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Physical units are fundamental to scientific computing. However, many finite element frameworks lack built-in support for dimensional analysis. In this work, we present a systematic framework for integrating physical units into the Unified…
We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…
The implementation of efficient multigrid preconditioners for elliptic partial differential equations (PDEs) is a challenge due to the complexity of the resulting algorithms and corresponding computer code. For sophisticated finite element…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering…
Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite element transformations in FInAT…
The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms…
An automated framework is presented for the numerical solution of optimal control problems with PDEs as constraints, in both the stationary and instationary settings. The associated code can solve both linear and non-linear problems, and…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov…
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
Federated Learning (FL) has emerged as a promising technique for edge devices to collaboratively learn a shared machine learning model while keeping training data locally on the device, thereby removing the need to store and access the full…
This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the…
This document summarizes the main concepts of the finite element (FE) theory and constitutive relations as implemented in the open-source code phase-field multiphysics materials simulator PHIMATS https://github.com/ahcomat/PHIMATS. PHIMATS…
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…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid-fluid free surface and a liquid-solid interface, with the three-phase contact line moving across the…
To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness…