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Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical…

We explore the recently-proposed Virtual Element Method (VEM) for numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the elasticity equations in three-dimensions and elaborate upon…

The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…

Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…

Approximating the solutions of boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process. This difficulty can be partly explained by the spectral…

Classical model-based imaging methods for ultrasound elasticity inverse problem require prior constraints about the underlying elasticity patterns, while finding the appropriate hand-crafted prior for each tissue type is a challenge. In…

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More…

Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…

Network Embedding is the task of learning continuous node representations for networks, which has been shown effective in a variety of tasks such as link prediction and node classification. Most of existing works aim to preserve different…

Despite their renowned predictive power on i.i.d. data, convolutional neural networks are known to rely more on high-frequency patterns that humans deem superficial than on low-frequency patterns that agree better with intuitions about what…

Neural networks are predominantly trained using gradient-based methods, yet in many applications their final predictions remain far from the accuracy attainable within the model's expressive capacity. We introduce Linearized Subspace…

We develop fast algorithms and robust software for convex optimization of two-layer neural networks with ReLU activation functions. Our work leverages a convex reformulation of the standard weight-decay penalized training problem as a set…

Pixel- and voxel-based representations of microstructures obtained from tomographic imaging methods is an established standard in computational materials science. The corresponding highly resolved, uniform discretitization in numerical…

Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…

This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions…

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method…

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We…

This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…

Meshing complex engineering domains is a challenging task. Arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains. The geometric property of the polyhedral meshes such as the…

Background: When using deep learning models, there are many possible vulnerabilities and some of the most worrying are the adversarial inputs, which can cause wrong decisions with minor perturbations. Therefore, it becomes necessary to…