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This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…
In this paper we focus on learning optimized partial differential equation (PDE) models for image filtering. In this approach, the grey-scaled images are represented by a vector field of two real-valued functions and the image restoration…
Design-space dimensionality reduction is essential to mitigate the cost of high-fidelity simulation-based optimization, especially when dealing with high-dimensional geometric parameterizations. Traditional linear techniques, such as…
In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well…
Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a…
Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics.…
The paper addresses Bayesian inferences in inverse problems with uncertainty quantification involving a computationally expensive forward map associated with solving a partial differential equations. To mitigate the computational cost, the…
This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for…
We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDEs) driven by space-time noise, for multiplicative and additive noise. We examine convergence of…
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based…
We present and analyze a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a…
Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…
The Bayesian uncertainty quantification technique has become well established in turbulence modeling over the past few years. However, it is computationally expensive to construct a globally accurate surrogate model for Bayesian inference…
Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these…
Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically enormous sizes of datasets for reliable conclusions. We develop an approach based on partial derivatives,…
This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction…
Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling…
Finite element analysis (FEA) has been widely used to generate simulations of complex and nonlinear systems. Despite its strength and accuracy, the limitations of FEA can be summarized into two aspects: a) running high-fidelity FEA often…
We propose a novel \textit{capsule} based deep encoder-decoder model for surrogate modeling and uncertainty quantification of systems in mechanics from sparse data. The proposed framework is developed by adapting Capsule Network (CapsNet)…
We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed…