Discrete Design Space Research Articles

Accelerated development of multi-component alloys in discrete design space using Bayesian multi-objective optimisation

Abstract Bayesian optimisation (BO) protocols grounded in active learning (AL) principles have gained significant recognition for their ability to efficiently optimize black-box objective functions. This capability is critical for advancing autonomous and high-throughput materials design and discovery processes. However, the application of these protocols in materials science, particularly in the design of novel alloys with multiple targeted properties, remains constrained by computational complexity and the absence of reliable and robust acquisition functions for multiobjective optimisation. Recent advancements have demonstrated that expected hypervolume-based geometrical acquisition functions outperform other multiobjective optimisation algorithms, such as Thompson Sampling Efficient Multiobjective optimisation and pareto efficient global optimisation (parEGO), in both performance and speed. This study evaluates several leading multiobjective BO acquisition functions–namely, parallel expected hypervolume improvement (qEHVI), noisy qEHVI, parallel parEGO, and parallel noisy parEGO (qNparEGO)–in optimizing the physical properties of multi-component alloys. Our findings highlight the superior performance of the qEHVI acquisition function in identifying the optimal Pareto front across 1-, 2-, and 3-objective aluminum alloy optimisation problems, all within a constrained evaluation budget and reasonable computational cost. Furthermore, we explore the impact of various surrogate model optimisation methods from both computational cost and efficiency perspectives. Finally, we demonstrate the effectiveness of a pool-based AL protocol in expediting the discovery process by executing multiple computational and experimental campaigns in each iteration. This approach is particularly advantageous for deployment in massively parallel high-throughput synthesis facilities and advanced computing architectures.

Geotechnical reliability‐based design optimization updating under changing design scenarios

AbstractIn the context of geotechnical uncertainty characterization, the reliability‐based design optimization (RBDO) provides an effective means to achieve the most balanced design outcome between the targeted safety and the risk‐based cost planning. As a common practice, the geotechnical RBDO is conducted on the premise of prescribed statistical information of those uncertain parameters. Nevertheless, the uncertain geotechnical parameters are usually subject to amendment resulting from further site investigation. In other words, changing design scenarios must be properly handled in advanced geotechnical RBDO. Thus, in this study, a novel reliability updating method using a sample reweighting technique that requires only a single simulation run is developed for RBDO. Simultaneously, the RBDO is readily extendable to RBDO updating under changing design scenarios while avoiding any further re‐evaluation of the performance functions. The proposed RBDO updating method allows for the consideration of a continuous design space encompassing an infinite number of feasible designs, whereas the existing simulation‐based method can only consider a finite number of feasible designs in a discrete design space. Moreover, the efficiency and flexibility of the proposed RBDO updating method are illustrated through a standard RBDO test problem and a practical rock slope design example, where the changing design scenarios of different means, standard deviations, correlation coefficient, and distributions are respectively studied.

Unifying the design space and optimizing linear and nonlinear truss metamaterials by generative modeling

The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials—truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.

Exact electronic states with shallow quantum circuits from global optimisation

Quantum computers promise to revolutionise molecular electronic simulations by overcoming the exponential memory scaling. While electronic wave functions can be represented using a product of fermionic unitary operators, the best ansatz for strongly correlated electronic systems is far from clear. In this contribution, we construct universal wave functions from gate-efficient, spin symmetry-preserving fermionic operators by introducing an algorithm that globally optimises the wave function in the discrete ansatz design and continuous parameter spaces. Our approach maximises the accuracy that can be obtained with near-term quantum circuits and provides a practical route for designing ansätze in the future. Numerical simulations for strongly correlated molecules, including water and molecular nitrogen, and the condensed-matter Hubbard model, demonstrate the improved accuracy of gate-efficient quantum circuits for simulating strongly correlated chemistry.

Design and Fabrication of Soft 3D Printed Sensors and Performance Analysis of the Soft Sensors in a C-leg as Sensing Element

In soft robotics, a recent challenge is to decrease the number of rigid components used tocreate entirely soft robots. A common rigid component used in soft robots is the rigid encoder, which should be replaced with a soft counterpart if possible. In this work, we de-sign and manufacture a soft sensor, which is embedded into a C-shaped leg of a soft, legged, miniature robot. Our main goal is to show that we can embed a soft sensor into and receive contact feedback from a soft C-shaped leg of our soft miniature quadruped. We test various sensor parameters using custom test setups to analyze the soft sensor performance. Our soft sensor design is iterated by experimentally investigating several sensor shape options. For the C-leg of the soft miniature quadruped, optimal sensor geometry and position for the sensor implementation are found from a discrete design space as the outcome of this work. We received feedback from the soft sensor and compared commercial encoder data to the soft sensor embedded C-leg data. We managed to detect the rotation speed of the C-leg with the accuracy of 87.5% on a treadmill and with the accuracy of %86.7 under free rotation of the C-leg. However, if connection loss occurs in the miniature slipring mechanism, the error percentage in estimating the rotational speed increases significantly.

Efficient decoupling-assisted evolutionary/metaheuristic framework for expensive reliability-based design optimization problems

Reliability-based design optimization (RBDO) algorithm is to minimize the objective under the probabilistic factors. While gradient-based and classical evolutionary RBDO algorithms provide promising performance on simple optimization problems, they are likely to perform poorly on challenging problems, including the multimodal functions, discrete design spaces, non-differential problems, etc. This paper proposes a unified framework to improve the performance of existing RBDO algorithms for complex RBDO problems. Our framework is based on three new strategies: generalized decoupling evolutionary and metaheuristic RBDO framework, particle’s memory saving strategy, and adaptive fractional-order equilibrium optimizer algorithm. The proposed algorithm is characterized by a decoupling strategy to enable the parallel operation of the inner reliability computation and outer deterministic optimization, a particle’s memory saving strategy to provide effective guidance from the previous iteration, and the adaptive fractional-order equilibrium optimizer algorithm to enhance the search efficiency and global convergence capacity. To evaluate the performance of the proposed algorithm, a wide range of experiments are conducted on different types of use cases. The experimental results demonstrate that our algorithm provides superior performance over other comparative algorithms.

RT-RCG: Neural Network and Accelerator Search Towards Effective and Real-time ECG Reconstruction from Intracardiac Electrograms.

There exists a gap in terms of the signals provided by pacemakers (i.e., intracardiac electrogram (EGM)) and the signals doctors use (i.e., 12-lead electrocardiogram (ECG)) to diagnose abnormal rhythms. Therefore, the former, even if remotely transmitted, are not sufficient for doctors to provide a precise diagnosis, let alone make a timely intervention. To close this gap and make a heuristic step towards real-time critical intervention in instant response to irregular and infrequent ventricular rhythms, we propose a new framework dubbed RT-RCG to automatically search for (1) efficient Deep Neural Network (DNN) structures and then (2) corresponding accelerators, to enable Real-Time and high-quality Reconstruction of ECG signals from EGM signals. Specifically, RT-RCG proposes a new DNN search space tailored for ECG reconstruction from EGM signals, and incorporates a differentiable acceleration search (DAS) engine to efficiently navigate over the large and discrete accelerator design space to generate optimized accelerators. Extensive experiments and ablation studies under various settings consistently validate the effectiveness of our RT-RCG. To the best of our knowledge, RT-RCG is the first to leverage neural architecture search (NAS) to simultaneously tackle both reconstruction efficacy and efficiency.

An Actor-Critic Method for Simulation-Based Optimization

In this work, we study simulation-based optimization, where the agent aims to select the best configuration from the design space with as few as possible iterations. Inspired by the success of deep reinforcement learning (DRL), we formulate the sampling process as policy searching and give a solving method from the perspective of policy iteration. Concretely, a surrogate model for predicting the performance of each configuration and a parameterized sampling policy are applied, which correspond to the critic and actor in actor-critic (AC) method, respectively. We further derive the updating rule and propose two algorithms for configuration selection in continuous and discrete design spaces, respectively. Finally, the algorithms are validated experimentally on 1) two toy examples to intuitively explain the principle and 2) two high-dimensional tasks to reveal the effectiveness in large-scale problems. The results show that the proposed algorithms can efficiently deal with large-scale problems and effectively eliminate sub-optimal configurations.

Minimax A‐, c‐, and I‐optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A‐, c‐, and I‐optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.

Thickness-based subdomian hybrid cellular automata algorithm for lightweight design of BIW under side collision

Body-in-white (BIW) is a typical frame structure formed by a great many thin-walled structures, and the optimal design of its lightweight and crashworthiness is a typical nonlinear dynamic response optimization problem with multiple design variables. Here, we propose a thickness-based subdomain hybrid cellular automata (T-SHCA) algorithm to solve the lightweight design of BIW under side collision, in which there are two loops: one is the outer loop to conduct crash finite element analysis, calculate output responses, and update internal energy density (IED) and target mass; the other is the inner loop to adjust cell thicknesses according to IEDs of current cell and its neighboring cells to make the actual mass of current cells converge to target mass. The concept of "subdomain cellular automata (SCA)" model was introduced, so that the T-SHCA can solve nonlinear dynamic response optimization in discrete and separated design spaces. The step IED target update rule and the cell thickness update rule based on a PID control strategy were implemented in the inner loop to improve the global optimal solution search capability and the robustness of the proposed algorithm, respectively. The thicknesses optimization of a BIW was carried out by the proposed method and a parallel efficient global optimization algorithm to verify its convergence and efficiency. The results show that the T-SHCA algorithm can be implemented to effectively solve nonlinear dynamic response structural optimization problem with many thickness variables in discrete and separated design spaces.

Optimal group testing designs for prevalence estimation combining imperfect and gold standard assays

We consider group testing studies where a relatively inexpensive but imperfect assay and a perfectly accurate but higher-priced assay are both available. The primary goal is to accurately estimate the prevalence of a trait of interest, with the error rates of the imperfect assay treated as nuisance parameters. Considering the costs for performing the two assays and enrolling subjects, we propose a $D_{s}$-optimal mixed design to provide maximal information about the prevalence. We show that extreme values for the cost of the perfect assay lead to designs in which only one of the two assays is used, but otherwise the optimal designs use both assays. We provide a guaranteed algorithm to efficiently build an optimal design on discrete design spaces. Our computational results also show the robustness of the proposed design.

Mixed-integer second-order cone optimization for composite discrete ply-angle and thickness topology optimization problems

Discrete variable topology optimization problems are usually solved by using solid isotropic material with penalization (SIMP), genetic algorithms (GA), or mixed-integer nonlinear optimization (MINLO). In this paper, we propose formulating discrete ply-angle and thickness topology optimization problems as a mixed-integer second-order cone optimization (MISOCO) problem. Unlike SIMP and GA methods, MISOCO efficiently finds the problem’s globally optimal solution. Furthermore, in contrast with existing MISOCO formulations of discrete ply-angle optimization problems, our reformulations allow the structure to change topology, consider the more realistic Tsai–Wu stress yield criteria constraint, and eliminate checkerboard patterns using simple linear constraints. We address two types of discrete ply-angle and thickness problems: a structural mass minimization problem and a compliance optimization problem where the objective is to maximize the structural stiffness. For each element, one first chooses if the element is present or not in the structure. One can then choose the element’s ply-angle and thickness from a finite set of possibilities for the former case. The discrete design space for ply-angle and thickness is a result of manufacturing limitations. To improve the problem’s MISOCO solution approach, we develop valid inequality constraints to tighten the continuous relaxation of the MISOCO reformulation. We compare the performance of various MISOCO solvers: Gurobi, CPLEX, and MOSEK to solve the MISOCO reformulation. We also use BARON to solve the original MINLO formulations of the problems. Our results show that solving the MISOCO problem’s formulation using MOSEK is the most efficient solution approach.

R-optimal designs for multi-response regression models with multi-factors

We investigate R-optimal designs for multi-response regression models with multi-factors, where the random errors in these models are correlated. Several theoretical results are derived for R-optimal designs, including scale invariance, reflection symmetry, line and plane symmetry, and dependence on the covariance matrix of the errors. All the results can be applied to linear and non-linear models. In addition, an efficient algorithm based on an interior point method is developed for finding R-optimal designs on discrete design spaces. The algorithm is very flexible, and can be applied to any multi-response regression model.

Minimax D-optimal designs for multivariate regression models with multi-factors

In multi-response regression models, the error covariance matrix is never known in practice. Thus, there is a need for optimal designs which are robust against possible misspecification of the error covariance matrix. In this paper, we approximate the error covariance matrix with a neighborhood of covariance matrices, in order to define minimax D-optimal designs that are robust against small departures from an assumed error covariance matrix. It is well known that the optimization problems associated with robust designs are non-convex, which makes it challenging to construct robust designs analytically or numerically, even for one-response regression models. We show that the objective function for the minimax D-optimal design is a difference of two convex functions, and develop a flexible algorithm for computing minimax D-optimal designs, which can be applied to any multi-response model with a discrete design space. We also derive several theoretical results for minimax D-optimal designs.

A new Newton metaheuristic algorithm for discrete performance-based design optimization of steel moment frames

In this paper a new and efficient metaheuristic algorithm is proposed for discrete performance-based seismic design optimization of steel moment frames. The proposed metaheuristic uses the Newton gradient-based method as its updating scheme in a population-based framework and therefore it is termed as Newton Metaheuristic Algorithm (NMA). In order to enable the NMA to effectively explore the discrete design space, a term containing the best solution found is added to the basic updating rule of the algorithm. In addition, a simple and efficient method is proposed in order to establish a balance between local and global search abilities of the proposed algorithm. The efficiency of the NMA is illustrated by presenting two benchmark discrete truss optimization problems. Moreover, three steel moment frames are optimized in the framework of performance-based design by the NMA and the results are compared with those of some recent metaheuristics. The performance of the algorithms is analyzed using statistical parametric and non-parametric tests indicating that NMA outperforms the other algorithms in literature.

Maximin power designs in testing lack of fit

In a previous article (Wiens, 1991) we established a maximin property, with respect to the power of the test for Lack of Fit, of the absolutely continuous uniform ‘design’ on a design space which is a subset of Rq with positive Lebesgue measure. Here we discuss some issues and controversies surrounding this result. We find designs which maximize the minimum power, over a broad class of alternatives, in discrete design spaces of cardinality N. We show that these designs are supported on the entire design space. They are in general not uniform for fixed N, but are asymptotically uniform as N→∞. Several examples with N fixed are discussed; in these we find that the approach to uniformity is very quick.

R-LRFD: Load and resistance factor design considering robustness

LRFD (Load and Resistance Factor Design) is becoming a design method of choice in geotechnical practice, in lieu of the factor of safety (FS)-based design approach. However, even with LRFD, the need to reduce the variation in the predicted performance of a geotechnical system (or a geotechnical structure), referred to herein as the system response under applied loads, is still apparent. This paper presents a novel approach for applying existing LRFD codes, with explicit consideration of design robustness, safety, and cost efficiency. The recently developed reliability-based robust geotechnical design (RGD) approach is modified such that it can be integrated with the standard LRFD approach. This modified RGD approach is termed R-LRFD, which stands for Robust Load and Resistance Factor Design. In R-LRFD, the robustness of the system response against the variation in uncertain parameters is explicitly considered. Unlike the reliability-based design (RBD), the user is not required to conduct a full statistical characterization of uncertain parameters. With R-LRFD, the gain in the robustness, as reflected by the reduction in the sensitivity of the system response to the recognized but unquantified uncertainty of input parameters, is accompanied by an increase in cost. Thus, the concepts of Pareto front and knee point are introduced to aid in making an informed design decision. Further, a simplified procedure is developed to identify the most preferred design (knee point) in the design space, which is generally solved with multi-objective optimization algorithms. The effectiveness and significance of the proposed R-LRFD approach are demonstrated with two examples: one is the design of drilled shaft in sand (illustrated with a discrete design space) and the other is the design of drilled shaft in clay (illustrated with a continuous design space).

Simulation-based fully Bayesian experimental design for mixed effects models

Bayesian inference has commonly been performed on nonlinear mixed effects models. However, there is a lack of research into performing Bayesian optimal design for nonlinear mixed effects models, especially those that require searches to be performed over several design variables. This is likely due to the fact that it is much more computationally intensive to perform optimal experimental design for nonlinear mixed effects models than it is to perform inference in the Bayesian framework. Fully Bayesian experimental designs for nonlinear mixed effects models are presented, which involve the use of simulation-based optimal design methods to search over both continuous and discrete design spaces. The design problem is to determine the optimal number of subjects and samples per subject, as well as the (near) optimal urine sampling times for a population pharmacokinetic study in horses, so that the population pharmacokinetic parameters can be precisely estimated, subject to cost constraints. The optimal sampling strategies, in terms of the number of subjects and the number of samples per subject, were found to be substantially different between the examples considered in this work, which highlights the fact that the designs are rather problem-dependent and can be addressed using the methods presented.

Computing A-optimal and E-optimal designs for regression models via semidefinite programming

In semidefinite programming (SDP), we minimize a linear objective function subject to a linear matrix being positive semidefinite. A powerful program, SeDuMi, has been developed in MATLAB to solve SDP problems. In this article, we show in detail how to formulate A-optimal and E-optimal design problems as SDP problems and solve them by SeDuMi. This technique can be used to construct approximate A-optimal and E-optimal designs for all linear and nonlinear regression models with discrete design spaces. In addition, the results on discrete design spaces provide useful guidance for finding optimal designs on any continuous design space, and a convergence result is derived. Moreover, restrictions in the designs can be easily incorporated in the SDP problems and solved by SeDuMi. Several representative examples and one MATLAB program are given.

Discrete Optimum Design for Truss Structures by Subset Simulation Algorithm

AbstractThis article deals with the design optimization of truss structures with discrete design variables, which remains quite a challenging task in structural design. A new discrete search strategy based on the recently developed subset simulation optimization algorithm is proposed in detail for this type of structural optimization. The discrete design variables are transformed into standard normal variable space to implement the sampling procedure in subset simulation optimization, while the optimization is processed in the discrete design space in the mean time. The performance of the proposed method is illustrated by four representative benchmark optimization problems. Comparisons are made with other well known stochastic optimization algorithms. It is found that the proposed method can produce optimum designs as good as or better than those of other stochastic optimization algorithms.