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Understanding stress distributions at grain boundaries in polycrystalline materials is crucial for predicting damaged nucleation sites. In high-purity materials, voids often nucleate at grain boundaries due to high stress from granular interactions and weakened atomic ordering. While traditional crystal plasticity models simulate grain-level mechanics, their high computational cost often prevents systematic identification of critical microstructural features and efficient forecast of extreme damage events. This paper addresses these challenges by developing a computationally efficient physics-assisted statistical modelling framework. The method starts by leveraging physical knowledge to hypothesize a broad set of microstructural factors influencing stress conditions. Causal inference is then applied to reveal the predominant features with physical explanations, leading to a parsimonious statistical model. A conditional Gaussian mixture model (CGMM) is employed when the identified relationship is utilized as a predictive model to quantify the uncertainty not readily explained by these features. Using body-centred cubic (BCC) tantalum as a representative material, a series of synthetic microstructures from single- to octu-crystal configurations are created. Results show that high-stress states strongly correlate with the elastic and plastic deformation capabilities and the directional misalignment of grain responses near boundaries. The statistical model achieves rapid and accurate forecasts, demonstrating its potential for analysing realistic polycrystalline materials. 

Discovering the underlying dynamics of complex systems from data is an important practical topic. Constrained optimization algorithms are widely utilized and lead to many successes. Yet, such purely data-driven methods may bring about incorrect physics in the presence of random noise and cannot easily handle the situation with incomplete data. In this paper, a new iterative learning algorithm for complex turbulent systems with partial observations is developed that alternates between identifying model structures, recovering unobserved variables, and estimating parameters. First, a causality-based learning approach is utilized for the sparse identification of model structures, which takes into account certain physics knowledge that is pre-learned from data. It has unique advantages in coping with indirect coupling between features and is robust to stochastic noise. A practical algorithm is designed to facilitate causal inference for high-dimensional systems. Next, a systematic nonlinear stochastic parameterization is built to characterize the time evolution of the unobserved variables. Closed analytic formula via efficient nonlinear data assimilation is exploited to sample the trajectories of the unobserved variables, which are then treated as synthetic observations to advance a rapid parameter estimation. Furthermore, the localization of the state variable dependence and the physics constraints are incorporated into the learning procedure. This mitigates the curse of dimensionality and prevents the finite time blow-up issue. Numerical experiments show that the new algorithm identifies the model structure and provides suitable stochastic parameterizations for many complex nonlinear systems with chaotic dynamics, spatiotemporal multiscale structures, intermittency, and extreme events. 

Predicting the process of porosity-based ductile damage in polycrystalline metallic materials is an essential practical topic. Ductile damage and its precursors are represented by extreme values in stress and material state quantities, the spatial probability density function (PDF) of which are highly non-Gaussian with strong fat tails. Traditional deterministic forecasts utilizing sophisticated continuum-based physical models generally lack in representing the statistics of structural evolution during material deformation. Computational tools which do represent complex structural evolution are typically expensive. The inevitable model error and the lack of uncertainty quantification may also induce significant forecast biases, especially in predicting the extreme events associated with ductile damage. In this paper, a data-driven statistical reduced-order modeling framework is developed to provide a probabilistic forecast of the deformation process of a polycrystal aggregate leading to porosity-based ductile damage with uncertainty quantification. The framework starts with computing the time evolution of the leading few moments of specific state variables from the spatiotemporal solution of full- field polycrystal simulations. Then a sparse model identification algorithm based on causation entropy, including essential physical constraints, is utilized to discover the governing equations of these moments. An approximate solution of the time evolution of the PDF is obtained from the predicted moments exploiting the maximum entropy principle. Numerical experiments based on polycrystal realizations of a representative body-centered cubic (BCC) tantalum illustrate a skillful reduced-order model in characterizing the time evolution of the non-Gaussian PDF of the von Mises stress and quantifying the probability of extreme events. The learning process also reveals that the mean stress is not simply an additive forcing to drive the higher-order moments and extreme events. Instead, it interacts with the latter in a strongly nonlinear and multiplicative fashion. In addition, the calibrated moment equations provide a reasonably accurate forecast when applied to the realizations outside the training data set, indicating the robustness of the model and the skill for extrapolation. Finally, an information-based measurement is employed to quantitatively justify that the leading four moments are sufficient to characterize the crucial highly non-Gaussian features throughout the entire deformation history considered.