tilt. Numerical simulations show two distinct parameter regimes, corresponding to free-running near-deterministic particles, and particles which are trapped in local minima of the potential with intermittent escape flights. Perturbation and asymptotic methods are developed to understand the drift velocity and diffusion coefficient in each parameter regime.     

On the theory of superfluidity

We investigate the properties of dispersion spectra of one-dimensional periodic Bose systems with repulsive interparticle interactions. These systems with sufficient large interactions can support metastable supercurrent states, which correspond to the local minima of the dispersion spectra at non-zero momenta. The existence of local minima in the spectra and the energy barriers, which separate the minima, can be explained in terms of Bose exchange symmetry. We extend our study to the case of higher dimensional Bose systems. We suggest that superfluidity could be understood as a Bose exchange effect.     

Networks of local minima in optical system optimization

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the local minima form a network in which all nodes are connected. Each link between two neighboring minima contains a saddle point with a Morse index of 1. For a simple global optimization search (the symmetric Cooke triplet), the network of the corresponding set of local minima is presented.     

Geometric approach to the dynamic glass transition

We numerically study the potential energy landscape of a fragile glassy system and find that the dynamic crossover corresponding to the glass transition is actually the effect of an underlying geometric transition caused by the vanishing of the instability index of saddle points of the potential energy. Furthermore, we show that the potential energy barriers connecting local glassy minima increase with decreasing energy of the minima, and we relate this behavior to the fragility of the system. Finally, we analyze the real space structure of activated processes by studying the distribution of particle displacements for local minima connected by simple saddles.     

Nonlinear Time Series Forecast Using Radial Basis Function Neural Networks

In the research of using Radial Basis Function Neural Network (RBF NN) forecasting nonlinear timeseries, we investigate how the different clusterings affect the process of learning and forecasting. We find that k-meansclustering is very suitable. In order to increase the precision we introduce a nonlinear feedback term to escape from thelocal minima of energy, then we use the model to forecast the nonlinear time series which are produced by Mackey-Glassequation and stocks. By selecting the k-means clustering and the suitable feedback term, much better forecasting resultsare obtained.     

Statistics of deep energy states in Coulomb glasses

We study the statistics of local energy minima in the configuration space and the energy relaxation due to activated hopping in a system of interacting electrons in a random environment. The distribution of the local minima is exponential, which is in agreement with extreme value statistics considerations. The relaxation of the system energy shows logarithmic time dependence reflecting the ultrametric structure of the system.     

A theoretical approach to the decay of unstable states providing a correct description both of short and long time behaviour

The time evolution of a delta-like distribution initially at the bottom of a well in a potential field having two minima is described in a complete way including both rearrangement and escape processes.     

Gels and glasses in a single system: evidence for an intricate free-energy landscape of glassy materials

In the free-energy landscape picture of glassy systems, their slow dynamics is due to a complicated free-energy landscape with many local minima. We show that for a colloidal glassy material multiple paths can be taken through the free-energy landscape. The evolution of the nonergodicity parameter shows two distinct master curves that we identify as gels and glasses. We show that for a range of colloid concentrations, the transition to nonergodicity can occur in either direction (gel or glass), accompanied by "hesitations" between the two. Thus, colloidal gels and glasses are merely global free-energy minima in the same free-energy landscape, and the paths leading to these minima can be complicated.     

Correlation length of quasiperiodic vortex lattices

We investigated vortex-lattice dynamics in superconducting Nb thin films with different quasiperiodic arrays of magnetic pinning centers. The mixed-state magnetoresistance exhibits minima for well-defined applied fields, related to matching effects between the vortex lattice and those arrays. The results show that critical matching can originate at a local scale. For fractal arrays, the vortex-lattice correlation length is longer and the minima are deeper, close to those of periodic arrays.