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Coarse graining techniques play an essential role in accelerating molecular simulations of systems with large length and time scales. Theoretically grounded bottom-up models are appealing due to their thermodynamic consistency with the…

In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational…

In this paper, we develop a multiscale finite element method for solving flows in fractured media. Our approach is based on Generalized Multiscale Finite Element Method (GMsFEM), where we represent the fracture effects on a coarse grid via…

In this paper, we propose a novel multiscale model reduction strategy tailored to address the Poisson equation within heterogeneous perforated domains. The numerical simulation of this intricate problem is impeded by its multiscale…

Deep neural networks can be roughly divided into deterministic neural networks and stochastic neural networks.The former is usually trained to achieve a mapping from input space to output space via maximum likelihood estimation for the…

In Hezaveh et al. 2017 we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational lensing systems. Here we demonstrate a method for…

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a…

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space-time…

The automated construction of coarse-grained models represents a pivotal component in computer simulation of physical systems and is a key enabler in various analysis and design tasks related to uncertainty quantification. Pertinent methods…

In this paper, we present a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis…

Assessing the predictive uncertainty of deep neural networks is crucial for safety-related applications of deep learning. Although Bayesian deep learning offers a principled framework for estimating model uncertainty, the common approaches…

Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic…

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying…

Uncertainty quantification for deep learning is a challenging open problem. Bayesian statistics offer a mathematically grounded framework to reason about uncertainties; however, approximate posteriors for modern neural networks still…

Recent studies have shown that ensemble approaches could not only improve accuracy and but also estimate model uncertainty in deep learning. However, it requires a large number of parameters according to the increase of ensemble models for…

Deep learning is a powerful tool to represent subgrid processes in climate models, but many application cases have so far used idealized settings and deterministic approaches. Here, we develop stochastic parameterizations with calibrated…

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use…

Machine learning can significantly improve performance for decision-making under uncertainty across a wide range of domains. However, ensuring robustness guarantees requires well-calibrated uncertainty estimates, which can be difficult to…

Statistical (machine learning) tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning…

Deep Ensembles, as a type of Bayesian Neural Networks, can be used to estimate uncertainty on the prediction of multiple neural networks by collecting votes from each network and computing the difference in those predictions. In this paper,…