Related papers: Finite Element Neural Network Interpolation. Part …
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…
Physics-informed neural networks (PINNs) have shown remarkable prospects in solving forward and inverse problems involving partial differential equations (PDEs). However, PINNs still face the challenge of high computational cost in solving…
We introduce a novel hybrid methodology combining classical finite element methods (FEM) with neural networks to create a well-performing and generalizable surrogate model for forward and inverse problems. The residual from finite element…
In this work, we introduce a hypergraph representation learning framework called Hypergraph Neural Networks (HNN) that jointly learns hyperedge embeddings along with a set of hyperedge-dependent embeddings for each node in the hypergraph.…
An emerging trend in deep learning research focuses on the applications of graph neural networks (GNNs) for mesh-based continuum mechanics simulations. Most of these learning frameworks operate on graphs wherein each edge connects two…
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…
We show how to construct the deep neural network (DNN) expert to predict quasi-optimal $hp$-refinements for a given computational problem. The main idea is to train the DNN expert during executing the self-adaptive $hp$-finite element…
We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ…
Visual image reconstruction from functional Magnetic Resonance Imaging (fMRI) is a fundamental task in brain decoding, providing a crucial pathway for understanding human perceptual mechanisms and developing advanced brain-computer…
In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first…
This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time dependent, multi-fidelity problems, and use the trained hybrid models to…
With growing concerns regarding data privacy and rapid increase in data volume, Federated Learning(FL) has become an important learning paradigm. However, jointly learning a deep neural network model in a FL setting proves to be a…
In this paper, we introduce a novel approach to neural learning: the Feature-Imitating-Network (FIN). A FIN is a neural network with weights that are initialized to reliably approximate one or more closed-form statistical features, such as…
In this paper, we address the design of lightweight deep learning-based edge detection. The deep learning technology offers a significant improvement on the edge detection accuracy. However, typical neural network designs have very high…
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…
Deep neural networks are powerful tools for solving nonlinear problems in science and engineering, but training highly accurate models becomes challenging as problem complexity increases. Non-convex optimization and sensitivity to…
This paper presents a proper generalized decomposition (PGD) based reduced-order model of hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-PGD method keeps both advantages of HiDeNN and PGD methods. The automatic…
In this paper, we propose a novel $hr$-adaptive finite element method, enhanced by neural networks, for parabolic equations. The main challenge of the conventional $h$-adaptive finite element method is interpolating the finite element…
Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent…
Physics-Informed Neural Networks (PINNs) seek to solve partial differential equations (PDEs) with deep learning. Mainstream approaches that deploy fully-connected multi-layer deep learning architectures require prolonged training to achieve…