Data-driven learning is generalized to consider history-dependent multi-fidelity data, while quantifying epistemic uncertainty and disentangling it from data noise (aleatoric uncertainty). This generalization is hierarchical and adapts to different learning scenarios: from training the simplest single-fidelity deterministic neural networks up to the proposed multi-fidelity variance estimation Bayesian recurrent neural networks. The proposed methodology is demonstrated by applying it to different data-driven constitutive modeling scenarios for history-dependent plasticity of elastoplastic biphasic materials that include multiple fidelities with and without aleatoric uncertainty (noise). The method accurately predicts the response and quantifies model error while also discovering the noise distribution (when present). The versatility and generality of the proposed method open opportunities for future real-world applications in diverse scientific and engineering domains; especially, the most challenging cases involving design and analysis under uncertainty.

Machine learning delivers strong predictive performance in scientific and engineering tasks when high-fidelity data are abundant. Yet, real-world models seldom quantify aleatoric (data) and epistemic (model) uncertainties, leading to overfitting on noisy inputs. In addition, collecting adequate high-fidelity data is often expensive or infeasible, whereas low-fidelity data are more accessible but less reliable. To address these challenges, this thesis proposes a general multi-fidelity Bayesian learning framework that enables trustworthy uncertainty disentanglement, and extends its application to constitutive modeling and design of recycled composite materials.

The thesis begins with an introduction to regression from a Bayesian perspective, establishing the connection between deterministic and probabilistic treatments, and covering models from linear to kernel and deep neural network regressions. This chapter provides the theoretical foundation for the remainder of the thesis by formalizing inference techniques, uncertainty quantification strategies, and model evaluation metrics.

Two core machine learning methods are developed to tackle the challenges of uncertainty estimation and multi-fidelity data fusion. First, a cooperative training scheme is proposed that combines a variance estimation neural network with a Bayesian mean neural network, enabling explicit disentanglement of aleatoric and epistemic uncertainties while improving mean prediction. The approach demonstrates strong scalability and generality across tasks and network architectures. Second, a practical multi-fidelity Bayesian learning framework is introduced, which fuses low- and high-fidelity data via a deterministic model, a transfer-learning module, and a Bayesian residual learner. This architecture balances expressiveness and computational efficiency, yielding robust predictions in both data-scarce and data-rich regimes.

To support data-driven mechanics, the proposed framework is extended to generalized constitutive modeling of history-dependent materials. A hierarchical learning scheme is developed that spans from single-fidelity deterministic networks to multi-fidelity Bayesian recurrent neural networks. This addresses two major limitations in data-driven modeling: the reliance on large, clean datasets and the lack of interpretability in neural network predictions.

Finally, the methodology is applied to the sustainable design of recycled composite polymers, where uncertainty arises from microstructural variability and the use of compatibilizers. Leveraging the cooperative training framework, both aleatoric and epistemic uncertainties are quantified, and a novel polymer design is optimized to achieve better expected performance and lower data variation.

Together, this thesis presents a unified and scalable framework for Bayesian learning under uncertainty and multi-fidelity conditions, with broad applicability in scientific computing, materials modeling, and sustainable engineering. The thesis concludes by summarizing key findings, discussing current limitations, and outlining future research directions.

The sampling method is an important numerical technique for solving reliability problems in engineering systems. However, the evaluation of the failure probability using classical sampling methods is time-consuming for complex engineering structure. To address this issue, this paper proposes a gradient optimization assisted bubble sampling method (GOBSM) to reduce the computational costs, which enhances the coverage range of bubbles, thereby improving the computational efficiency without sacrificing the accuracy. Furthermore, the bubble gradient iterative algorithm is developed to efficiently construct bubbles. Eight complex numerical examples are tested for assessing the failure probability, and the results demonstrate the performance of GOBSM.

This paper proposes a new active learning method named as optimum-pursuing method (OPM) from the viewpoint of optimization theory, which aims to provide an effective tool for solving constrained optimization and reliability-based design optimization (RBDO) problems with low computation cost. It uses the cheap Kriging metamodel to replace the expensive physical response. The novelty of the proposed OPM primarily lies in two aspects. First, the OPM utilizes the advantage of the optimization theory rather than sampling technology. By using the augmented Lagrangian approach, it comprehensively considers the objective, constraints, and their relations, thereby automatic identification of important region in the vicinity of the optimum. Second, the accordingly optimum-pursuing function consists of three parts: Kriging mean, Kriging standard deviation, and merit function. Also, the target reliability surface is further considered to enhance the local accuracy of the reliability analysis. The performance of OPM is tested for both deterministic optimization and problems, in which two mathematical and three real-world engineering examples are selected to showcase the feasibility and validity. The results demonstrate that OPM is promising for solving both deterministic optimization and RBDO problems by comparing with the well-known active learning methods.

Multi-objective Stochastic Linear bandit (MOSLB) plays a critical role in the sequential decision-making paradigm, however, most existing methods focus on the Pareto dominance among different objectives without considering any priority. In this paper, we study bandit algorithms under mixed Pareto-lexicographic orders, which can reflect decision makers’ preferences. We adopt the Grossone approach to deal with these orders and develop the notion of Pareto-lexicographic optimality to evaluate the learners’ performance. Our work represents a first attempt to address these important and realistic orders in bandit algorithms. To design algorithms under these orders, the upper confidence bound (UCB) policy and the prior free lexicographical filter are adapted to approximate the optimal arms at each round. Moreover, the framework of the algorithms involves two stages in pursuit of the balance between exploration and exploitation. Theoretical analysis as well as numerical experiments demonstrate the effectiveness of our algorithms.

Data-driven modeling in mechanics is evolving rapidly based on recent machine learning advances, especially on artificial neural networks. As the field matures, new data and models created by different groups become available, opening possibilities for cooperative modeling. However, artificial neural networks suffer from catastrophic forgetting, i.e. they forget how to perform an old task when trained on a new one. This hinders cooperation because adapting an existing model for a new task affects the performance on a previous task trained by someone else. The authors developed a continual learning method that addresses this issue, applying it here for the first time to solid mechanics. In particular, the method is applied to recurrent neural networks to predict history-dependent plasticity behavior, although it can be used on any other architecture (feedforward, convolutional, etc.) and to predict other phenomena. This work intends to spawn future developments on continual learning that will foster cooperative strategies among the mechanics community to solve increasingly challenging problems. We show that the chosen continual learning strategy can sequentially learn several constitutive laws without forgetting them, using less data to achieve the same error as standard (non-cooperative) training of one law per model.

In this paper, a variable-fidelity constrained lower confidence bound (VF-CLCB) criterion is presented for computationally expensive constrained optimization problems (COPs) with two levels of fidelity. In VF-CLCB, the hierarchical Kriging model is adopted to model the objective and inequality constraints. Two infill sampling functions are developed based on the objective and the constraints, respectively, and an adaptive selection strategy is set to select the elite sample points. Moreover, based on the VF-CLCB criterion, a parallel optimization method noted as PVF-CLCB is subsequently developed to accelerate the optimization process. In PVF-CLCB, a VF influence function is defined to approximately evaluate the estimation error of the hierarchical Kriging models, based on which multiple promising points can be determined at each iteration. In addition, an allocation strategy is proposed to distribute the computation resources between the objective- and constraint-oriented functions properly. Lastly, the proposed VF-CLCB and PVF-CLCB approaches are compared with the alternative methods on 12 benchmark numerical cases, and their significant superiority in solving computationally expensive COPs is verified. Furthermore, the proposed methods are employed to optimize the global stability of the stiffened cylindrical shell, and the optimum structure is yielded.