Related papers: Solving Partial Differential Equations with Neural…
Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…
Neural network based methods have emerged as a promising paradigm for scientific computing, yet they face critical bottlenecks in high frequency function approximation and partial differential equation (PDE) solving.
Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance,…
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…
We propose an approach to solving partial differential equations (PDEs) using a set of neural networks which we call Neural Basis Functions (NBF). This NBF framework is a novel variation of the POD DeepONet operator learning approach where…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…
Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a…
Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…
Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…
In recent years, Physics-Informed Neural Networks (PINNs) have become a representative method for solving partial differential equations (PDEs) with neural networks. PINNs provide a novel approach to solving PDEs through optimization…
Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…