Traditional Bayesian Optimization Research Articles

Towards a Digital Twin Framework in Additive Manufacturing: Machine Learning and Bayesian Optimization for Time Series Process Optimization

Laser directed-energy deposition (DED) offers notable advantages in additive manufacturing (AM) for producing intricate geometries and facilitating material functional grading. However, inherent challenges such as material property inconsistencies and part variability persist, predominantly due to its layer-wise fabrication approach. Critical to these challenges is heat accumulation during DED, influencing the resultant material microstructure and properties. Although closed-loop control methods for managing heat accumulation and temperature regulation are prevalent in DED literature, few approaches integrate real-time monitoring, physics-based modeling, and control simultaneously in a cohesive framework. To address this, we present a digital twin (DT) framework for real-time model predictive control of process parameters of the DED for achieving a specific process design objective. To enable its implementation, we detail the development of a surrogate model utilizing Long Short-Term Memory (LSTM)-based machine learning which uses Bayesian Inference to predict temperatures across various spatial locations of the DED-built part. This model offers real-time predictions of future temperature states. In addition, we introduce a Bayesian Optimization (BO) method for Time Series Process Optimization (BOTSPO). Its foundational principles align with traditional BO, and its novelty lies in our unique time series process profile generator with a reduced dimensional representation. BOTSPO is used for dynamic process optimization in which we deploy BOTSPO to determine the optimal laser power profile, aiming to achieve desired mechanical properties in a DED build. The identified profile establishes a process trajectory that online process optimizations aim to match or exceed in performance. This paper elucidates components of the digital twin framework, advocating its prospective consolidation into a comprehensive digital twin system for AM.

Enhancing LC×LC separations through multi-task Bayesian optimization

Method development in comprehensive two-dimensional liquid chromatography (LC×LC) is a challenging process. The interdependencies between the two dimensions and the possibility of incorporating complex gradient profiles, such as multi-segmented gradients or shifting gradients, make trial-and-error method development time-consuming and highly dependent on user experience. Retention modeling and Bayesian optimization (BO) have been proposed as solutions to mitigate these issues. However, both approaches have their strengths and weaknesses. On the one hand, retention modeling, which approximates true retention behavior, depends on effective peak tracking and accurate retention time and width predictions, which are increasingly challenging for complex samples and advanced gradient assemblies. On the other hand, Bayesian optimization may require many experiments when dealing with many adjustable parameters, as in LC×LC. Therefore, in this work, we investigate the use of multi-task Bayesian optimization (MTBO), a method that can combine information from both retention modeling and experimental measurements. The algorithm was first tested and compared with BO using a synthetic retention modeling test case, where it was shown that MTBO finds better optima with fewer method-development iterations than conventional BO. Next, the algorithm was tested on the optimization of a method for a pesticide sample and we found that the algorithm was able to improve upon the initial scanning experiments. Multi-task Bayesian optimization is a promising technique in situations where modeling retention is challenging, and the high number of adjustable parameters and/or limited optimization budget makes traditional Bayesian optimization impractical.

Asynchronous Multi-Information Source Bayesian Optimization

Abstract Resource management in engineering design seeks to optimally allocate while maximizing the performance metrics of the final design. Bayesian optimization (BO) is an efficient design framework that judiciously allocates resources through heuristic-based searches, aiming to identify the optimal design region with minimal experiments. Upon recommending a series of experiments or tasks, the framework anticipates their completion to augment its knowledge repository, subsequently guiding its decisions toward the most favorable next steps. However, when confronted with time constraints or other resource challenges, bottlenecks can hinder the traditional BO’s ability to assimilate knowledge and allocate resources with efficiency. In this work, we introduce an asynchronous learning framework designed to utilize idle periods between experiments. This model adeptly allocates resources, capitalizing on lower fidelity experiments to gather comprehensive insights about the target objective function. Such an approach ensures that the system progresses uninhibited by the outcomes of prior experiments, as it provisionally relies on anticipated results as stand-ins for actual outcomes. We initiate our exploration by addressing a basic problem, contrasting the efficacy of asynchronous learning against traditional synchronous multi-fidelity BO. We then employ this method to a practical challenge: optimizing a specific mechanical characteristic of a dual-phase steel.

CAPBO: A cost‐aware parallelized Bayesian optimization method for chemical reaction optimization

AbstractBayesian optimization employs probabilistic surrogate models to effectively address expensive and time‐consuming closed‐loop chemical experimental design. However, traditional Bayesian optimization focuses on reducing the number of iterations and follows an inherently sequential process (with one new data point sampled in each iteration), which is an inefficient means of exploiting and characterizing reactions using parallel microreactors. In this article, we present an approach that overcomes this issue by considering two essential factors: experimental cost sensitivity and parallelization. We propose a novel cost‐aware parallelized Bayesian optimization method (CAPBO) in which the goal is changed from reducing the number of iterations to reducing the experimental cost, with quantitative experimental speed‐up achieved through parallelization. The combination of these two items greatly fits the need of reaction optimization, leading to better optimization effects. Benchmarking and case studies demonstrate that the proposed method significantly outperforms traditional Bayesian optimization and considerably improves the optimization efficiency while maintaining the simplicity of the framework.

No-regret constrained Bayesian optimization of noisy and expensive hybrid models using differentiable quantile function approximations

This paper investigates the problem of efficient constrained global optimization of hybrid models that are a composition of a known white-box function and an expensive multi-output black-box function subject to noisy observations, which often arises in real-world science, engineering, manufacturing, and control applications. We propose a novel method, Constrained Upper Quantile Bound (CUQB), to solve such problems that directly exploits the composite structure of the objective and constraint functions that we show leads substantially improved sampling efficiency. CUQB is a conceptually simple, deterministic approach that avoids the constraint approximations used by previous methods. Although the CUQB acquisition function is not available in closed form, we propose a novel differentiable sample average approximation that enables it to be efficiently maximized. We further derive bounds on the cumulative regret and constraint violation under a non-parametric Bayesian representation of the black-box function. Since these bounds depend sublinearly on the number of iterations under some regularity assumptions, we establish explicit bounds on the convergence rate to the optimal solution of the original constrained problem. In contrast to most existing methods, CUQB further incorporates a simple infeasibility detection scheme, which we prove triggers in a finite number of iterations when the original problem is infeasible (with high probability given the Bayesian model). Numerical experiments on several test problems, including environmental model calibration and real-time optimization of a reactor system, show that CUQB significantly outperforms traditional Bayesian optimization in both constrained and unconstrained cases. Furthermore, compared to other state-of-the-art methods that exploit composite structure, CUQB achieves competitive empirical performance while also providing substantially improved theoretical guarantees.

Faux-Data Injection Optimization for Accelerating Data-Driven Discovery of Materials

Artificial intelligence is now extensively being used to optimize and discover novel materials through data-driven search. The search space for the material to be discovered is usually so large, that it renders manual optimization impractical. This is where data-driven search and optimization enables us to resourcefully locate an optimal or acceptable material configuration with desirable target properties. One such prominent data-driven optimization technique is Bayesian optimization (BO). Among the mechanics of a BO is the use of a machine learning (ML) model that learns about the scope of the problem through data being acquired on the fly. In this way a BO becomes more informative, directing the search more exquisitely by providing informative suggestions for locating a suitable material candidate for further evaluation. The candidate material is suggested by proposing parameters such as its composition and configuration, which are then evaluated either by physically synthesizing the material and testing its properties or through computational methods such as through density functional theory (DFT). DFT enables researchers to exploit massively parallel architectures such as high-performance computing (HPC) which a traditional BO might not be able to fully leverage due to their typical sequential data-acquisition bottleneck. Here, we tackle such shortcomings of BO and maximize the utilization of HPC by enabling BO to suggest multiple candidate material suggestions for DFT evaluations at once, which can then be distributed in multiple compute nodes of an HPC. We achieve this objective through a batch optimization technique based on faux-data injection in the BO loop. In the approach at each candidate suggestion from a typical BO loop, we “predict” the outcome, instead of running the actual experiment or DFT calculation, forming a “faux-data-point” and injecting it back to update an ML model. The next BO suggestion is therefore conditioned on the actual data as well as faux-data, to yield the next candidate data-point suggestion. The objective of this methodology is to simulate a time-consuming sequential data-gathering process and approximate the next k-potential candidates, quickly. All these k-potential candidates can then be distributed to run in parallel in an HPC. Our objective in this work is to test the theory if faux-data injection methodology enables us accelerate our data-driven material discovery workflow. To this end, we execute computational experiments by utilizing organic–inorganic halide perovskites as a case study since the optimality of the results can be easily verified from our previous work. To evaluate the performance, we propose a metric that considers and consolidates acceleration along with the quality of the results such as the best value reached in the process. We also utilize a different performance indicator for situations where the desired outcome is not material with optimal properties but rather a material whose properties satisfy some minimum requirements. We use these performance indicators to compare this BO-based faux-data injection method (FDI-BO) with different baselines. The results show that based on our design constraints, the FDI-BO approach enabled us to obtain around two- to sixfold acceleration on average compared to the sequential BO.

Fast Bayesian optimization of Needle-in-a-Haystack problems using zooming memory-based initialization (ZoMBI)

Needle-in-a-Haystack problems exist across a wide range of applications including rare disease prediction, ecological resource management, fraud detection, and material property optimization. A Needle-in-a-Haystack problem arises when there is an extreme imbalance of optimum conditions relative to the size of the dataset. However, current state-of-the-art optimization algorithms are not designed with the capabilities to find solutions to these challenging multidimensional Needle-in-a-Haystack problems, resulting in slow convergence or pigeonholing into a local minimum. In this paper, we present a Zooming Memory-Based Initialization algorithm, entitled ZoMBI, that builds on conventional Bayesian optimization principles to quickly and efficiently optimize Needle-in-a-Haystack problems in both less time and fewer experiments. The ZoMBI algorithm demonstrates compute time speed-ups of 400× compared to traditional Bayesian optimization as well as efficiently discovering optima in under 100 experiments that are up to 3× more highly optimized than those discovered by similar methods.

Bayesian Optimization Based on Student’s T Process for Microstrip Antenna Design

Bayesian Optimization (BO) is an efficient global optimization algorithm, which is widely used in the field of engineering design. The probabilistic surrogate model and acquisition function are the two keys to the algorithm. Building an efficient probabilistic surrogate model and designing a collection function with excellent exploring capabilities can improve the performance of BO algorithm, allowing it to find the optimal value of the objective function with fewer iterations. Due to the characteristics of small samples and non-parametric derivation of the Gaussian Process (GP), traditional BO algorithms usually use the GP as a surrogate model. Compared with the GP, the Student’s T Process (STP) retains the excellent properties of GP, and has more flexible posterior variance and stronger robustness. In this paper, STP is used as the surrogate model in BO algorithm, the hyperparameters of the model are optimized by STP, and the estimation strategy function (EST) is improved based on the posterior output of the optimized STP, thus realizing the improved BO algorithm based on the STP. To verify the performance of the proposed algorithm, numerical experiments are designed to compare the performances of the traditional BO algorithm, which includes the lower confidence bound function (LCB) and EST as acquisition function respectively and GP as the surrogate model, and the proposed BO algorithm with STP as the surrogate model and LCB, expected improvement function (EI), expected regret minimization function (ERM) as acquisition function respectively. The results show that the proposed algorithm in this paper performs well when finding the global minimum of multimodal functions. Based on the developed algorithm in this paper, the resonant frequency of printed dipole antenna and E-shaped antenna is modeled and optimized, which further confirms the good design ability and design accuracy of the BO algorithm proposed in this paper.

Gradient-Enhanced Bayesian Optimization via Acquisition Ensembles with Application to Reinforcement Learning

Bayesian optimization (BO) has shown great promise as a data-efficient strategy for the global optimization of expensive, black-box functions in a plethora of control applications. Traditional BO is derivative-free, as it solely relies on observations of a performance function to find its optimum. Recently, so-called first-order BO methods have been proposed that additionally exploit gradient information of the performance function to accelerate convergence. First-order BO methods mostly utilize standard acquisition functions, while indirectly using gradient information in the kernel structure to learn more accurate probabilistic surrogates for the performance function. In this work, we present a gradient-enhanced BO method that directly exploits performance function (zeroth-order) and its corresponding gradient (first-order) evaluations in the acquisition function. To this end, a novel gradient-based acquisition function is proposed that can identify stationary points of the performance optimization problem. We then leverage ideas from multi-objective optimization to develop an effective strategy for finding query points that optimally tradeoff between a zeroth-order acquisition function and the proposed gradient-based acquisition function. We show how the proposed acquisition-ensemble gradient-enhanced BO (AEGEBO) method enables accelerating convergence of policy-based reinforcement learning by combining noisy observations of the reward function and its gradient that can be directly estimated from closed-loop data. The performance of AEGBO is compared to standard BO and the well-known REINFORCE algorithm on a benchmark LQR problem, for which we consistently observe significantly improved performance over a limited data budget.

Multi-agent Black-box Optimization using a Bayesian Approach to Alternating Direction Method of Multipliers*

Bayesian optimization (BO) is a powerful black-box optimization framework that looks to efficiently learn the global optimum of an unknown system by systematically trading-off between exploration and exploitation. However, the use of BO as a tool for coordinated decision-making in multi-agent systems with unknown structure has not been widely studied. This paper investigates a black-box optimization problem over a multi-agent network coupled via shared variables or constraints, where each subproblem is formulated as a BO that uses only its local data. The proposed multi-agent BO (MABO) framework adds a penalty term to traditional BO acquisition functions to account for coupling between the subsystems without data sharing. We derive a suitable form for this penalty term using alternating directions method of multipliers (ADMM), which enables the local decision-making problems to be solved in parallel (and potentially asynchronously). The effectiveness of the proposed MABO method is demonstrated on an intelligent transport system for fuel efficient vehicle platooning.

An Integrated Method of Bayesian Optimization and D-Optimal Design for Chemical Experiment Optimization

The smart chemical laboratory has recently emerged as a promising trend for future chemical research, where experiment optimization is of vital importance. The traditional Bayesian optimization (BO) algorithm focuses on exploring the dependent variable space while overlooking the independent variable space. Consequently, the BO algorithm suffers from becoming stuck at local optima, which severely deteriorates the optimization performance, especially with bad-quality initial points. Herein, we propose a novel stochastic framework of Bayesian optimization with D-optimal design (BODO) by integrating BO with D-optimal design. BODO can balance the exploitation in the dependent variable space and the exploration in the independent variable space. We highlight the excellent performance of BODO even with poor initial points on the benchmark alpine2 function. Meanwhile, BODO demonstrates a better average objective function value than BO on the benchmark Summit SnAr chemical process, showing its advantage in chemical experiment optimization and potential application in future chemical experiments.

CircuitBot: Learning to survive with robotic circuit drawing.

Robots with the ability to actively acquire power from surroundings will be greatly beneficial for long-term autonomy and to survive in uncertain environments. In this work, a scenario is presented where a robot has limited energy, and the only way to survive is to access the energy from an unregulated power source. With no wires or resistors available, the robot heuristically learns to maximize the input voltage on its system while avoiding potential obstacles during the connection. CircuitBot is a 6 DOF manipulator capable of drawing circuit patterns with graphene-based conductive ink, and it uses a state-of-the-art continuous/categorical Bayesian Optimization to optimize the placement of conductive shapes and maximize the energy it receives. Our comparative results with traditional Bayesian Optimization and Genetic algorithms show that the robot learns to maximize the voltage within the smallest number of trials, even when we introduce obstacles to ground the circuit and steal energy from the robot. As autonomous robots become more present, in our houses and other planets, our proposed method brings a novel way for machines to keep themselves functional by optimizing their own electric circuits.

COBALT: COnstrained Bayesian optimizAtion of computationaLly expensive grey-box models exploiting derivaTive information

Many engineering problems involve the optimization of computationally expensive models for which derivative information is not readily available. The Bayesian optimization (BO) framework is a particularly promising approach for solving these problems, which uses Gaussian process (GP) models and an expected utility function to systematically tradeoff between exploitation and exploration of the design space. BO, however, is fundamentally limited by the black-box model assumption that does not take into account any underlying problem structure. In this paper, we propose a new algorithm, COBALT, for constrained grey-box optimization problems that combines multivariate GP models with a novel constrained expected utility function whose structure can be exploited by state-of-the-art nonlinear programming solvers. COBALT is compared to traditional BO on seven test problems including the calibration of a genome-scale bioreactor model to experimental data. Overall, COBALT shows very promising performance on both unconstrained and constrained test problems.

Reinforcement Learning based Multi‐Step Look‐Ahead Bayesian Optimization

This paper considers the situation where data‐based optimization is to be performed but data sampling is limited due to high cost and time. Such situations demand highly efficient data‐sampling and utilization and Bayesian optimization (BO) is the most commonly used method as it allows users to balance between exploration and exploitation in deciding where to sample next in the design space. However, the standard acquisition functions used in Bayesian optimization such as the expected improvement have been criticized for being greedy and myopic in many situations. To address the limitation of the standard acquisition functions of BO due to its near-sighted nature, this paper suggests a novel reinforcement learning based method which enables multi‐step lookahead Bayesian optimization. Several benchmark functions are tested to compare the performance of the RL based method against the traditional BO methods using expected improvement and its rollout-based extensions. The proposed method outperformed popular Bayesian optimization methods in the case study.

Bayesian optimization with reference models: A case study in MPC for HVAC central plants

We present a framework for exploiting reference models in Bayesian optimization (BO). Our approach is motivated by a model predictive control (MPC) tuning application for central heating, ventilation, and air conditioning (HVAC) plants. Evaluating the closed-loop performance of MPC by trial-and-error is time-consuming (e.g., a closed-loop simulation can involve solving thousands of optimization problems). The BO algorithm can accelerate this process by treating the functional relationship between the closed-loop performance of MPC and its tuning parameters (e.g., constraint back off terms and objective weights) as a black-box model and by systematically navigating the parameter space to maximize MPC performance. Traditional BO algorithms have been recently developed to tune controllers but such frameworks tend not to use pre-existing (prior) system information. In this work, we propose to incorporate such information in the form of reference models and we show that such models can be constructed in creative ways. Specifically, we propose to construct a reference model by evaluating the closed-loop performance of a low-complexity MPC controller. We show that the use of a reference model changes the goal of BO from learning the objective function to learning the residual function (error between the reference and the objective function), which is a much easier task. Specifically, we show that the use of a reference model can help BO to more judiciously sample the parameter estimate and more rapidly discover regions where the solution exists. This feature significantly reduces the number of function evaluations and can help avoid getting stuck in local minima. The effectiveness of our BO framework is demonstrated using a central HVAC plant study with realistic data. The reference model is built by training a kriging model based on data collected for 21 low-fidelity, closed-loop MPC simulations. Using this reference model, we show that our BO method can find the optimal constraint back-off terms within 3 high-fidelity, closed-loop simulations (total computation time of 20 h). In contrast, a traditional BO approach without a reference model requires 14 high-fidelity, closed simulations (total time of 28 h). Our approach thus reduces the computational time by 28%.

Green machine learning via augmented Gaussian processes and multi-information source optimization

Searching for accurate machine and deep learning models is a computationally expensive and awfully energivorous process. A strategy which has been recently gaining importance to drastically reduce computational time and energy consumed is to exploit the availability of different information sources, with different computational costs and different “fidelity,” typically smaller portions of a large dataset. The multi-source optimization strategy fits into the scheme of Gaussian Process-based Bayesian Optimization. An Augmented Gaussian Process method exploiting multiple information sources (namely, AGP-MISO) is proposed. The Augmented Gaussian Process is trained using only “reliable” information among available sources. A novel acquisition function is defined according to the Augmented Gaussian Process. Computational results are reported related to the optimization of the hyperparameters of a Support Vector Machine (SVM) classifier using two sources: a large dataset—the most expensive one—and a smaller portion of it. A comparison with a traditional Bayesian Optimization approach to optimize the hyperparameters of the SVM classifier on the large dataset only is reported.

SMF-BO-2CoGP: A Sequential Multi-Fidelity Constrained Bayesian Optimization Framework for Design Applications

Abstract Bayesian optimization (BO) is an efiective surrogate-based method that has been widely used to optimize simulation-based applications. While the traditional Bayesian optimization approach only applies to single-fidelity models, many realistic applications provide multiple levels of fidelity with various computational complexity and predictive capability. In this work, we propose a multi-fidelity Bayesian optimization method for design applications with both known and unknown constraints. The proposed framework, called sMF-BO-2CoGP, is built on a multi-level CoKriging method to predict the objective function. An external binary classifier, which we approximate using a separate CoKriging model, is used to distinguish between feasible and infeasible regions. The sMF-BO-2CoGP method is demonstrated using a series of analytical examples, and a fiip-chip application for design optimization to minimize the deformation due to warping under thermal loading conditions.

Bayesian Optimization for Categorical and Category-Specific Continuous Inputs

Many real-world functions are defined over both categorical and category-specific continuous variables and thus cannot be optimized by traditional Bayesian optimization (BO) methods. To optimize such functions, we propose a new method that formulates the problem as a multi-armed bandit problem, wherein each category corresponds to an arm with its reward distribution centered around the optimum of the objective function in continuous variables. Our goal is to identify the best arm and the maximizer of the corresponding continuous function simultaneously. Our algorithm uses a Thompson sampling scheme that helps connecting both multi-arm bandit and BO in a unified framework. We extend our method to batch BO to allow parallel optimization when multiple resources are available. We theoretically analyze our method for convergence and prove sub-linear regret bounds. We perform a variety of experiments: optimization of several benchmark functions, hyper-parameter tuning of a neural network, and automatic selection of the best machine learning model along with its optimal hyper-parameters (a.k.a automated machine learning). Comparisons with other methods demonstrate the effectiveness of our proposed method.

Context knowledge-aided sparse bayesian learning method for bathymetric sonar

Sparse Bayesian learning has recently become successful in many compressed sensing problems. However, their performance critically relies on the appropriate tuning of numerous hyperparameters, because they directly control the statistical distribution and have a significant impact on the performance of the model being trained. Compared with manual tuning, grid search and random search, Bayesian optimization is a better hyperparameters intelligent optimization strategy. Nevertheless, traditional Bayesian optimization treats each problem as an independent black box. In this study, the hyperparameters tuning process is further improved because the contextual knowledge can be formulated into problem domain that is easier to optimize. For the bathymetric sonar, it is generally believed that the underwater acoustic environment changes slowly, which can be draw inspiration from the property and incorporated into the Bayesian optimization framework as additional prior information. This extension could yield efficient optimization without requiring too many iterative steps to converge. Simulation and tank experiment are conducted, and the results demonstrate that the proposed method has a lower computational cost in comparison with the conventional sparse Bayesian learning method.

Pre-training the deep generative models with adaptive hyperparameter optimization

The performance of many machine learning algorithms depends crucially on the hyperparameter settings, especially in Deep Learning. Manually tuning the hyperparameters is laborious and time consuming. To address this issue, Bayesian optimization (BO) methods and their extensions have been proposed to optimize the hyperparameters automatically. However, they still suffer from highly computational expense when applying to deep generative models (DGMs) due to their strategy of the black-box function optimization. This paper provides a new hyperparameter optimization procedure at the pre-training phase of the DGMs, where we avoid combining all layers as one black-box function by taking advantage of the layer-by-layer learning strategy. Following this procedure, we are able to optimize multiple hyperparameters in an adaptive way by using Gaussian process. In contrast to the traditional BO methods, which mainly focus on the supervised models, the pre-training procedure is unsupervised where there is no validation error can be used. To alleviate this problem, this paper proposes a new holdout loss, the free energy gap, which takes into account both factors of the model fitting and over-fitting. The empirical evaluations demonstrate that our method not only speeds up the process of hyperparameter optimization, but also improves the performances of DGMs significantly in both the supervised and unsupervised learning tasks.