Analytical Benchmark Problems for Multifidelity Optimization Methods

The paper presents a collection of analytical benchmark problems specifically selected to provide a set of stress tests for the assessment of multifidelity optimization methods. In addition, the paper discusses a comprehensive ensemble of metrics and criteria recommended for the rigorous and meaningful assessment of the performance of multifidelity strategies and algorithms

Efficient Algorithms for Computationally Expensive Multifidelity Optimization Problems

Multifidelity optimization problems refer to a class of problems where one is presented with a physical system or mathematical model that can be represented in different levels of fidelity. The term “fidelity” refers to the accuracy of representation, where higher fidelity estimates are more accurate and expensive, while lower fidelity estimates are inaccurate, albeit cheaper. Most common iterative solvers such as those employed in computational fluid dynamics (CFD), finite element analysis (FEA), computational electromagnetics (CEM) etc. can be run with different fine/course meshes or residual error thresholds to yield estimates in various fidelities. In the event an optimization exercise requires their use, it is possible to invoke analysis in various fidelities for different solutions during the course of search. Multifidelity optimization algorithms are the special class of algorithms that are able to deal with analysis in various levels of fidelity. In this thesis, two novel multifidelity optimization algorithms have been developed. The first is to deal with bilevel optimization problems and the second is to deal with robust optimization problems involving iterative solvers. Bilevel optimization problems are particularly challenging as the optimum of an upper level (UL) problem is sought subject to the optimality of a nested lower level (LL) problem. Due to the inherent nested nature, naive implementations consume very significant number of UL and LL evaluations. The proposed multifidelity approach controls the rigour of LL optimization exercise for any given UL solution during the course of search as opposed to undertaking exhaustive LL optimization for every UL solution. Robust optimization problems are yet another class of problems where numerous solutions need to be assessed since the intent is to identify solutions that have both good performance and is also insensitive to unavoidable perturbations in the variable values. Computing the latter metric requires evaluation of numerous solutions in the vicinity of the given solution and not all solutions are worthy of such computation. The proposed multifidelity approach considers pre-converged simulations as lower fidelity estimates and uses them to reduce the computational overhead. While multi-objective optimization problems have long been in existence, there has been limited attempts in the past to deal with problems where the objectives can be independently computed. For example, the weight of a structure and the maximum stress in the structure are two objectives that can be independently computed. For such classes of problems, an efficient algorithm should ideally evaluate either one or both objectives as opposed of always evaluating both objectives. A novel algorithm is introduced that is capable of selectively evaluating the objectives of the infill solutions. The approach exploits principles of non-dominance and sparse subset selection to facilitate decomposition and through maximization of probabilistic dominance (PD) measure, identifies the infill solutions. Thereafter, for each of these infill solutions, one or more objectives are evaluated based on evaluation status of its closest neighbor and the probability of improvement along each objective. Finally, there has been significant research interest in recent years to develop efficient algorithms to deal with multimodal, multi-objective optimization problems (MMOPs). Such problems are particulatly challenging as there is a need to identify well distributed and well converged solutions in the objective space along with diverse solutions in the variable space. Existing algorithms for MMOPs still require prohibitive number of function evaluations (often in several thousands). The algorithms are typically embedded with sophisticated, customized mechanisms that require additional parameters to manage the diversity and convergence in the variable and the objective spaces. A steady-state evolutionary algorithm is introduced in this thesis for solving MMOPs, with a simple design and no additional user-defined parameters that need tuning. All the developments listed above have been studied using well established benchmarks and real-world examples. The results have been compared with existing state-of-the-art approaches to substantiate the benefits

Branched Latent Neural Maps

We introduce Branched Latent Neural Maps (BLNMs) to learn finite dimensional input-output maps encoding complex physical processes. A BLNM is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNMs leverage latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent generalization properties with small training datasets and short training times on a single processor. Indeed, their generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections significantly reduce the number of tunable parameters. We show the capabilities of BLNMs in a challenging test case involving electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a 1D Purkinje network for fast conduction and a 3D heart-torso geometry. Specifically, we trained BLNMs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale and organ-level. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNM, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The resulting mean square error is on the order of $10^{-4}$ on a test dataset comprised of 50 electrophysiology simulations. In the online phase, the BLNM allows for 5000x faster real-time simulations of cardiac electrophysiology on a single core standard computer and can be used to solve inverse problems via global optimization in a few seconds of computational time

Democratizing machine learning

Modelle des maschinellen Lernens sind zunehmend in der Gesellschaft verankert, oft in Form von automatisierten Entscheidungsprozessen. Ein wesentlicher Grund dafür ist die verbesserte Zugänglichkeit von Daten, aber auch von Toolkits für maschinelles Lernen, die den Zugang zu Methoden des maschinellen Lernens für Nicht-Experten ermöglichen. Diese Arbeit umfasst mehrere Beiträge zur Demokratisierung des Zugangs zum maschinellem Lernen, mit dem Ziel, einem breiterem Publikum Zugang zu diesen Technologien zu er- möglichen. Die Beiträge in diesem Manuskript stammen aus mehreren Bereichen innerhalb dieses weiten Gebiets. Ein großer Teil ist dem Bereich des automatisierten maschinellen Lernens (AutoML) und der Hyperparameter-Optimierung gewidmet, mit dem Ziel, die oft mühsame Aufgabe, ein optimales Vorhersagemodell für einen gegebenen Datensatz zu finden, zu vereinfachen. Dieser Prozess besteht meist darin ein für vom Benutzer vorgegebene Leistungsmetrik(en) optimales Modell zu finden. Oft kann dieser Prozess durch Lernen aus vorhergehenden Experimenten verbessert oder beschleunigt werden. In dieser Arbeit werden drei solcher Methoden vorgestellt, die entweder darauf abzielen, eine feste Menge möglicher Hyperparameterkonfigurationen zu erhalten, die wahrscheinlich gute Lösungen für jeden neuen Datensatz enthalten, oder Eigenschaften der Datensätze zu nutzen, um neue Konfigurationen vorzuschlagen. Darüber hinaus wird eine Sammlung solcher erforderlichen Metadaten zu den Experimenten vorgestellt, und es wird gezeigt, wie solche Metadaten für die Entwicklung und als Testumgebung für neue Hyperparameter- Optimierungsmethoden verwendet werden können. Die weite Verbreitung von ML-Modellen in vielen Bereichen der Gesellschaft erfordert gleichzeitig eine genauere Untersuchung der Art und Weise, wie aus Modellen abgeleitete automatisierte Entscheidungen die Gesellschaft formen, und ob sie möglicherweise Individuen oder einzelne Bevölkerungsgruppen benachteiligen. In dieser Arbeit wird daher ein AutoML-Tool vorgestellt, das es ermöglicht, solche Überlegungen in die Suche nach einem optimalen Modell miteinzubeziehen. Diese Forderung nach Fairness wirft gleichzeitig die Frage auf, ob die Fairness eines Modells zuverlässig geschätzt werden kann, was in einem weiteren Beitrag in dieser Arbeit untersucht wird. Da der Zugang zu Methoden des maschinellen Lernens auch stark vom Zugang zu Software und Toolboxen abhängt, sind mehrere Beiträge in Form von Software Teil dieser Arbeit. Das R-Paket mlr3pipelines ermöglicht die Einbettung von Modellen in sogenan- nte Machine Learning Pipelines, die Vor- und Nachverarbeitungsschritte enthalten, die im maschinellen Lernen und AutoML häufig benötigt werden. Das mlr3fairness R-Paket hingegen ermöglicht es dem Benutzer, Modelle auf potentielle Benachteiligung hin zu über- prüfen und diese durch verschiedene Techniken zu reduzieren. Eine dieser Techniken, multi-calibration wurde darüberhinaus als seperate Software veröffentlicht.Machine learning artifacts are increasingly embedded in society, often in the form of automated decision-making processes. One major reason for this, along with methodological improvements, is the increasing accessibility of data but also machine learning toolkits that enable access to machine learning methodology for non-experts. The core focus of this thesis is exactly this – democratizing access to machine learning in order to enable a wider audience to benefit from its potential. Contributions in this manuscript stem from several different areas within this broader area. A major section is dedicated to the field of automated machine learning (AutoML) with the goal to abstract away the tedious task of obtaining an optimal predictive model for a given dataset. This process mostly consists of finding said optimal model, often through hyperparameter optimization, while the user in turn only selects the appropriate performance metric(s) and validates the resulting models. This process can be improved or sped up by learning from previous experiments. Three such methods one with the goal to obtain a fixed set of possible hyperparameter configurations that likely contain good solutions for any new dataset and two using dataset characteristics to propose new configurations are presented in this thesis. It furthermore presents a collection of required experiment metadata and how such meta-data can be used for the development and as a test bed for new hyperparameter optimization methods. The pervasion of models derived from ML in many aspects of society simultaneously calls for increased scrutiny with respect to how such models shape society and the eventual biases they exhibit. Therefore, this thesis presents an AutoML tool that allows incorporating fairness considerations into the search for an optimal model. This requirement for fairness simultaneously poses the question of whether we can reliably estimate a model’s fairness, which is studied in a further contribution in this thesis. Since access to machine learning methods also heavily depends on access to software and toolboxes, several contributions in the form of software are part of this thesis. The mlr3pipelines R package allows for embedding models in so-called machine learning pipelines that include pre- and postprocessing steps often required in machine learning and AutoML. The mlr3fairness R package on the other hand enables users to audit models for potential biases as well as reduce those biases through different debiasing techniques. One such technique, multi-calibration is published as a separate software package, mcboost

The stochastic aeroelastic response analysis of helicopter rotors using deep and shallow machine learning

This paper addresses the influence of manufacturing variability of a helicopter rotor blade on its aeroelastic responses. An aeroelastic analysis using finite elements in spatial and temporal domains is used to compute the helicopter rotor frequencies, vibratory hub loads, power required and stability in forward flight. The novelty of the work lies in the application of advanced data-driven machine learning (ML) techniques, such as convolution neural networks (CNN), multi-layer perceptron (MLP), random forests, support vector machines and adaptive Gaussian process (GP) for capturing the nonlinear responses of these complex spatio-temporal models to develop an efficient physics-informed ML framework for stochastic rotor analysis. Thus, the work is of practical significance as (i) it accounts for manufacturing uncertainties, (ii) accurately quantifies their effects on nonlinear response of rotor blade and (iii) makes the computationally expensive simulations viable by the use of ML. A rigorous performance assessment of the aforementioned approaches is presented by demonstrating validation on the training dataset and prediction on the test dataset. The contribution of the study lies in the following findings: (i) The uncertainty in composite material and geometric properties can lead to significant variations in the rotor aeroelastic responses and thereby highlighting that the consideration of manufacturing variability in analyzing helicopter rotors is crucial for assessing their behaviour in real-life scenarios. (ii) Precisely, the substantial effect of uncertainty has been observed on the six vibratory hub loads and the damping with the highest impact on the yawing hub moment. Therefore, sufficient factor of safety should be considered in the design to alleviate the effects of perturbation in the simulation results. (iii) Although advanced ML techniques are harder to train, the optimal model configuration is capable of approximating the nonlinear response trends accurately. GP and CNN followed by MLP achieved satisfactory performance. Excellent accuracy achieved by the above ML techniques demonstrates their potential for application in the optimization of rotors under uncertainty

SAFE: Scale-Adaptive Fitness Evaluation method for expensive optimization problems

The key challenge of expensive optimization problems (EOP) is that evaluating the true fitness value of the solution is computationally expensive. A common method to deal with this issue is to seek for a less expensive surrogate model to replace the original expensive objective function. However, this method also brings in model approximation error. To efficiently solve the EOP, a novel scale-adaptive fitness evaluation (SAFE) method is proposed in this article to directly evaluate the true fitness value of the solution on the original objective function. To reduce the computational cost, the SAFE method uses a set of evaluation methods (EM) with different accuracy scales to cooperatively complete the fitness evaluation process. The basic idea is to adopt the low-accuracy scale EM to fast locate promising regions and utilize the high-accuracy scale EM to refine the solution accuracy. To this aim, two EM switch strategies are proposed in the SAFE method to adaptively control the multiple EMs according to different evolutionary stages and search requirements. Moreover, a neighbor best-based evaluation (NBE) strategy is also put forward to evaluate the solution according to its nearest high-quality evaluated solution, which can further reduce computational cost. Extensive experiments are carried out on the case study of crowdshipping scheduling problem in the smart city to verify the effectiveness and efficiency of the proposed SAFE method, and to investigate the effects of the two EM switch strategies and the NBE strategy. Experimental results show that the proposed SAFE method achieves better solution quality than some baseline and state-of-the-art algorithms, indicating an efficient method for solving EOP with a better balance between solution accuracy and computational cost

Machine Learning in Aerodynamic Shape Optimization

Machine learning (ML) has been increasingly used to aid aerodynamic shape optimization (ASO), thanks to the availability of aerodynamic data and continued developments in deep learning. We review the applications of ML in ASO to date and provide a perspective on the state-of-the-art and future directions. We first introduce conventional ASO and current challenges. Next, we introduce ML fundamentals and detail ML algorithms that have been successful in ASO. Then, we review ML applications to ASO addressing three aspects: compact geometric design space, fast aerodynamic analysis, and efficient optimization architecture. In addition to providing a comprehensive summary of the research, we comment on the practicality and effectiveness of the developed methods. We show how cutting-edge ML approaches can benefit ASO and address challenging demands, such as interactive design optimization. Practical large-scale design optimizations remain a challenge because of the high cost of ML training. Further research on coupling ML model construction with prior experience and knowledge, such as physics-informed ML, is recommended to solve large-scale ASO problems

Development of numerical procedures for turbomachinery optimizaion

This Doctoral Thesis deals with high speed turbomachinery optimization and all those tools employed in the optimization process, mainly the optimization algorithm, the parameterization framework and the automatic CFD-based optimization loop. Optimization itself is not just a mean to improve the performance of a generic system, but can be a powerful instigator that helps gaining insight on the physic phenomena behind the observed improvements. As for the optimization engine, a novel surrogate-assisted (SA) genetic algorithm for multi-objective optimization problems, namely GeDEA-II-K, was developed. GeDEA-II-K is grounded on the cooperation between a genetic algorithm, namely GeDEA-II, and the Kriging methodology, with the aim at speeding up the optimization process by taking advantage of the surrogate model. The comparison over two- and three-objective test functions revealed the effectiveness of GeDEA-II-K approach. In order to carry out high speed turbomachinery optimizations, an automatic CFD-based optimization loop built around GeDEA-II-K was constructed. The loop was realized for a UNIX/Linux cluster environment in order to exploit the computational resources of parallel computing. Among the tools, a dedicated parameterization framework for 2D airfoils and 3D blades has been designed based on the displacement filed approach. The effectiveness of both the CFD-based automatic loop and the parameterization was verified on two real-life multi-objective optimization problems: the 2D shape optimization of a supersonic compressor cascade and the 3D shape optimization of the NASA Rotor 67. To better understand the outcomes of the optimization process, a wide section has been dedicated to supersonic flows and their behavior when forced to work throughout compressor cascades. The results obtained surely have demonstrated the effectiveness of the optimization approach, and even more have given deep insight on the physic of supersonic flows in the high speed turbomachinery applications that were studied