Related papers: Multi-scale Physical Representations for Approxima…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed…
Machine learning (ML) and deep learning (DL) techniques have gained significant attention as reduced order models (ROMs) to computationally expensive structural analysis methods, such as finite element analysis (FEA). Graph neural network…
In this paper, a physics-informed multiresolution wavelet neural network (PIMWNN) method is proposed for solving partial differential equations (PDEs). This method uses the multiresolution wavelet neural network (MWNN) to approximate…
Graph Neural Networks (GNNs) have become a prevailing tool for learning physical dynamics. However, they still encounter several challenges: 1) Physical laws abide by symmetry, which is a vital inductive bias accounting for model…
Many physical systems can be best understood as sets of discrete data with associated relationships. Where previously these sets of data have been formulated as series or image data to match the available machine learning architectures,…
Graph Neural Networks (GNNs) and differential equations (DEs) are two rapidly advancing areas of research that have shown remarkable synergy in recent years. GNNs have emerged as powerful tools for learning on graph-structured data, while…
Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based…
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
We propose a physics-informed machine learning framework called P-DivGNN to reconstruct local stress fields at the micro-scale, in the context of multi-scale simulation given a periodic micro-structure mesh and mean, macro-scale, stress…
Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be…
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional…
Graph Neural Networks (GNNs) have established themselves as the preferred methodology in a multitude of domains, ranging from computer vision to computational biology, especially in contexts where data inherently conform to graph…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
This paper considers the approximation of spatial convolution with a given radial integral kernel. Previous studies have demonstrated that approximating spatial convolution using a system of partial differential equations (PDEs) can…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing…