Points In Parameter Space Research Articles

Variational quantum classifiers through the lens of the Hessian.

In quantum computing, the variational quantum algorithms (VQAs) are well suited for finding optimal combinations of things in specific applications ranging from chemistry all the way to finance. The training of VQAs with gradient descent optimization algorithm has shown a good convergence. At an early stage, the simulation of variational quantum circuits on noisy intermediate-scale quantum (NISQ) devices suffers from noisy outputs. Just like classical deep learning, it also suffers from vanishing gradient problems. It is a realistic goal to study the topology of loss landscape, to visualize the curvature information and trainability of these circuits in the existence of vanishing gradients. In this paper, we calculate the Hessian and visualize the loss landscape of variational quantum classifiers at different points in parameter space. The curvature information of variational quantum classifiers (VQC) is interpreted and the loss function’s convergence is shown. It helps us better understand the behavior of variational quantum circuits to tackle optimization problems efficiently. We investigated the variational quantum classifiers via Hessian on quantum computers, starting with a simple 4-bit parity problem to gain insight into the practical behavior of Hessian, then thoroughly analyzed the behavior of Hessian’s eigenvalues on training the variational quantum classifier for the Diabetes dataset. Finally, we show how the adaptive Hessian learning rate can influence the convergence while training the variational circuits.

Topological charge distributions of an interacting two-spin system

Quantum systems are often described by parameter-dependent Hamiltonians. Points in parameter space where two levels are degenerate can carry a topological charge. Here we theoretically study an interacting two-spin system where the degeneracy points form a nodal loop or a nodal surface in the magnetic parameter space, similarly to such structures discovered in the band structure of topological semimetals. We determine the topological charge distribution along these degeneracy geometries. We show that these non-point-like degeneracy patterns can be obtained not only by fine-tuning, but they can be stabilized by spatial symmetries. Since simple spin systems such as the one studied here are ubiquitous in condensed-matter setups, we expect that our findings, and the physical consequences of these nontrivial degeneracy geometries, are testable in experiments with quantum dots, molecular magnets, and adatoms on metallic surfaces.

Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

Nonuniqueness of scattering amplitudes at special points

We point out little discussed phenomenon in elementary quantum mechanics. In one-dimensional potential scattering problems, the scattering amplitudes are not uniquely determined at special points in parameter space. We examine a few explicit examples. We also discuss the relation with the pole-skipping phenomena recently found in holographic duality. In the holographic pole-skipping, the retarded Green's functions are not uniquely determined at imaginary Matsubara frequencies. It turns out that this universality comes from the fact that the corresponding potential scattering problem has the angular momentum potential.

A model-independent analysis of b{rightarrow }smu ^{+}mu ^{-}transitions with GAMBIT \u2019s FlavBit

The search for flavour-changing neutral current effects in B-meson decays is a powerful probe of physics beyond the Standard Model. Deviations from SM behaviour are often quantified by extracting the preferred values of the Wilson coefficients of an operator product expansion. We use the FlavBit module of the GAMBIT package to perform a simultaneous global fit of the Wilson coefficients C_7, C_9, and C_{10} using a combination of all current data on b{rightarrow }smu ^{+}mu ^{-} transitions. We further extend previous analyses by accounting for the correlated theoretical uncertainties at each point in the Wilson coefficient parameter space, rather than deriving the uncertainties from a Standard Model calculation. We find that the best fit deviates from the SM value with a significance of 6.6sigma . The largest deviation is associated with a vector coupling of muons to b and s quarks.

Cellular Automata for excitable media on a Complex Network: The effect of network disorder in the collective dynamics

Complex adaptive systems can be modelled with Networks and Cellular Automata (CA). In the present work, we study the Greenberg–Hastings (GH) cellular automata running in the Watts–Strogatz (WS) network model. We are interested in finding the conditions under which the system operates near a critical point. We introduce the notion of leverage point in such a simple kind of model: a point in parameter space, at criticality, such that we can change the collective state of the system with a minimal effort. Within our proposed framework, the system’s response to changes in disorder is maximal at the leverage point. The GH CA includes a transmission coefficient r that sets the threshold size in the dynamics. We evaluate numerically the critical transmission coefficient rc as a function of the average coordination number of the network K and of the rewiring probability p, where p controls the fluctuations in the coordination number. There is an interval of values in the transmission coefficient r for which the collective state of the system depends on network disorder. This interval narrows as the average coordination number increases and only within it we can tune for criticality by changing disorder alone. Our results are relevant for systems that operate at criticality in order to increase their dynamic range or to operate under optimal information-processing conditions.

Predicting Limit Cycle Boundaries Deep Inside Parameter Space of a 2D Biochemical Nonlinear Oscillator Using Renormalization Group

Formation and study of periodic orbits in phase space in the case of nonlinear oscillators have been a topic of much interest in the recent past. In the current work, a method to go deep inside the limit cycle zone on one side of the bifurcation curve of a 2D non-Lienard biochemical oscillator has been introduced. It is discussed how such an introduction facilitates predicting the boundaries of limit cycles at various points of parameter space, nearly accurately, by the use of perturbative Renormalization Group. Sel’kov model of Glycolytic oscillator has been chosen as the base model to introduce the method.

Derivation of coarse-grained force fields for buckling-induced topological defects of liquid crystals.

Microscopic details of buckling-induced topological defects are required for molecular design of smectic liquid crystals to control buckling instability of the layers. In this study, we present a multiobjective optimization method to derive the coarse-grained (CG) force fields with sufficiently precise buckling characteristics including the molecular details for molecular dynamics (MD) simulations. We perform CGMD simulations of buckling deformation at sample points in the CG force field parameter space, from which the response surfaces of objective functions such as the scalar orientational order parameters, critical angles of layer collapse, and radial distribution functions are estimated. Since not all objective functions can be optimized simultaneously, we use a genetic algorithm to calculate the Pareto set of optimal solutions. We select the models with different molecular head-tail symmetries to study buckling deformation. The extracted CG model successfully reproduces the buckling deformation in terms of the collapse of smectic layers through the generation of dislocations with dipole disclinations. We also find that the molecular symmetry is a dominant factor to control the class of the buckling-induced dislocations.

Optimal template banks

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Vorono\"i) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

Spatial control over near-critical-point operation ensures fidelity of ParABS-mediated DNA partition

In bacteria, most low-copy-number plasmid and chromosomally encoded partition systems belong to the tripartite ParABS partition machinery. Despite the importance in genetic inheritance, the mechanisms of ParABS-mediated genome partition are not well understood. Combining theory and experiment, we provided evidence that the ParABS system—DNA partitioning in vivo via the ParA-gradient-based Brownian ratcheting—operates near a transition point in parameter space (i.e., a critical point), across which the system displays qualitatively different motile behaviors. This near-critical-point operation adapts the segregation distance of replicated plasmids to the half length of the elongating nucleoid, ensuring both cell halves to inherit one copy of the plasmids. Further, we demonstrated that the plasmid localizes the cytoplasmic ParA to buffer the partition fidelity against the large cell-to-cell fluctuations in ParA level. The spatial control over the near-critical-point operation not only ensures both sensitive adaptation and robust execution of partitioning but also sheds light on the fundamental question in cell biology: how do cells faithfully measure cellular-scale distance by only using molecular-scale interactions?

Curvature effects and radial homoclinic snaking

Abstract In this work, we revisit some general results on the dynamics of circular fronts between homogeneous states and the formation of localized structures in two dimensions (2D). We show how the bifurcation diagram of axisymmetric structures localized in radius fits within the framework of collapsed homoclinic snaking. In 2D, owing to curvature effects, the collapse of the snaking structure follows a different scaling that is determined by the so-called nucleation radius. Moreover, in the case of fronts between two symmetry-related states, the precise point in parameter space to which radial snaking collapses is not a ‘Maxwell’ point but is determined by the curvature-driven dynamics only. In this case, the snaking collapses to a ‘zero surface tension’ point. Near this point, the breaking of symmetry between the homogeneous states tilts the snaking diagram. A different scaling law is found for the collapse of the snaking curve in each case. Curvature effects on axisymmetric localized states with internal structure are also discussed, as are cellular structures separated from a homogeneous state by a circular front. While some of these results are well understood in terms of curvature-driven dynamics and front interactions, a proper mathematical description in terms of homoclinic trajectories in a radial spatial dynamics description is lacking.

The BACCO simulation project: exploiting the full power of large-scale structure for cosmology

ABSTRACT We present the BACCO project, a simulation framework specially designed to provide highly-accurate predictions for the distribution of mass, galaxies, and gas as a function of cosmological parameters. In this paper, we describe our main suite of gravity-only simulations ($L\sim 2\,$ Gpc and 43203 particles) and present various validation tests. Using a cosmology-rescaling technique, we predict the non-linear mass power spectrum over the redshift range 0 < z < 1.5 and over scales $10^{-2} \lt k/(\, h\, {\rm Mpc}^{-1}) \lt 5$ for 800 points in an eight-dimensional cosmological parameter space. For an efficient interpolation of the results, we build an emulator and compare its predictions against several widely-used methods. Over the whole range of scales considered, we expect our predictions to be accurate at the $2{{\ \rm per\ cent}}$ level for parameters in the minimal Lambda cold dark matter model and to $3{{\ \rm per\ cent}}$ when extended to dynamical dark energy and massive neutrinos. We make our emulator publicly available under http://www.dipc.org/bacco

Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability

Abstract An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.

Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations

The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold:1.To demonstrate that neural networks can be efficient aggregators of whole families of parametric solutions to physical problems, trained using data created with traditional, trusted numerical methods such as finite elements. Advantages include extremely fast evaluation of pressure and velocity at any point in physical and parameter space (asymptotically, ∼3 μs/query), and data compression (the network requires 99% less storage space compared to its own training data).2.To demonstrate that the neural networks can accurately interpolate between finite element solutions in parameter space, allowing them to be instantly queried for pressure and velocity field solutions to problems for which traditional simulations have never been performed.3.To introduce an active learning algorithm, so that during training, a finite element solver can automatically be queried to obtain additional training data in locations where the neural network's predictions are in most need of improvement, thus autonomously acquiring and efficiently distributing training data throughout parameter space. In addition to the obvious utility of Item 2, above, we demonstrate an application of the network in rapid parameter sweeping, very precisely predicting the degree of narrowing in a tube which would result in a 50% increase in end-to-end pressure difference at a given flow rate. This capability could have applications in both medical diagnosis of arterial disease, and in computer-aided design.

The semialgebraic geometry of saturated optimal designs for the Bradley–Terry model

Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a D-optimal design.

Nonlinear Charge- and Flux-Tunable Cavity Derived From an Embedded Cooper-Pair Transistor

We introduce the cavity-embedded Cooper pair transistor (cCPT), a device which behaves as a highly nonlinear microwave cavity whose resonant frequency can be tuned both by charging a gate capacitor and by threading flux through a SQUID loop. We characterize this device and find excellent agreement between theory and experiment. A key difficulty in this characterization is the presence of frequency fluctuations comparable in scale to the cavity linewidth, which deform our measured resonance circles in accordance with recent theoretical predictions [Brock et al., Phys. Rev. Applied 14, 054026 (2020)]. By measuring the power spectral density of these frequency fluctuations at carefully chosen points in parameter space, we find that they are primarily a result of the $1/f$ charge and flux noise common in solid state devices. Notably, we also observe key signatures of frequency fluctuations induced by quantum fluctuations in the cavity field via the Kerr nonlinearity.

Topological bifurcations and reconstruction of travelling waves

This paper is devoted to periodic traveling waves solving Lie–Poisson equations based on the Virasoro group. We show that the reconstruction of any such solution can be carried out exactly, regardless of the underlying Hamiltonian (which need not be quadratic), provided the wave belongs to the coadjoint orbit of a uniform profile. Equivalently, the corresponding “fluid particle motion” is integrable. Applying this result to the Camassa–Holm equation, we express the drift of particles in terms of parameters labeling periodic peakons and exhibit orbital bifurcations: points in parameter space where the drift velocity varies discontinuously, reflecting a sudden change in the topology of Virasoro orbits.

State consistence of data-driven reduced order models for parametric aeroelastic analysis

This paper investigates the state consistence of parametric data-driven reduced order models (ROMs) in a state-space form obtained by various system identification methods, including autoregressive exogenous (ARX) and subspace identification (N4SID), for aeroelastic analysis in varying flight conditions. The target flight envelop is first partitioned into discrete grid points, on each of which an aerodynamic ROM is constructed using system identification to capture the dependence of the generalized aerodynamic force on the generalized displacement of structural modes. High-fidelity aeroelastic modal perturbation simulations are used to generate the ROM training and verification data. Aerodynamic ROMs not on the grid point are obtained by interpolating those at neighboring grid points. Through a thorough analysis of the model coefficients and pole migration, it is found that only the ARX-based aerodynamic ROM preserves the state consistence, and hence, allowing direct interpolation of system matrices at the non-grid point and rapid aerodynamic ROM database development in the entire flight parameter space. In contrast, N4SID-based ROM destroys the state consistence and yields physically meaningless results when ROMs are interpolated. The origin of the difference in the state consistence caused by both methods is also discussed. The interpolated ARX aerodynamic ROMs coupled with the structural ROM for parametric aeroelastic analysis exhibit excellent agreement with the high fidelity full order model (mostly <5% relative error) and salient computational efficiency.

Asymptotic behaviour of the posterior distribution in approximate Bayesian computation

Approximate Bayesian computation (ABC) is a popular technique for approximating likelihoods and is often used in parameter estimation when the likelihood functions are analytically intractable. In the context of Hidden Markov Models (HMMs), we analyze the asymptotic behavior of the posterior distribution in ABC based Bayesian parameter estimation. In particular we show that Bernstein-von Mises type results still hold but that the resulting posterior is biased in the sense that it concentrates around a point in parameter space that differs from the true parameter value. Furthermore we obtain precise rates for the size of this bias with respect to a natural accuracy parameter of the ABC method. Finally we discuss, via a numerical example, the implications of our results for the practical implementation of ABC.

Sequential Design of Computer Experiments for the Computation of Bayesian Model Evidence

Given a set of competing models of some phenomenon together with measurement data, Bayesian model selection (BMS) is a process of finding the model that is the best candidate for being the true data-generating process. BMS relies on the computation of Bayesian model evidence, which is defined as the marginal likelihood of the measurement data (i.e., the average likelihood over a model's parameter space). In this article, we introduce a new method for computing Bayesian model evidence. Our method consists of three key elements. First, all competing model functions are emulated by Gaussian processes. Model evaluations for training the Gaussian processes are chosen one by one in a sequential manner. Second, a model-time allocation strategy decides how many model evaluations are spent on each of the competing models. Third, a sequential sampling strategy selects design points in each model's parameter space. In numerical experiments, the method shows a speed-up of more than 1,000 compared to Monte Carlo estimation. While, in lower-dimensional cases, the use of Gaussian processes alone is very effective, in higher-dimensional cases, the model-time allocation strategy and the sampling strategy become more important as they focus the effort on the right model and in the right areas of the parameter domains.