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Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE…

Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all…

Scientific computing using deep learning has seen significant advancements in recent years. There has been growing interest in models that learn the operator from the parameters of a partial differential equation (PDE) to the corresponding…

Physics-informed neural networks (PINNs) have lately received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). Most fully connected network-based PINNs use automatic…

Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of…

The great success of Physics-Informed Neural Networks (PINN) in solving partial differential equations (PDEs) has significantly advanced our simulation and understanding of complex physical systems in science and engineering. However, many…

Time-independent Partial Differential Equations (PDEs) on large meshes pose significant challenges for data-driven neural PDE solvers. We introduce a novel graph rewiring technique to tackle some of these challenges, such as aggregating…

This work addresses the accurate and efficient simulation of physical phenomena governed by parametric Partial Differential Equations (PDEs) characterized by varying boundary conditions, where parametric instances modify not only the…

Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…

We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an…

Enhancing neural networks with knowledge of physical equations has become an efficient way of solving various physics problems, from fluid flow to electromagnetism. Graph neural networks show promise in accurately representing irregularly…

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…

This paper introduces a novel two-stream deep model based on graph convolutional network (GCN) architecture and feed-forward neural networks (FFNN) for learning the solution of nonlinear partial differential equations (PDEs). The model aims…

Graph neural network (GNN) is a promising approach to learning and predicting physical phenomena described in boundary value problems, such as partial differential equations (PDEs) with boundary conditions. However, existing models…

We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of…

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…

Graph Neural Networks (GNNs) have demonstrated remarkable success in modeling complex relationships in graph-structured data. A recent innovation in this field is the family of Differential Equation-Inspired Graph Neural Networks (DE-GNNs),…

Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional…