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Physics-Informed Neural Networks (PINNs) have emerged as a tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual…

Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…

Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential…

With rapid progress in deep learning, neural networks have been widely used in scientific research and engineering applications as surrogate models. Despite the great success of neural networks in fitting complex systems, two major…

Deep neural networks (DNNs) provide useful models of visual representational transformations. We present a method that enables a DNN (student) to learn from the internal representational spaces of a reference model (teacher), which could be…

We consider the problem of 3D seismic inversion from pre-stack data using a very small number of seismic sources. The proposed solution is based on a combination of compressed-sensing and machine learning frameworks, known as…

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning…

Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for…

Deep neural networks (DNNs) have achieved significant success in a variety of real world applications, i.e., image classification. However, tons of parameters in the networks restrict the efficiency of neural networks due to the large model…

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional…

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…

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their…

We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation (PDE) forms. A comparative analysis between…

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and…

Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or…

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…

We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…

Weight pruning methods for deep neural networks (DNNs) have been investigated recently, but prior work in this area is mainly heuristic, iterative pruning, thereby lacking guarantees on the weight reduction ratio and convergence time. To…

In this paper, we consider the use of structure learning methods for probabilistic graphical models to identify statistical dependencies in high-dimensional physical processes. Such processes are often synthetically characterized using PDEs…

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…