Multiple Minima Research Articles

Parameter Determination – Mathematical Model of Proton Exchange Member Fuel Cell (PEMFC) using Big Bang Big-Crunch (BB-BC) Optimization

This paper proposes a numerically simple physics based optimization (Met Heuristic) method to determine parameters of Proton Exchange Member Fuel cell (PEMFC) .A cell stack of series connected PEMFC to meet the required electrical power ratings is the usual arrangement .Each cell being operated at various operating temperatures provide individual terminal voltages .The terminal voltage measurements are made for n number of series connected individual cells. An objective function is constructed as sum squared error of voltages measured practically with that available by model upon cell parameter estimates. The obtained mathematical model after minimizing the objective function can be integrated for electrical simulation purpose of Electrical Vehicles and smart Grid co –simulation studies. The objective function being non –convex has multiple minima and escaping without being stuck in local minima is a challenge to any Met Heuristic method. Most of Met Heuristic methods are tuning factor dependent while the proposed method in this paper is numerically simple and practical, the proposed method is tested for cell stack of 35 cells and results are also compared with other algorithms and an electrical model circuit is obtained. Keywords: Fuell cell, Voltage efficacy, Manns model, semi empirical values, Big –Bang Big-crunch method

Direct stellarator coil design using global optimization: application to a comprehensive exploration of quasi-axisymmetric devices

Many stellarator coil design problems are plagued by multiple minima, where the locally optimal coil sets can sometimes vary substantially in performance. As a result, solving a coil design problem a single time with a local optimization algorithm is usually insufficient and better optima likely do exist. To address this problem, we propose a global optimization algorithm for the design of stellarator coils and outline how to apply box constraints to the physical positions of the coils. The algorithm has a global exploration phase that searches for interesting regions of design space and is followed by three local optimization algorithms that search in these interesting regions (a ‘global-to-local’ approach). The first local algorithm (phase I), following the globalization phase, is based on near-axis expansions and finds stellarator coils that optimize for quasisymmetry in the neighbourhood of a magnetic axis. The second local algorithm (phase II) takes these coil sets and optimizes them for nested flux surfaces and quasisymmetry on a toroidal volume. The final local algorithm (phase III) polishes these configurations for an accurate approximation of quasisymmetry. Using our global algorithm, we study the trade-off between coil length, aspect ratio, rotational transform and quality of quasi-axisymmetry. The database of stellarators, which comprises approximately 200 000 coil sets, is available online and is called QUASR, for ‘quasi-symmetric stellarator repository’.

ANÁLISIS COMPARATIVO DE METAHEURÍSTICAS DE BASE BIOLÓGICA

This article reviews the metaheuristics published in the literature, emphasizing their usefulness in solving complex optimization problems. The review highlights inspiration's relevance in the metaheuristics design, being the main classification in multiple taxonomies existing in the literature. After reviewing the state of the art, six of the most relevant metaheuristics used to solve problems of various types (engineering, logistics, economics, data science, ...) were selected. This selection of metaheuristics will be subjected to an analysis of their performance using a set of problems selected from different authors. The problems selected for this analysis include problems with a single minimum or multiple minima, different sizes in terms of dimensions, and different types of mathematical functions such as polynomial, trigonometric, or exponential. The analysis offers a discussion of which scenarios are the best for each metaheuristic, analyzing aspects such as the ability of metaheuristics to explore and escape local minima. The article concludes by summarizing which metaheuristic is best for each type of problem. Keywords: Metaheuristics, benchmark, optimization problems, biological-based metaheuristics

Attractor neural networks with double well synapses.

It is widely believed that memory storage depends on activity-dependent synaptic modifications. Classical studies of learning and memory in neural networks describe synaptic efficacy either as continuous or discrete. However, recent results suggest an intermediate scenario in which synaptic efficacy can be described by a continuous variable, but whose distribution is peaked around a small set of discrete values. Motivated by these results, we explored a model in which each synapse is described by a continuous variable that evolves in a potential with multiple minima. External inputs to the network can switch synapses from one potential well to another. Our analytical and numerical results show that this model can interpolate between models with discrete synapses which correspond to the deep potential limit, and models in which synapses evolve in a single quadratic potential. We find that the storage capacity of the network with double well synapses exhibits a power law dependence on the network size, rather than the logarithmic dependence observed in models with single well synapses. In addition, synapses with deeper potential wells lead to more robust information storage in the presence of noise. When memories are sparsely encoded, the scaling of the capacity with network size is similar to previously studied network models in the sparse coding limit.

