Computational modeling of heterogeneous materials increasingly relies on multiscale simulations, which typically leverage homogenization theory for scale coupling. These simulations are prohibitively expensive and memory-intensive, especially when modeling damage and fracture in large 3D components such as cast metallic alloys. To address these challenges, this thesis presents a physics-constrained deep learning model that surrogates microscale simulations within a mechanistic data-driven framework, accurately predicting effective microstructural responses under irreversible elasto-plastic hardening and softening deformations. The model's architecture is based on damage mechanics, with a new loss component to enhance thermodynamical consistency. Using mechanistic reduced-order models (ROMs) to generate training data, it achieves high accuracy on unseen deformation paths and can be embedded in 3D multiscale simulations with fracture. Numerical experiments show that the model presented in this thesis is more accurate than pure data-driven models and more efficient than mechanistic reduced-order models.

Extending the application of deep learning models in computational mechanics, this thesis also explores their use in unsupervised settings where no labeled data is available. Deep neural networks (DNNs) as physics-informed models are increasingly used to solve partial differential equations (PDEs) that naturally arise while modeling a wide range of systems and physical phenomena. However, the accuracy of such DNNs decreases as the PDE complexity increases and they also suffer from spectral bias as they tend to learn the low-frequency solution characteristics. To address these issues, Parametric Grid Convolutional Attention Networks (PGCANs) are introduced in this work that can solve PDE systems without leveraging any labeled data in the domain. The main idea of PGCAN is to parameterize the input space with a grid-based encoder whose parameters are connected to the output via a DNN decoder that leverages attention to prioritize feature training. The encoder provides a localized learning ability and uses convolution layers to avoid overfitting and improve information propagation rate from the boundaries to the interior of the domain. The performance of PGCAN on a wide range of PDE systems are tested and show that it effectively addresses spectral bias and provides more accurate solutions compared to competing methods.

Emulation or surrogate modeling is an indispensable part of many scientific and engineering disciplines. However, most emulators such as Gaussian processes (GPs) or deep neural networks (DNNs) are exclusively developed for interpolation/regression and have limited generalization power. In this thesis we present two approaches for improved and scalable emulation of physical systems.

In the context of finding governing dynamics for a physical system, we introduce evolutionary Gaussian processes (EGPs). EGPs are a systematic integration of GPs and evolutionary programming (EP) that boost the performance of GPs by the automatic discovery of free-form symbolic bases that regress the data. As we demonstrate via examples that include a host of analytical functions as well as an engineering problem on materials modeling, EGP can improve the performance of ordinary GPs in terms of not only extrapolation, but also interpolation/regression and numerical stability.

In the context of surrogating solvers for PDEs, we introduce a transferable framework for solving intial conditions and boundary value problems (IBVPs) via DNNs which can be trained once and used forever for various domains of unseen sizes, shapes, and boundary conditions. We present the genomic flow network (GFNet), a neural network that can infer the solution of an IBVP subject to arbitrary initial and boundary conditions on a square domain called genome. Furthermore, we propose mosaic flow (MF) predictor, a novel iterative algorithm that assembles the GFNet's inferences for IBVPs on large domains with unseen sizes and shapes while preserving the spatial regularity of the solution.

In many applications in engineering and sciences analysts have simultaneous access to multiple data sources. In such cases, the overall cost of acquiring information can be reduced via data fusion or multi-fidelity (MF) modeling where one leverages inexpensive low-fidelity (LF) sources to reduce the reliance on expensive high-fidelity (HF) data. In this thesis we present two main contributions in the field of data fusion and its application in engineering. In particular, we introduce: (1) a novel neural network (NN) architecture for data fusion under uncertainty, namely Probabilistic Neural Data Fusion (Pro-NDF), and (2) a MF calibration scheme based on Latent Map Gaussian Processes (LMGPs) for fracture modelling of metallic components via reduced-order models (ROMs). In the context of building an emulator for data fusion under uncertainty, we introduce Pro-NDF, a novel NN architecture that converts MF modeling into a nonlinear manifold learning problem. Pro-NDF inversely learns non-trivial (e.g., non-additive and nonhierarchical) biases of the LF sources in an interpretable and visualizable manifold where each data source is encoded via a low-dimensional distribution. This probabilistic manifold quantifies model form uncertainties such that LF sources with small bias are encoded close to the HF source. Through a set of analytic and engineering examples, we demonstrate that our approach provides a high predictive power while quantifying various sources of uncertainty.

In the context of fracture modelling of metallic materials with microscopic pores, we propose a data-driven framework that integrates a mechanistic ROM with a MF calibration scheme based on LMGPs. The proposed ROM offers computational speedup compared to direct numerical simulations (DNS) by solving a reduced-order representation of the governing equations and reducing the degrees of freedom via clustering. Since clustering affects local strain fields and hence the fracture response, we employ LMGPs to calibrate the damage parameters of an ROM as a function of microstructure and clustering, i.e., fidelity level, such that the ROM (i.e., LF source) faithfully emulates DNS (i.e., HF source). We demonstrate the application of our MF framework in predicting the damage behavior of a multiscale metallic component with spatially varying porosity. Our results indicate that microstructural porosity can significantly affect the performance of macro-components and hence must be considered in the design process.

Gaussian processes (GPs) are ubiquitously used in science and engineering as metamodels. Standard GPs, however, can only handle numerical or quantitative variables. In this paper, we introduce latent map Gaussian processes (LMGPs) that inherit the attractive properties of GPs but are also applicable to mixed data that have both quantitative and qualitative inputs. The core idea behind LMGPs is to learn a low-dimensional manifold where all qualitative inputs are represented by some quantitative features. To learn this manifold, we first assign a unique prior vector representation to each combination of qualitative inputs. We then use a linear map to project these priors on a manifold that characterizes the posterior representations. As the posteriors are quantitative, they can be used in any standard correlation function such as the Gaussian. Hence, the optimal map and the corresponding manifold can be efficiently learned by maximizing the Gaussian likelihood function. Through a wide range of analytical and real-world examples, we demonstrate the advantages of LMGPs over state-of-the-art methods in terms of accuracy and versatility. In particular, we show that LMGPs can handle variable-length inputs and provide insights into how qualitative inputs affect the response or interact with each other. We also provide a neural network interpretation of LMGPs and study the effect of prior latent representations on their performance. Lastly, we demonstrate that LMGP can be applied to data assimilation problems in which data from multiple sources (e.g., simulations, mathematical models and real-world experiments) are fused to improve prediction performance.

Inverse problems, which involve identifying the system input(s) which produce desired output(s), encompass a wide class of tasks which are of great interest in scientific and engineering applications. Tackling such problems is challenging and requires leveraging all available information about the system, which in such applications typically consists of high-fidelity (HF) data, such as direct observations or experiments, which is expensive to obtain, and low-fidelity (LF) data, such as computer models or physics-based simulators, which is cheaper to obtain but is less accurate and often comes with additional parameters separate from the system inputs which must be tuned. A common approach to facilitate inverse problems in such contexts is multi-fidelity modeling (MFM), which comprises a wide variety of techniques that fuse data from multiple sources and may also calibrate (i.e., inversely estimate the tuning parameters of) the LF sources (we refer to these tasks collectively as data fusion). However, tackling inverse problems via modern MFM techniques is challenging and often requires the use of domain-specific frameworks. Additionally, existing MFM techniques are limited in the number of data sources they can incorporate or calibrate, the types of data they can process, and do not quantify uncertainty (which is especially important in inverse problems and in low-data contexts) or otherwise aid in system identification beyond mimicking the input-output behavior. In this dissertation, we develop data fusion approaches which address these gaps by leveraging manifold learning to facilitate data fusion and aid in uncertainty quantification and system identification. The contributions of this work comprise four research tasks which we briefly discuss below.

Existing data fusion approaches are typically limited in the number of sources they can accommodate and often do not provide tools for uncertainty quantification or system identification. My first contribution consists of developing data fusion via latent map Gaussian processes (LMGPs), which address this by recasting data fusion as a latent space learning problem and are applicable to a wide variety of problems. Relationships between data sources are learned and represented in a low-dimensional manifold which aids in learning, and provides a visualization of the relationships between the HF and LF sources, and enables handling an arbitrary number of data sources. The kernel is also extended to enable calibration.

My second contribution consists of demonstrating LMGP's abilities by inversely designing microstructures to be used for evaporative porous cooling, which is a challenging problem that balances competing objectives of fluid flow through pores and heat transfer through the solid phase. Microstructures are typically represented as voxel arrays which are too high-dimensional to design directly, and can be rendered at a range of sizes with cost and accuracy of property extraction increasing with resolution. To address this, we develop parameterizations via spectral density functions or SDFs which represent structures via a tractable number of parameters, and explore these parameters to generate microstructures at various resolutions which are then fused via LMGPs, providing a balance between cost and accuracy. The fitted LMGPs are then used in a multi-objective optimization loop to find a Pareto front of designs which maximize the objectives relevant to cooling performance.

While LMGPs are a powerful data fusion approach, the are based on GPs which do not scale well to large or high-dimensional data. My third contribution is the development of probabilistic neural data fusion or Pro-NDF, which employs the same manifold learning concepts behind LMGPs using neural networks (NNs) instead of GPs as they provide improved flexibility and the ability to scale to large data. Pro-NDF splits learning tasks into blocks which learn low-dimensional manifolds which represent the relationships between data sources and categorical variables. It employs probabilistic BNNs to learn the source manifold and represents the output as a parameterized distribution, quantifying and separating uncertainties, and is trained via a novel loss function incorporating a proper scoring rule. Pro-NDF's flexibility and scalability are demonstrated on analytic examples and real-world datasets.

Unlike LMGP, Pro-NDF only enables MFM and not calibration. I address this through my fourth contribution, which develops probabilistic neural calibration or Pro-NC which is an extension of Pro-NDF with greatly improved capabilities that allows simultaneous calibration of any number of LF sources. Pro-NDF employs manifolds to learn the relationships between data sources and categorical combinations as well as the effects of the calibration inputs on the LF sources. It represents learned calibration parameters and the output via distributions, again providing separable uncertainty estimates. This work also extends the loss function for Pro-NDF to enable calibration and develops an algorithm for weighting loss terms that facilitates accurate multi-task learning. Pro-NC is validated on a number of analytic examples, and will be applied to a complex material science problem in the future.

Bayesian optimization (BO) is increasingly employed in critical applications such as materials design and drug discovery. An increasingly popular strategy in BO is to forgo the sole reliance on high-fidelity data and instead use an ensemble of information sources which provide inexpensive low-fidelity data. The overall premise of this strategy is to reduce the total sampling costs by querying inexpensive low-fidelity sources whose data are correlated with high-fidelity samples. In this thesis, we propose a multi-fidelity cost-aware BO framework that significantly outperforms the state-of-the-art technologies in terms of efficiency, consistency, and robustness. We demonstrate the advantages of our framework on analytic and engineering problems and argue that these benefits stem from our two main contributions: (1) we develop a novel acquisition function for multi-fidelity cost-aware BO that safeguards the convergence against the biases of low-fidelity data, and (2) we tailor a newly developed emulator for multi-fidelity BO which enables us to not only simultaneously learn from an ensemble of multi-fidelity datasets, but also identify the severely biased low-fidelity sources that should be excluded from BO.