Related papers: A generic finite element framework on parallel tre…
Although deep learning has demonstrated remarkable capability in learning from unstructured data, modern tree-based ensemble models remain superior in extracting relevant information and learning from structured datasets. While several…
In order to learn distributed port-Hamiltonian systems (dPHS) using Gaussian processes (GPs), the partitioned finite element method (PFEM) is combined with the Gp-dPHS method. By following a late lumping approach, the discretization of the…
We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously…
Recent work showed that compiling functional programs to use dense, serialized memory representations for recursive algebraic datatypes can yield significant constant-factor speedups for sequential programs. But serializing data in a…
Machine learning models deployed in real-world settings must operate under evolving data distributions and constrained computational resources. This challenge is particularly acute in non-stationary domains such as energy time series,…
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…
In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on H(div) and H(curl). The proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor…
Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…
We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework…
In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid…
We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices…
A robust $hp$-adaptive finite element framework is presented for the investigation of static cracks in materials characterized by complex, pointwise density variations. Within such heterogeneous media, the equilibrium equation governed by…
We present an anisotropic $hp-$mesh adaptation strategy using a continuous mesh model for discontinuous Petrov-Galerkin (DPG) finite element schemes with optimal test functions, extending our previous work on $h-$adaptation. The proposed…
Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can…
We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on…
We present an MPI-parallel algorithm for the in-situ visualization of computational data that is built around a distributed linear forest-of-octrees data structure. Such octrees are frequently used in element-based numerical simulations;…
We present a high-level domain-specific language (DSL) interface to drive an adaptive incomplete $k$-d tree-based framework for finite element (FEM) solutions to PDEs. This DSL provides three key advances: (a) it abstracts out the…
This work describes a concise algorithm for the generation of triangular meshes with the help of standard adaptive finite element methods. We demonstrate that a generic adaptive finite element solver can be repurposed into a triangular mesh…
Neural networks and tree ensembles are state-of-the-art learners, each with its unique statistical and computational advantages. We aim to combine these advantages by introducing a new layer for neural networks, composed of an ensemble of…
We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut…