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In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling…
Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely…
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…
We present a novel framework combining Deep Operator Networks (DeepONets) with Physics-Informed Neural Networks (PINNs) to solve partial differential equations (PDEs) and estimate their unknown parameters. By integrating data-driven…
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…
Partial differential equations (PDEs) are an essential computational kernel in physics and engineering. With the advance of deep learning, physics-informed neural networks (PINNs), as a mesh-free method, have shown great potential for fast…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and…
One of the most popular recent areas of machine learning predicates the use of neural networks augmented by information about the underlying process in the form of Partial Differential Equations (PDEs). These physics-informed neural…
Solving partial differential equations (PDEs) can be prohibitively expensive using traditional numerical methods. Deep learning-based surrogate models typically specialize in a single PDE with fixed parameters. We present a framework for…
Deep neural networks (DNNs), especially physics-informed neural networks (PINNs), have recently become a new popular method for solving forward and inverse problems governed by partial differential equations (PDEs). However, these methods…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs). However, finding a set of neural network parameters that lead to fulfilling a PDE can be challenging and…
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.…
Many types of physics-informed neural network models have been proposed in recent years as approaches for learning solutions to differential equations. When a particular task requires solving a differential equation at multiple…
Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…
The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous…