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The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time-consuming. To address this problem,…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Phase-field models have been widely used to investigate the phase transformation phenomena. However, it is difficult to solve the problems numerically due to their strong nonlinearities and higher-order terms. This work is devoted to…

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…

Pointer Network (PtrNet) is a specific neural network for solving Combinatorial Optimization Problems (COPs). While PtrNets offer real-time feed-forward inference for complex COPs instances, its quality of the results tends to be less…

Inspired by the relation between deep neural network (DNN) and partial differential equations (PDEs), we study the general form of the PDE models of deep neural networks. To achieve this goal, we formulate DNN as an evolution operator from…

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seed numerical solutions for a class of partial differential equations instead of a single partial differential equation,…

Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires…

Physics-informed neural networks (PINNs) solve time-dependent partial differential equations (PDEs) by learning a mesh-free, differentiable solution that can be evaluated anywhere in space and time. However, standard space--time PINNs take…

Information networks are ubiquitous and are ideal for modeling relational data. Networks being sparse and irregular, network embedding algorithms have caught the attention of many researchers, who came up with numerous embeddings algorithms…

In recent years, deep learning-based methods have been proposed for solving inverse scattering problems (ISPs), but most of them heavily rely on data and suffer from limited generalization capabilities. In this paper, a new solving scheme…

Learning the solution of partial differential equations (PDEs) with a neural network is an attractive alternative to traditional solvers due to its elegance, greater flexibility and the ease of incorporating observed data. However, training…

Predicting the evolution of systems that exhibit spatio-temporal dynamics in response to external stimuli is a key enabling technology fostering scientific innovation. Traditional equations-based approaches leverage first principles to…

Modern sensing and metrology systems now stream terabytes of heterogeneous, high-dimensional (HD) data profiles, images, and dense point clouds, whose natural representation is multi-way tensors. Understanding such data requires regression…

We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differential equations in…

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…

Tensor neural networks (TNNs) have demonstrated their superiority in solving high-dimensional problems. However, similar to conventional neural networks, TNNs are also influenced by the Frequency Principle, which limits their ability to…