Related papers: M2N: Mesh Movement Networks for PDE Solvers
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
A unifying moving mesh method is developed for general $m$-dimensional geometric objects in $d$-dimensions ($d \ge 1$ and $1\le m \le d$) including curves, surfaces, and domains. The method is based on mesh equidistribution and alignment…
A fast method is presented for adaptive moving mesh generation in multi-dimensions using a domain decomposition parabolic Monge-Amp\`ere approach. The domain decomposition procedure employed here is non-iterative and involves splitting the…
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…
Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local…
In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces,…
Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances…
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…
Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural…
The nonlinear Schr\"{o}dinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a…
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…
Recent advances in geometric deep-learning introduce complex computational challenges for evaluating the distance between meshes. From a mesh model, point clouds are necessary along with a robust distance metric to assess surface quality or…
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…
Solving Partial Differential Equations (PDEs) is a cornerstone of engineering and scientific research. Traditional methods for PDE solving are cumbersome, relying on manual setup and domain expertise. While Physics-Informed Neural Network…
An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve…
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…
Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space--time domain but dominate the approximation error. This makes fixed…
We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality…