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Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the…

Machine learning solvers for partial differential equations (PDEs) have attracted growing interest. However, most existing approaches, such as neural network solvers, rely on stochastic training, which is inefficient and typically requires…

Recently, physics-informed neural networks (PINNs) and their variants have gained significant popularity as a scientific computing method for solving partial differential equations (PDEs), whereas accuracy is still its main shortcoming.…

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for…

Nonlinear differential equations and systems play a crucial role in modeling systems where time-dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties…

Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…

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…

In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for…

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work,…

In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs) emerges to be a promising method for solving both forward and inverse PDE…

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…

This study focuses on the solution of partial differential equations (PDEs) by using physics-informed neural networks (PINNs). The Newell-Whitehead-Segel (NWS) equation and the Allen-Cahn equation belong to fundamental PDEs used mostly in…

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…

Physics informed neural network (PINN) based solution methods for differential equations have recently shown success in a variety of scientific computing applications. Several authors have reported difficulties, however, when using PINNs to…

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on…

The growing demand for accurate, efficient, and scalable solutions in computational mechanics highlights the need for advanced operator learning algorithms that can efficiently handle large datasets while providing reliable uncertainty…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…

The Gaussian process (GP) is a widely used probabilistic machine learning method with implicit uncertainty characterization for stochastic function approximation, stochastic modeling, and analyzing real-world measurements of nonlinear…