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Providing fast and accurate solutions to partial differential equations is a problem of continuous interest to the fields of applied mathematics and physics. With the recent advances in machine learning, the adoption learning techniques in…

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently.…

Differentiable solvers for the linear assignment problem (LAP) have attracted much research attention in recent years, which are usually embedded into learning frameworks as components. However, previous algorithms, with or without learning…

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in…

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…

The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to…

Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central…

Generative adversarial network (GAN) has been shown to be useful in various applications, such as image recognition, text processing and scientific computing, due its strong ability to learn complex data distributions. In this study, a…

Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…

We present a unified convergence theory for gradient-based training of neural network methods for partial differential equations (PDEs), covering both physics-informed neural networks (PINNs) and the Deep Ritz method. For linear PDEs, we…

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a…

The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards…

Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances…

Graph convolutional networks (GCNs) allow us to learn topologically-aware node embeddings, which can be useful for classification or link prediction. However, they are unable to capture long-range dependencies between nodes without adding…

In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…

The inverse problem of supervised reconstruction of depth-variable (time-dependent) parameters in a neural ordinary differential equation (NODE) is considered, that means finding the weights of a residual network with time continuous…

We present the design and optimization of a linear solver on General Purpose GPUs for the efficient and high-throughput evaluation of the marginalized graph kernel between pairs of labeled graphs. The solver implements a preconditioned…

In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…

We analyze a hybrid method that enriches coarse grid finite element solutions with fine scale fluctuations obtained from a neural network. The idea stems from the Deep Neural Network Multigrid Solver (DNN-MG), (Margenberg et al., J Comput…

Background: Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency…