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The superior performance of ensemble methods with infinite models are well known. Most of these methods are based on optimization problems in infinite-dimensional spaces with some regularization, for instance, boosting methods and convex…

Sparsification of neural networks is one of the effective complexity reduction methods to improve efficiency and generalizability. We consider the problem of learning a one hidden layer convolutional neural network with ReLU activation…

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…

Large-scale non-convex sparsity-constrained problems have recently gained extensive attention. Most existing deterministic optimization methods (e.g., GraSP) are not suitable for large-scale and high-dimensional problems, and thus…

We present a novel method for using Neural Networks (NNs) for finding solutions to a class of Partial Differential Equations (PDEs). Our method builds on recent advances in Neural Radiance Field research (NeRFs) and allows for a NN to…

Approximate solutions of partial differential equations (PDEs) obtained by neural networks are highly affected by hyper parameter settings. For instance, the model training strongly depends on loss function design, including the choice of…

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…

Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…

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…

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…

In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural…

Graph Neural Networks (GNNs) have emerged as potent tools for predicting outcomes in graph-structured data. Despite their efficacy, a significant drawback of GNNs lies in their limited ability to provide robust uncertainty estimates, posing…

Scalability properties of deep neural networks raise key research questions, particularly as the problems considered become larger and more challenging. This paper expands on the idea of conditional computation introduced by Bengio, et.…

The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…

Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters…

Dataless neural networks represent a paradigm shift in applying neural architectures to combinatorial optimization problems, eliminating the need for training datasets by encoding problem instances directly into network parameters. Despite…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…