Catastrogenesis with unstable ALPs as the origin of the NANOGrav 15 yr gravitational wave signal

In post-inflation axion-like particle (ALP) models, a stable domain wall network forms if the model's potential has multiple minima. This system must annihilate before dominating the Universe's energy density, producing ALPs and gravitational waves (a process we dub “catastrogenesis,” or “creation via annihilation”). We examine the possibility that the gravitational wave background recently reported by NANOGrav is due to catastrogenesis. For the case of ALP decay into two photons, we identify the region of ALP mass and coupling, just outside current limits, compatible with the NANOGrav signal.

Multisoliton complex systems with explicit superpotential interactions

We consider scalar field theories in dimensions with N fields which interact through a potential , which is defined in terms of an explicit superpotential W. We construct W for any N in terms of a known superpotential w for a single-scalar model, such as that for the sine-Gordon equation or the ϕ 4 model, leading to an expression for V which has multiple minima that supports solitons. Static solitons which minimize the total energy in each soliton sector appear as solutions of first-order Bogomolny equations, which have a gradient structure. These are identical in form to equations which arise in the context of synchronization phenomena in complex systems, with the space and time variables interchanged. The sine-Gordon superpotential, for example, leads to an explicit periodic superpotential W for N scalar fields, with associated Bogomolny equations that are equivalent to the well-known Kuramoto equations which describe the synchronization of identical phase oscillators on the unit circle. The known asymptotic properties of the Kuramoto system, for both positive and negative coupling constants, ensure that finite-energy solitons exist for any given set of intermediate values imposed at the origin. Besides the models derived from the sine-Gordon equation, we investigate ϕ 4 and ϕ 6 models with N scalar fields and show numerically that solitons again exist over a wide range of parameters. We also derive general properties of the elementary meson excitations of the system, in particular we show that meson-soliton bound states exist over a restricted range of mass parameters with respect to an exact solution of the ϕ 6 system for N = 3.

Mpemba effect for a Brownian particle trapped in a single well potential.

The Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piecewise linear single well potential that is devoid of any other metastable minima, we show the existence of the Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects.

Linear Stability of a Combined Convective Flow in an Annulus

Linear stability analysis of a combined convective flow in an annulus is performed in the paper. The base flow is generated by two factors: (a) different constant wall temperatures and (b) heat release as a result of a chemical reaction that takes place in the fluid. The nonlinear boundary value problem for the distribution of the base flow temperature is analyzed using bifurcation analysis. The linear stability problem is solved numerically using a collocation method. Two separate cases are considered: Case 1 (non-zero different constant wall temperatures) and Case 2 (zero wall temperatures). Numerical calculations show that the development of instability is different for Cases 1 and 2. Multiple minima on the marginal stability curves are found for Case 1 as the Prandtl number increases. Concurrence between local minima leads to the selection of the global minimum in such a way that a finite jump in the value of the wave number is observed for some values of the Prandtl number. All marginal stability curves for Case 2 have one minimum in the range of the Prandtl numbers considered. The corresponding critical values of the Grashof number decrease monotonically as the Prandtl number grows.

Do we need delocalised wavefunctions for the excited state dynamics of 1,1-difluoroethylene?

In this work, we set up a model Hamiltonian to study the excited state quantum dynamics of 1,1-difluoroethylene, a molecule that has equivalent atoms exchanged by a torsional symmetry operation leading to equivalent minima on the potential energy surface. In systems with many degrees of freedom where the minimum energy geometry is not unique, the ground state wavefunction will be delocalised among multiple minima. In this small test system, we probe the excited state dynamics considering localised (in a single minimum) and delocalised (spread over among multiple minima) wavefunctions and check whether this choice would influence the final outcome of the quantum dynamics calculations. Our molecular Hamiltonian comprises seven electronic states, including valence and Rydberg states, computed with the MS-CASPT2 method and projected onto the vibrational coordinates of the twelve normal modes of 1,1-difluoroethylene in its vibrational ground state. This Hamiltonian has been symmetrised along the torsional degree of freedom to make both minima completely equivalent and the model is supported by the excellent agreement with the experimental absorption spectrum. Quantum dynamics results show that the different initial conditions studied do not appreciably affect the excited state populations or the absorption spectrum when the dynamics is simulated assuming a delta pulse excitation.

Regularization when modeling with biased simulation data as a prior

Embedding physical knowledge in system identification increases the generalization capabilities of the identified models. For complex engineering systems, such as a process plant, the most complete and detailed quantitative description of the existing physical and structural knowledge is often provided by a simulator. We describe the procedure of fusing simulated data with measurement data via L2 regularization for models that are linear in the parameters. We characterize how the MSE minimization problem in this framework is nontrivial, and show that for certain realizations of the data there is no unique minimum of the MSE w.r.t. the regularization parameter. In these cases the MSE can even increase to larger values than both the variance and the bias, which is counter-intuitive. We show how this issue appears less frequently with more data, even though multiple minima can occur for any realization of the data. However, we show also that the Stein effect is present regardless, so that it is always possible to decrease the MSE with careful selection of the regularization parameter, i.e., information fusion may always be beneficial.

Modeling magnetic multipolar phases in density functional theory

Multipolar magnetic phases in correlated insulators represent a great challenge for density functional theory (DFT) due to the coexistence of intermingled interactions, typically spin-orbit coupling, crystal field and complex noncollinear and high-rank intersite exchange, creating a complected configurational space with multiple minima. Although the $+\mathrm{U}$ correction to DFT allows, in principle, the modeling of such magnetic ground states, its results strongly depend on the initially symmetry breaking, constraining the nature of order parameter in the converged $\mathrm{DFT}+\mathrm{U}$ solution. As a rule, $\mathrm{DFT}+\mathrm{U}$ calculations starting from a set of initial on-site magnetic moments result in a conventional dipolar order. A more sophisticated approach is clearly needed in the case of magnetic multipolar ordering, which is revealed by a null integral of the magnetization density over spheres centered on magnetic atoms, but with nonzero local contributions. Here we show how such phases can be efficiently captured using an educated constrained initialization of the on-site density matrix, which is derived from the multipolar-ordered ground state of an ab initio effective Hamiltonian. Various properties of such exotic ground states, like their one-electron spectra, become therefore accessible by all-electron $\mathrm{DFT}+\mathrm{U}$ methods. We assess the reliability of this procedure on the ferro-octupolar ground state recently predicted in ${\mathrm{Ba}}_{2}M{\mathrm{OsO}}_{6}\phantom{\rule{0.28em}{0ex}}(\mathrm{M}=\mathrm{Ca},\mathrm{Mg},\mathrm{Zn})$ [Phys. Rev. Lett. 127, 237201 (2021)].

Efficient ensemble for image-based identification of Pneumonia utilizing deep CNN and SGD with warm restarts

Childhood pneumonia, the leading cause of children mortality globally, is most commonly diagnosed based on the radiographic data, which requires radiologic interpretation of X-ray images. With recent advancements in the field of deep learning, the convolutional neural networks (CNN) have proven to be able to achieve great performance in medical image segmentation, analysis and classification tasks. However, developing and training methods utilizing CNN is still a complex and time-consuming process with several open issues — generalization, demand for large datasets, and high time complexity. The optimization objective in the training of CNN models has multiple minima, which do not necessarily generalize well and may result in poor performance. Ensemble methods are commonly employed to address the generalization issue, but they require a group of diverse models and are generally even more time-consuming. To address the issues of generalization, dataset size and time complexity, we developed an ensemble method based on stochastic gradient descent with warm restarts (SGDRE) that exploits the generalization capabilities of ensemble methods and SGD with warm restarts mechanism, which is adopted to obtain a diverse group of classifiers, necessary for ensemble, in one single training process, spending the same or less training time than a single CNN classification model. The SGDRE method has been trained on publicly available pediatric chest X-ray images dataset and evaluated using 10-fold cross-validation approach. The experimental results show a significant improvement of SGDRE over the two compared baseline methods. With an achieved test accuracy of 96.26% and AUC of 95.15%, the proposed method proved to be a very competitive classification method.

