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Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by…

Physics-Informed Neural Networks (PINNs) represent a groundbreaking paradigm in scientific computing, seamlessly integrating the robust framework of deep learning with fundamental physical laws. This paper meticulously applies the standard…

Physics-informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and…

Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network…

Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…

In this paper, we compute numerical approximations of the minimal surfaces, an essential type of Partial Differential Equation (PDE), in higher dimensions. Classical methods cannot handle it in this case because of the Curse of…

Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations…

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with…

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their…

Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free,…

The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…

Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large…

Many Partial Differential Equations (PDEs) do not have analytical solution, and can only be solved by numerical methods. In this context, Physics-Informed Neural Networks (PINN) have become important in the last decades, since it uses a…

Solving partial differential equations (PDEs) has been indispensable in scientific and engineering applications. Recently, deep learning methods have been widely used to solve high-dimensional problems, one of which is the physics-informed…

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…

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential…

Since the introduction of deep learning for solving partial differential equations (PDEs), there has been growing interest in real-time system responses, where the kernel function plays a key role. Physics-informed neural networks (PINNs),…

The concepts and techniques of physics-informed neural networks (PINNs) is studied and limitations are identified to make it efficient to approximate dynamical equations. Potential working research domains are explored for increasing the…

For hyperbolic conservation laws, traditional methods and physics-informed neural networks (PINNs) often encounter difficulties in capturing sharp discontinuities and maintaining temporal consistency. To address these challenges, we…