Uncertainty Quantification for Material Science and Engineering

Much of materials science research is a “small data” problem – an experimental or computational result may contain many data points, perhaps an entire space or time field of outputs, but there might only be several dozens of these outputs. So materials scientists cannot rely on techniques of analysis that are data hungry, requiring hundreds of thousands of training datasets. We discuss recent ideas in employing Gaussian process surrogate models, a methodology that provides good predictive capability based on relatively modest data needs, and which comes with objective measures of credibility in those predictions. In particular, we discuss how Gaussian processes are being extended to handle fields of inputs and/or outputs.

This work presents a parameter-free method for estimating materials novelty along chemical and structural axes using mutual information informed density functions. The approach quantifies novelty by analyzing how MI changes with distance between materials, establishing objective criteria for determining meaningful neighborhoods without requiring predetermined parameters. We demonstrate the method's effectiveness using two case studies: a control dataset of materials with varying degrees of similarity and a practical application analyzing lithium-containing compounds from the GNOME dataset relative to known materials. The method successfully identifies meaningful patterns of novelty in both chemical and structural domains while providing interpretable results that align with materials science intuition. This framework offers researchers a quantitative tool for assessing candidate materials against existing knowledge bases and could support more informed selection of synthesis targets in materials discovery campaigns.

The sublimation enthalpy, Δ⁢Hsub, is a key thermodynamic parameter governing the phase transformation of a substance between its solid and gas phases. This transformation is at the core of many important materials' purification, deposition, and etching processes. While Δ⁢Hsub can be measured experimentally and estimated computationally, these approaches have their own different challenges. Here, we develop a machine learning (ML) approach to rapidly predict Δ⁢Hsub from data generated using density functional theory (DFT). We further demonstrate how combining ML and DFT methods with active learning can be efficient in exploring the materials space, expanding the coverage of the computed dataset, and systematically improving the ML predictive model of Δ⁢Hsub. With an error of ∼15 kJ/mol in instantaneous predictions of Δ⁢Hsub, the ML model developed in this work will be useful for the community.

Ti/TiN coatings are utilized in a wide variety of engineering applications due to their superior properties such as high hardness and toughness. Doping Al into Ti/TiN can also enhance properties and lead to even higher performance. Therefore, studying the atomic-level behavior of the TiAl/TiAlN interface is important. However, due to the large number of possible combinations for the 50 mol% Al-doped Ti/TiN system, it is time-consuming to use the DFT-based Monte Carlo method to find the optimal TiAl/TiAlN system with a high work of adhesion. In this study, we use a graph convolutional neural network as an interatomic potential, combined with reinforcement learning, to improve the efficiency of finding optimal structures with a high work of adhesion. By inspecting the features of structures in neural networks, we found that the optimal structures follow a certain pattern of doping Al near the interface. The electronic structure and bonding analysis indicate that the optimal TiAl/TiAlN structures have higher bonding strength. We

Material properties are governed by both chemical composition and microstructure. With advancements in additive manufacturing, we can now engineer metamaterials that tune properties through microstructural design, without altering composition or fabrication parameters. By assembling metamaterial microstructures aperiodically, we create heterogeneous metamaterial systems (HMS) capable of accommodating spatially varying property requirements. Despite their potential, designing HMS remains challenging due to the high-dimensional design space, complex structure-property relationships, mixed-type design variables, and multiscale interactions. In this talk, we will introduce data-driven frameworks that accelerate the multiscale design of HMS. We will cover database generation, surrogate modeling, generative unit-cell design, and multiscale optimization, all centered around latent variable representations. We will present the development of various latent variable models, such as latent variable Gaussian processes and variational autoencoders, to learn low-dimensional, interpretable representations of complex microstructures, enabling effective cross-scale property modeling and efficient inverse design. By integrating these models into multiscale topology optimization, we develop data-driven frameworks that simultaneously design macro- and microstructures in HMS, enabling precise control over spatial property distributions. They provide unprecedented design flexibility and efficiency, enabling applications in deformation and vibration control, fracture resistance, and strain cloaking.

In this talk, we explore cutting-edge advancements in computational statistics and machine learning to tackle key challenges in contemporary materials science. At the core of this discussion is the grand challenge of multi-scale, multi-physics materials design, with far-reaching implications for industries such as manufacturing, aerospace, defense, and energy. We will cover recent developments in multi-fidelity and forward uncertainty quantification using polynomial chaos expansion, Bayesian optimization leveraging various Gaussian process regression methods, and stochastic modeling for microstructure evolution. Additionally, we will discuss reduced-order modeling for crystal plasticity finite element methods, generative models for large-scale microstructure reconstruction, and applications in additive manufacturing. The talk will conclude with a forward-looking perspective on multi-scale materials digital twins and their role in future advancements.

Multidisciplinary concurrent materials, geometry and manufacturing process optimization involves many computational challenges such as high-dimensionality associated with location dependency, material heterogeneity, multi-modal information, and nonlinear material behaviors such as large deformations and plasticity. The recent growth of using physics-based machine learning creates opportunities for incorporating data-driven methodologies with physical models into design. Furthermore, digital twin is an emerging technology in the era of Industry 4.0 that holds promises for real time optimization of manufacturing processes and quality control. We will present in this talk a digital twin framework with uncertainty quantification that facilitates a bidirectional information exchange between virtual and physical systems in complex manufacturing processes. Using laser directed-energy deposition (DED) as an example in additive manufacturing (AM), we will first present the development of a time-series machine learning (ML) model of DED process to predict temperatures across various spatial locations of the DED-built part while taking dynamic processing conditions as inputs. With the uncertainty quantification using Monte Carlo dropout methods and a reduced dimensional representation, we introduce a Bayesian Optimization (BO) method for Time Series Process Optimization. We will then present a simultaneous multi-step Robust Model Predictive Control (R-MPC) framework for real-time decision-making, using a multi-variate deep neural network (DNN), Time-Series Dense Encoder (TiDE), as the surrogate model, and quantile prediction to account for uncertainty associated with process disturbances. TiDE allows one-shot forward propagation and auto-differentiation for rapid decisions therefore proactive control over melt pool temperatures, while mitigating porosity defects by regulating laser power to maintain melt pool depth. Overall, the proposed R-MPC framework offers as a powerful tool for future Digital Twin applications and real-time

Engineering, material science, and biological models generally have a number of parameters which are nonidentifiable in the sense that they are not uniquely determined by measured responses. Furthermore, the computational cost of high-fidelity simulation codes often precludes their direct use for Bayesian model calibration and uncertainty propagation. In this presentation, we will discuss techniques to isolate influential parameters for subsequent surrogate model construction, Bayesian inference and uncertainty propagation. For parameter selection, we will discuss advantages and shortcomings of global sensitivity analysis to isolate influential inputs and detail the use of parameter subset selection and active subspace techniques as an alternative. We will also discuss the manner in which Bayesian calibration on active subspaces can be used to quantify uncertainties in physical parameters. These techniques will be illustrated for models arising in nuclear power plant design and quantitative systems pharmacology (QSP), as well as models for transductive materials

Virtually every real-world materials optimization task involves considering multiple properties of interest, weighing trade-offs of between experiment value and cost, and optimizing many tunable parameters at once. While traditional design of experiments is often used, Bayesian optimization is a dramatically more efficient alternative in many cases. Basic and advanced topics related to Bayesian Optimization will be discussed (as well as specific application examples such as high-dimensional optimization). Applying state-of-the-art algorithms to materials tasks isn’t trivial, even for veteran materials informatics practitioners. Additionally, Python libraries can be cumbersome to learn and use serving as a barrier to entry for interested users. To address these challenges, we also present Honegumi, an interactive script generator for materials-relevant Bayesian optimization using the Ax Platform.

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.

Microstructures of grains are stochastic in nature and evolve throughout the life time of processing and usage. Different models such as molecular dynamics, phase field, and kinetic Monte Carlo have been developed to simulate and predict the grain evolution. Physics-based simulations are computationally expensive for large material systems. Pure data-driven models require a large amount of training data and lack the extrapolation capability. In this work, a physics-constrained Bayesian neural network (PCBNN) is proposed to predict the probabilistic distributions of grains during the evolution. Two PCBNN models are developed. One is based on the Hamiltonian in Potts models as the physical loss, the other is based on the Fokker-Planck equation that describes the evolution of two-point correlations. The results show that the proposed physics-informed machine learning approach helps alleviate the data sparsity issue and improve the prediction accuracy.

We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic behavior with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system. The proposed method is applied to a simple one-dimensional system to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.