Physical characteristics of frozen hydrometeors inferred with parameter estimation

Abstract. Frozen hydrometeors are found in a huge range of shapes and sizes, with variability on much smaller scales than those of typical model grid boxes or satellite fields of view. Neither models nor in situ measurements can fully describe this variability, so assumptions have to be made in applications including atmospheric modelling and radiative transfer. In this work, parameter estimation has been used to optimise six different assumptions relevant to frozen hydrometeors in passive microwave radiative transfer. This covers cloud overlap, convective water content and particle size distribution (PSD), the shapes of large-scale snow and convective snow, and an initial exploration of the ice cloud representation (particle shape and PSD combined). These parameters were simultaneously adjusted to find the best fit between simulations from the European Centre for Medium-range Weather Forecasts (ECMWF) assimilation system and near-global microwave observations covering the frequency range 19 to 190 GHz. The choices for the cloud overlap and the convective particle shape were particularly well constrained (or identifiable), and there was even constraint on the cloud ice PSD. The practical output is a set of improved assumptions to be used in version 13.0 of the Radiative Transfer for TOVS microwave scattering package (RTTOV-SCATT), taking into account newly available particle shapes such as aggregates and hail, as well as additional PSD options. The parameter estimation explored the full parameter space using an efficient assumption of linearly additive perturbations. This helped illustrate issues such as multiple minima in the cost function, and non-Gaussian errors, that would make it hard to implement the same approach in a standard data assimilation system for weather forecasting. Nevertheless, as modelling systems grow more complex, parameter estimation is likely to be a necessary part of the development process.

Challenges for machine learning force fields in reproducing potential energy surfaces of flexible molecules.

Dynamics of flexible molecules are often determined by an interplay between local chemical bond fluctuations and conformational changes driven by long-range electrostatics and van der Waals interactions. This interplay between interactions yields complex potential-energy surfaces (PESs) with multiple minima and transition paths between them. In this work, we assess the performance of the state-of-the-art Machine Learning (ML) models, namely, sGDML, SchNet, Gaussian Approximation Potentials/Smooth Overlap of Atomic Positions (GAPs/SOAPs), and Behler-Parrinello neural networks, for reproducing such PESs, while using limited amounts of reference data. As a benchmark, we use the cis to trans thermal relaxation in an azobenzene molecule, where at least three different transition mechanisms should be considered. Although GAP/SOAP, SchNet, and sGDML models can globally achieve a chemical accuracy of 1 kcal mol-1 with fewer than 1000 training points, predictions greatly depend on the ML method used and on the local region of the PES being sampled. Within a given ML method, large differences can be found between predictions of close-to-equilibrium and transition regions, as well as for different transition mechanisms. We identify key challenges that the ML models face mainly due to the intrinsic limitations of commonly used atom-based descriptors. All in all, our results suggest switching from learning the entire PES within a single model to using multiple local models with optimized descriptors, training sets, and architectures for different parts of the complex PES.

Spiral magnetic field and bound states of vortices in noncentrosymmetric superconductors

We discuss the unconventional magnetic response and vortex states arising in noncentrosymmetric superconductors with chiral octahedral and tetrahedral ($O$ or $T$) symmetry. We microscopically derive Ginzburg-Landau free energy. It is shown that due to spin-orbit and Zeeman coupling magnetic response of the system can change very significantly with temperature. For sufficiently strong coupling this leads to a crossover from type-1 superconductivity at elevated temperature to vortex states at lower temperature. The external magnetic field decay in such superconductors does not have the simple exponential law. We show that in the London limit, magnetic field can be solved in terms of complex force-free fields $\vec{W}$, which are defined by $\nabla \times \vec{W} = \text{const} \vec{W}$. Using that we demonstrate that the magnetic field of a vortex decays in spirals. Because of such behavior of the magnetic field, the intervortex and vortex-boundary interaction becomes non-monotonic with multiple minima. This implies that vortices form bound states with other vortices, antivortices, and boundaries.

Data-driven analysis of the number of Lennard\u2013Jones types needed in a force field

Force fields used in molecular simulations contain numerical parameters, such as Lennard–Jones (LJ) parameters, which are assigned to the atoms in a molecule based on a classification of their chemical environments. The number of classes, or types, should be no more than needed to maximize agreement with experiment, as parsimony avoids overfitting and simplifies parameter optimization. However, types have historically been crafted based largely on chemical intuition, so current force fields may contain more types than needed. In this study, we seek the minimum number of LJ parameter types needed to represent the key properties of organic liquids. We find that highly competitive force field accuracy is obtained with minimalist sets of LJ types; e.g., two H types and one type apiece for C, O, and N atoms. We also find that the fitness surface has multiple minima, which can lead to local trapping of the optimizer.

Bayesian system ID: optimal management of parameter, model, and measurement uncertainty

We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. Specifically, we compare estimators of future system behavior derived from the Bayesian posterior of a learning problem to several commonly used least squares-based optimization objectives used in system ID. Our comparisons indicate that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods that include differentially constrained least squares and least squares reconstructions of discrete time steppers like dynamic mode decomposition (DMD). These properties allow it to be both more sensitive to new data and less affected by multiple minima --- overall yielding a more robust approach. Our theoretical results indicate that least squares and regularized least squares methods like dynamic mode decomposition and sparse identification of nonlinear dynamics (SINDy) can be derived from the probabilistic formulation by assuming noiseless measurements. We also analyze the computational complexity of a Gaussian filter-based approximate marginal Markov Chain Monte Carlo scheme that we use to obtain the Bayesian posterior for both linear and nonlinear problems. We then empirically demonstrate that obtaining the marginal posterior of the parameter dynamics and making predictions by extracting optimal estimators (e.g., mean, median, mode) yields orders of magnitude improvement over the aforementioned approaches. We attribute this performance to the fact that the Bayesian approach captures parameter, model, and measurement uncertainties, whereas the other methods typically neglect at least one type of uncertainty.

Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization

We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.

The Cost-Sorted Distance Method for Identifying Minima Within Firefly Optimization Results: Application to Engineering Design

Abstract This paper provides a detailed description of the cost-sorted distance (CSD) method for visually and computationally identifying objective function minima within clustered population-based optimization results. CSD requires sorting the design vector population by cost and computing Euclidean distances between each pair of designs. It may be applied in conjunction with any population-based optimization method (e.g., particle swarm, genetic algorithm, simulated annealing, ant colony, firefly), but it is naturally compatible with the firefly algorithm (FA) because FA also requires the distances between each pair of design vectors and benefits from cost-sorting the population (the computational work required for CSD is a byproduct of FA). A modified FA is presented that uses CSD to more thoroughly search near potential minima and a systematic method for tuning the algorithm to reliably identify multiple minima is documented. The tuned algorithm's efficacy is demonstrated using a class of benchmark problems and a “real world” electromechanical design problem, where the identification of attractive design alternatives can be challenging.

Statistical mechanics of interacting metabolic networks.

We cast the metabolism of interacting cells within a statistical mechanics framework considering both the actual phenotypic capacities of each cell and its interaction with its neighbors. Reaction fluxes will be the components of high-dimensional spin vectors, whose values will be constrained by the stochiometry and the energy requirements of the metabolism. Within this picture, finding the phenotypic states of the population turns out to be equivalent to searching for the equilibrium states of a disordered spin model. We provide a general solution of this problem for arbitrary metabolic networks and interactions. We apply this solution to a simplified model of metabolism and to a complex metabolic network, the central core of Escherichia coli, and demonstrate that the combination of selective pressure and interactions defines a complex phenotypic space. We also present numerical results for cells fixed in a grid. These results reproduce the qualitative picture discussed for the mean-field model. Cells may specialize in producing or consuming metabolites complementing each other, and this is described by an equilibrium phase space with multiple minima, like in a spin-glass model.