Gradient-free Optimization Method Research Articles

Equivalence of cost concentration and gradient vanishing for quantum circuits: an elementary proof in the Riemannian formulation

Abstract The optimization of quantum circuits can be hampered by a decay of average gradient amplitudes with increasing system size. When the decay is exponential, this is called the barren plateau problem. Considering explicit circuit parametrizations (in terms of rotation angles), it has been shown in Arrasmith et al (2022 Quantum Sci. Technol. 7 045015) that barren plateaus are equivalent to an exponential decay of the variance of cost-function differences. We show that the issue is particularly simple in the (parametrization-free) Riemannian formulation of such optimization problems and obtain a tighter bound for the cost-function variance. An elementary derivation shows that the single-gate variance of the cost function is strictly equal to half the variance of the Riemannian single-gate gradient, where we sample variable gates according to the uniform Haar measure. The total variances of the cost function and its gradient are then both bounded from above by the sum of single-gate variances and, conversely, bound single-gate variances from above. So, decays of gradients and cost-function variations go hand in hand, and barren plateau problems cannot be resolved by avoiding gradient-based in favor of gradient-free optimization methods.

Slime mold algorithm for topology optimization: metagratings inverse design

In this paper we discuss the use of a metaheuristic (MH) gradient-free optimization method, specifically the slime mold algorithm (SMA), combined with the topology optimization (TO) method to design metasurfaces using a spectral modal method. The motivation behind using a MH approach comes from the drawbacks associated with traditional gradient-based methods. Normally, gradient-based methods require the calculation of the electromagnetic (EM) field at certain nodes within the computation domain. However, in spectral modal methods, this is unnecessary since these methods can compute the EM response without the need for field component values. Second, optimizing metagratings often involves a multimodal objective function with multiple local minimums and gradient-based methods might struggle with finding the global optimum. So to overcome these drawbacks, we propose using a MH approach, specifically the slime mold algorithm (SMA). We apply SMA to a metasurface design, especially in the context of TO and spectral methods, which is relatively unexplored. By coupling both TO with SMA, we successfully design metagratings capable of deflecting incident waves into a desired transmission angle.

Inverse Problem of Parameter Identification for Extended Compositional Gradient Model

In this study we address simulation of initial distribution of fluid composition in hydrocarbon reservoirs, particularly in scenarios where slow vertical segregation in a low-permeability near-critical reservoir results in an incomplete relaxation to the quasi-equilibrium state. To effectively analyze these complex scenarios, we propose an extended compositional gradient model that takes into account incomplete system relaxation. Practical application of the method involves solution of the forward problem of compositional profiling, with the model parameters adjusted according to real fluid samples. We formulate the inverse problem of parameter identification for the extended compositional gradient model and solve it using gradient-free optimization methods. The adjusted parameters are specific enthalpies of fluid components and the relaxation parameter. The overarching goal of parameter identification is to closely match the predictions of the model with real data, thereby enhancing the reliability of forecasts regarding distribution of fluid composition. We believe that the results of this study will significantly influence both understanding and simulation techniques for distribution of hydrocarbon components in natural reservoirs, particularly in situations characterized by incomplete system relaxation.

Transmission–reflection-integrated multifunctional metasurfaces for multiplexed acoustic wave manipulation

Considering the extraordinary wavefront modulation properties, acoustic metasurfaces have been extensively utilized to achieve powerful wave-manipulation functionalities. The next-generation acoustic metasurfaces are urgently required to encode more information capacity and process an increasing number of signal channels in a compact device, which needs more degrees of freedom for multifunctional wavefront modulation. In this study, subwavelength monolayer transmission–reflection-integrated metasurfaces (TRIMs) are systemically designed through a gradient-free topology optimization method to simultaneously realize diverse acoustic functions, such as beam steering, focusing, splitting, and diffusion, in reflection mode and transmission mode. Both numerical and experimental results demonstrate the desired wave-manipulation performance of the metasurfaces. In addition, a dual-frequency multiplexed TRIM is also numerically achieved for exploring the integration of multiple degrees of freedom and tunable function switching, which promise many unprecedented applications in integrated medical imaging, underwater wireless telecommunications, on-chip signal processing, etc.

Comparative study on gradient-free optimization methods for inverse source-term estimation of radioactive dispersion from nuclear accidents

In this study, we rigorously assess the performance of three gradient-free optimization algorithms—Ensemble Kalman Inversion (EKI), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA)—for estimating source terms in diverse radionuclide release scenarios. Our analysis encompasses both single and multiple sources with varying radionuclide compositions, delving into the influence of decay constants and radioactivity on source estimation accuracy. Although estimating a single radionuclide from a single source exhibits outstanding results, estimating multiple radionuclides from a single source proves more arduous due to the limited information available for discerning gamma dose rates. Contrary to expectations, increasing the number of observation stations does not consistently improve the likelihood of finding accurate solutions in ill-posed inverse problems. Impressively, under our simulation settings, EKI demonstrates competitive performance in terms of convergence, accuracy, and runtime compared to PSO and GA, with GPU parallelization further bolstering computational efficiency. We explore strategies for enhancing source term estimation, including incorporating prior information, applying uncertainty removal techniques, and optimizing observation placement. Additionally, this study underscores the intricate role of relative error in determining multi-radionuclide estimation accuracy from gamma dose measurements. By employing the Gaussian plume model under steady-state conditions, our research lays the groundwork for future applications of Lagrangian dispersion models with real-time data integration. The insights gleaned from our study promise to advance environmental radioactivity monitoring and catalyze the development of cutting-edge, real-time source estimation technologies in full-scale systems.

Collaborative optimization of composite structure and fiber orientation through real-time correction of deep neural network (DNN) models with elite samples

Gradient-free optimization methods can result in significant computational costs when solving complex structural design problems for composite materials. To this end, this article presents a machine learning-based co-optimization method for composite material structure and fiber orientation. In this approach, DNN are utilized as surrogate models for the optimization problem. Equilibrium optimizer is employed to find real-time optimal solution of the DNN. Subsequently, elite samples are generated based on this optimal solution and used to update the DNN until convergence is achieved. During the post-processing stage, B-spline functions are applied to smooth the density and fiber orientation of the optimized results.

Circularly polarized printed dual port MIMO antenna with polarization diversity optimized by machine learning approach for 5G NR n77/n78 frequency band applications

In this communication, a planar dual port multiple input multiple output antenna of size 1.2λ0 × 0.6λ0 × 0.008λ0 with LHCP/RHCP features is reported for the fifth-generation new radio n77/n78 sub-6 GHz wireless applications band. The single unit of the proposed design consists of a modified L-shape rectangular radiator with Z-shape slot loaded DGS. The defected ground structure is optimized through machine learning algorithms to achieve the maximum ARBW (output) by Right Shifting (RS) and left shifting (LS) the DGS and obtaining input features. The performance metric for ANN with ADAM optimizer was found to be optimal with MSE and R2 of 0.99 and 0.82, respectively. ANNs can leverage gradient information to guide the optimization process. This enables faster convergence towards optimal solutions compared to popular GAs and PSO, which are often gradient-free optimization methods. The MIMO configuration is achieved by creating a mirror image of the single unit about the x-axis. The salient features of the proposed design are (a) Impedance bandwidth (IBW) of 3.0–4.2 GHz covering the n77/n78 band, (b) 3-dB axial ratio bandwidth (ARBW) of the 2.6–3.9 GHz (c) Port-1 is generating RHCP while Port-2 is generating LHCP, results in polarization diversity. Different diversity performance parameters (ECC < 0.005, DG ~ 9.99 dB, and MEG < 3 dB) are in the optimum range confirming the proposed configuration as a suitable design for a MIMO radiator.

Geometric Constellation Shaping for Fiber-Optic Channels via End-to-End Learning

End-to-end learning has become a popular method to optimize a constellation shape of a communication system. When the channel model is differentiable, end-to-end learning can be applied with conventional backpropagation algorithm for optimization of the shape. A variety of optimization algorithms have also been developed for end-to-end learning over a non-differentiable channel model. In this paper, we compare gradient-free optimization method based on the cubature Kalman filter, model-free optimization and backpropagation for end-to-end learning on a fiber-optic channel modeled by the split-step Fourier method. The results indicate that the gradient-free optimization algorithms provide a decent replacement to backpropagation in terms of performance at the expense of computational complexity. Furthermore, the quantization problem of finite bit resolution of the digital-to-analog and analog-to-digital converters is addressed and its impact on geometrically shaped constellations is analysed. Here, the results show that when optimizing a constellation with respect to mutual information, a minimum number of quantization levels is required to achieve shaping gain. For generalized mutual information, the gain is maintained throughout all of the considered quantization levels. Also, the results implied that the autoencoder can adapt the constellation size to the given channel conditions.

Multi-Lane Differential Variable Speed Limit Control via Deep Neural Networks Optimized by an Adaptive Evolutionary Strategy.

In advanced transportation-management systems, variable speed limits are a crucial application. Deep reinforcement learning methods have been shown to have superior performance in many applications, as they are an effective approach to learning environment dynamics for decision-making and control. However, they face two significant difficulties in traffic-control applications: reward engineering with delayed reward and brittle convergence properties with gradient descent. To address these challenges, evolutionary strategies are well suited as a class of black-box optimization techniques inspired by natural evolution. Additionally, the traditional deep reinforcement learning framework struggles to handle the delayed reward setting. This paper proposes a novel approach using covariance matrix adaptation evolution strategy (CMA-ES), a gradient-free global optimization method, to handle the task of multi-lane differential variable speed limit control. The proposed method uses a deep-learning-based method to dynamically learn optimal and distinct speed limits among lanes. The parameters of the neural network are sampled using a multivariate normal distribution, and the dependencies between the variables are represented by a covariance matrix that is optimized dynamically by CMA-ES based on the freeway's throughput. The proposed approach is tested on a freeway with simulated recurrent bottlenecks, and the experimental results show that it outperforms deep reinforcement learning-based approaches, traditional evolutionary search methods, and the no-control scenario. Our proposed method demonstrates a 23% improvement in average travel time and an average of a 4% improvement in CO, HC, and NOx emission.Furthermore, the proposed method produces explainable speed limits and has desirable generalization power.

Performance comparison of optimization methods on variational quantum algorithms

Variational quantum algorithms (VQAs) offer a promising path toward using near-term quantum hardware for applications in academic and industrial research. These algorithms aim to find approximate solutions to quantum problems by optimizing a parametrized quantum circuit using a classical optimization algorithm. A successful VQA requires fast and reliable classical optimization algorithms. Understanding and optimizing how off-the-shelf optimization methods perform in this context is important for the future of the field. In this work, we study the performance of four commonly used gradient-free optimization methods: SLSQP, COBYLA, CMA-ES, and SPSA, at finding ground-state energies of a range of small chemistry and material science problems. We test a telescoping sampling scheme (where the accuracy of the cost-function estimate provided to the optimizer is increased as the optimization converges) on all methods, demonstrating mixed results across our range of optimizers and problems chosen. We further hyperparameter tune two of the four optimizers (CMA-ES and SPSA) across a large range of models and demonstrate that with appropriate hyperparameter tuning, CMA-ES is competitive with and sometimes outperforms SPSA (which is not observed in the absence of hyperparameter tuning). Finally, we investigate the ability of an optimizer to beat the `sampling noise floor' given by the sampling noise on each cost-function estimate provided to the optimizer. Our results demonstrate the necessity for tailoring and hyperparameter-tuning known optimization techniques for inherently-noisy variational quantum algorithms and that the variational landscape that one finds in a VQA is highly problem- and system-dependent. This provides guidance for future implementations of these algorithms in the experiment.

A deep kernel method for lithofacies identification using conventional well logs

How to fit a properly nonlinear classification model from conventional well logs to lithofacies is a key problem for machine learning methods. Kernel methods (e.g., KFD, SVM, MSVM) are effective attempts to solve this issue due to abilities of handling nonlinear features by kernel functions. Deep mining of log features indicating lithofacies still needs to be improved for kernel methods. Hence, this work employs deep neural networks to enhance the kernel principal component analysis (KPCA) method and proposes a deep kernel method (DKM) for lithofacies identification using well logs. DKM includes a feature extractor and a classifier. The feature extractor consists of a series of KPCA models arranged according to residual network structure. A gradient-free optimization method is introduced to automatically optimize parameters and structure in DKM, which can avoid complex tuning of parameters in models. To test the validation of the proposed DKM for lithofacies identification, an open-sourced dataset with seven conventional logs (GR, CAL, AC, DEN, CNL, LLD, and LLS) and lithofacies labels from the Daniudi Gas Field in China is used. There are eight lithofacies, namely clastic rocks (pebbly, coarse, medium, and fine sandstone, siltstone, mudstone), coal, and carbonate rocks. The comparisons between DKM and three commonly used kernel methods (KFD, SVM, MSVM) show that (1) DKM (85.7%) outperforms SVM (77%), KFD (79.5%), and MSVM (82.8%) in accuracy of lithofacies identification; (2) DKM is about twice faster than the multi-kernel method (MSVM) with good accuracy. The blind well test in Well D13 indicates that compared with the other three methods DKM improves about 24% in accuracy, 35% in precision, 41% in recall, and 40% in F1 score, respectively. In general, DKM is an effective method for complex lithofacies identification. This work also discussed the optimal structure and classifier for DKM. Experimental results show that (m1,m2,0) is the optimal model structure and linear SVM is the optimal classifier. (m1,m2,0) means there are m1 KPCAs, and then m2 residual units. A workflow to determine an optimal classifier in DKM for lithofacies identification is proposed, too.

Topological dimensionality reduction-based machine learning for efficient gradient-free 3D topology optimization

Powerful gradient-free topology optimization methods are needed for structural design concerning complex responses. In this paper, a novel gradient-free optimization method is proposed by integrating the material-field series expansion topological parameterization and the deep neural networks, providing two-fold advances: firstly, it generally reduces the massive topological design variables to fewer than 200, while keeps the capability to represent relative complex 3D topologies and clear boundaries; secondly, by constructing a sequential neural network surrogate model, it sufficiently explores the reduced design space and is capable of handling multi-peak and discontinuous optimization problems. The effectiveness of this method is illustrated via several design problems, among which the optimized material effective bulk modulus achieves 98% of the H-S bound and the highly-nonlinear peak weld stress in a phone dropping process is decreased by 16.59%. This method reduces the computational time by 1–4 orders of magnitude compared with the coarse-mesh-based gradient-free methods, and it is the first time to successfully conduct gradient-free 3D topology optimization with thousands of finite elements. The method’s ease of implementation and compatibility with various simulation software, brings topology optimization into complex industrial applications and proves that gradient-free technology represents an effective optimization benchmark for improving structural performance.

Binary Interaction Methods for High Dimensional Global Optimization and Machine Learning

In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. For the resulting kinetic optimization methods, convergence to the global minimizer is guaranteed for a large class of functions under appropriate parameter constraints that do not depend on the dimension of the problem. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. Algorithmic implementations inspired by the well-known direct simulation Monte Carlo methods in kinetic theory are derived and discussed. Several examples on prototype test functions for global optimization are reported including an application to machine learning.

Personalized On-Device E-Health Analytics With Decentralized Block Coordinate Descent.

Actuated by the growing attention to personal healthcare and the pandemic, the popularity of E-health is proliferating. Nowadays, enhancement on medical diagnosis via machine learning models has been highly effective in many aspects of e-health analytics. Nevertheless, in the classic cloud-based/centralized e-health paradigms, all the data will be centrally stored on the server to facilitate model training, which inevitably incurs privacy concerns and high time delay. Distributed solutions like Decentralized Stochastic Gradient Descent (D-SGD) are proposed to provide safe and timely diagnostic results based on personal devices. However, methods like D-SGD are subject to the gradient vanishing issue and usually proceed slowly at the early training stage, thereby impeding the effectiveness and efficiency of training. In addition, existing methods are prone to learning models that are biased towards users with dense data, compromising the fairness when providing E-health analytics for minority groups. In this paper, we propose a Decentralized Block Coordinate Descent (D-BCD) learning framework that can better optimize deep neural network-based models distributed on decentralized devices for E-health analytics. As a gradient-free optimization method, Block Coordinate Descent (BCD) mitigates the gradient vanishing issue and converges faster at the early stage compared with the conventional gradient-based optimization. To overcome the potential data scarcity issues for users' local data, we propose similarity-based model aggregation that allows each on-device model to leverage knowledge from similar neighbor models, so as to achieve both personalization and high accuracy for the learned models. Benchmarking experiments on three real-world datasets illustrate the effectiveness and practicality of our proposed D-BCD, where additional simulation study showcases the strong applicability of D-BCD in real-life E-health scenarios.

Gradient-Free Optimization in Thermoacoustics: Application to a Low-Order Model

Abstract Machine learning and automatized routines for parameter optimization have experienced a surge in development in the past years, mostly caused by the increasing availability of computing capacity. Gradient-free optimization can avoid cumbersome theoretical studies as input parameters are purely adapted based on output data. As no knowledge about the objective function is provided to the algorithms, this approach might reveal unconventional solutions to complex problems that were out of scope of classical solution strategies. In this study, the potential of these optimization methods on thermoacoustic problems is examined. The optimization algorithms are applied to a generic low-order thermoacoustic can-combustor model with several fuel injectors at different locations. We use three optimization algorithms – the well established downhill simplex method, the recently proposed explorative gradient method, and an evolutionary algorithm – to find optimal fuel distributions across the fuel lines while maintaining the amount of consumed fuel constant. The objective is to have minimal pulsation amplitudes. We compare the results and efficiency of the gradient-free algorithms. Additionally, we employ model-based linear stability analysis to calculate the growth rates of the dominant thermoacoustic modes. This allows us to highlight general and thermoacoustic-specific features of the optimization methods and results. The findings of this study show the potential of gradient-free optimization methods on combustor design for tackling thermoacoustic problems, and motivate further research in this direction.

Data-driven model for hydraulic fracturing design optimization. Part II: Inverse problem

We describe a stacked model for predicting the cumulative fluid production for an oil well with a multistage-fracture completion based on a combination of Ridge Regression and CatBoost algorithms. The model is developed based on an extended digital field data base of reservoir, well and fracturing design parameters. The database now includes more than 5000 wells from 23 oilfields of Western Siberia (Russia), with 6687 fracturing operations in total. Starting with 387 parameters characterizing each well, including construction, reservoir properties, fracturing design features and production, we end up with 38 key parameters used as input features for each well in the model training process. The model demonstrates physically explainable dependencies plots of the target on the design parameters (number of stages, proppant mass, average and final proppant concentrations and fluid rate). We developed a set of methods including those based on the use of Euclidean distance and clustering techniques for offset wells selection (search for similar wells in terms of certain metrics), which is useful for a field engineer to analyze earlier fracturing treatments on similar wells. These approaches are also adapted for obtaining the margings for optimization parameters for a particular pilot well, as part of the field testing campaign of the methodology. An inverse problem (selecting an optimum set of fracturing design parameters to maximize production) is formulated as optimizing a high dimensional black box approximation function constrained by boundaries and solved with four different optimization methods: surrogate-based optimization (pSeven), sequential least squares programming, particle swarm optimization and differential evolution. A recommendation system containing all of the above methods is designed to advise a production stimulation engineer on an optimized fracturing design. • Workflow for data-drive fracturing design optimization is developed based on digital database and production forecast model. • Inverse problem is solved with several gradient-free optimization methods (with probability of improvement) and surrogate-based optimization. • Methodology for offset wells selection is presented as an algorithm of optimization based on best practices on similar wells. • Recommendation system containing these methods is designed to assist production stimulation engineer.

A Consensus-Based Global Optimization Method with Adaptive Momentum Estimation

Objective functions in large-scale machine-learning and artificial intelligence applications often live in high dimensions with strong non-convexity and massive local minima. First-order methods, such as the stochastic gradient method and Adam, are often used to find global minima. Recently, the consensus-based optimization (CBO) method has been introduced as one of the gradient-free optimization methods and its convergence is proven with dimension-dependent parameters, which may suffer from the curse of dimensionality. By replacing the isotropic geometric Brownian motion with the component-wise one, the latest improvement of the CBO method is guaranteed to converge to the global minimizer with dimension-independent parameters, although the initial data need to be well-chosen. In this paper, based on the CBO method and Adam, we propose a consensus-based global optimization method with adaptive momentum estimation (Adam-CBO). Advantages of the Adam-CBO method include: (1) capable of finding global minima of non-convex objective functions with high success rates and low costs; (2) can handle non-differentiable activation functions and thus approximate low-regularity functions with better accuracy. The former is verified by approximating the $1000$ dimensional Rastrigin function with $100\%$ success rate at a cost only growing linearly with respect to the dimensionality. The latter is confirmed by solving a machine learning task for partial differential equations with low-regularity solutions where the Adam-CBO method provides better results than the state-of-the-art method Adam. A linear stability analysis is provided to understand the asymptotic behavior of the Adam-CBO method.

A Gradient-Free Topology Optimization Strategy for Continuum Structures with Design-Dependent Boundary Loads

In this paper, the topology optimization of continuum structures with design-dependent loads is studied with a gradient-free topology optimization method in combination with adaptive body-fitted finite element mesh. The material-field series-expansion (MFSE) model represents the structural topology using a bounded material field with specified spatial correlation and provides a crisp structural boundary description. This feature makes it convenient to identify the loading surface for the application of the design-dependent boundary loads and to generate a body-fitted mesh for structural analysis. Using the dimension reduction technique, the number of design variables is significantly decreased, which enables the use of an efficient Kriging-based algorithm to solve the topology optimization problem. The effectiveness of the proposed method is demonstrated using several numerical examples, among which a design problem with geometry and contact nonlinearity is included.

A consensus-based global optimization method for high dimensional machine learning problems

We improve recently introduced consensus-based optimization method, proposed in [R. Pinnau, C. Totzeck, O. Tse, S. Martin, Math. Models Methods Appl. Sci. 27 (2017) 183–204], which is a gradient-free optimization method for general non-convex functions. We first replace the isotropic geometric Brownian motion by the component-wise one, thus removing the dimensionality dependence of the drift rate, making the method more competitive for high dimensional optimization problems. Secondly, we utilize the random mini-batch ideas to reduce the computational cost of calculating the weighted average which the individual particles tend to relax toward. For its mean-field limit – a nonlinear Fokker-Planck equation – we prove, in both time continuous and semi-discrete settings, that the convergence of the method, which is exponential in time, is guaranteed with parameter constraints independent of the dimensionality. We also conduct numerical tests to high dimensional problems to check the success rate of the method.

Topological design of microstructures using periodic material-field series-expansion and gradient-free optimization algorithm

Light-weight cellular materials with periodic repetitive microstructures are widely used in various fields due to their superior mechanical/multi-physical performances. As the microstructural design problem is known to have multiple local minima, most gradient-based topology optimization methods significantly depend on the initial guess of the microstructural geometry, thus requiring the designer’s experiences. This paper presents an effective gradient-free framework for periodic microstructure design, which exhibits powerful global searching capabilities and requires no sensitivity information. The proposed framework combines the material-field series-expansion (MFSE) topology representation of periodic microstructures and the sequential Kriging-based optimization algorithm. The MFSE method decouples the topological representation from the finite element discretization, and describes a relatively complex microstructural topology with high-quality boundary description using a greatly reduced number of design variables. Based on the Kriging surrogate model, a solution scheme is suggested to solve material microstructural topology optimization successively in a sequence of sub-optimization problems with self-adaptive design spaces. With the present gradient-free optimization method, high-performance cellular materials that approach the H-S upper bound with porosities from 0.2 to 0.6, or that achieve negative Poisson’s ratios of -0.94 in the principal directions for materials with square symmetry, are obtained without prior knowledge of the optimum microstructural topology.