相关论文: Neural Galerkin Schemes with Active Learning for H…

Large-scale or high-resolution geologic models usually comprise a huge number of grid blocks, which can be computationally demanding and time-consuming to solve with numerical simulators. Therefore, it is advantageous to upscale geologic…

Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing…

We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall…

Most of the neural networks (NNs) learned via state-of-the-art machine learning techniques are black-box models. For a widespread success of machine learning in science and engineering, it is important to develop new NN architectures to…

We study different aspects of active learning with deep neural networks in a consistent and unified way. i) We investigate incremental and cumulative training modes which specify how the newly labeled data are used for training. ii) We…

We present a novel and mathematically transparent approach to function approximation and the training of large, high-dimensional neural networks, based on the approximate least-squares solution of associated Fredholm integral equations of…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

Deep Neural Networks (DNNs) often rely on very large datasets for training. Given the large size of such datasets, it is conceivable that they contain certain samples that either do not contribute or negatively impact the DNN's…

The success of the current wave of artificial intelligence can be partly attributed to deep neural networks, which have proven to be very effective in learning complex patterns from large datasets with minimal human intervention. However,…

Even though deep neural networks have shown tremendous success in countless applications, explaining model behaviour or predictions is an open research problem. In this paper, we address this issue by employing a simple yet effective method…

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of…

Whilst the partial differential equations that govern the dynamics of our world have been studied in great depth for centuries, solving them for complex, high-dimensional conditions and domains still presents an incredibly large…

First we show that physics-informed neural networks are not suitable for a large class of parabolic partial differential equations including the Fokker-Planck equation. Then we devise an algorithm to compute solutions of the Fokker-Planck…

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a $Q$-Wiener process and focus on noise-induced transitions and escape. We argue that proving a…

We introduce a new training algorithm for deep neural networks that utilize random complex exponential activation functions. Our approach employs a Markov Chain Monte Carlo sampling procedure to iteratively train network layers, avoiding…

Understanding the advantages of deep neural networks trained by gradient descent (GD) compared to shallow models remains an open theoretical challenge. In this paper, we introduce a class of target functions (single and multi-index Gaussian…

Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by…

Achievement of solutions in Navier-Stokes equation is one of challenging quests, especially for its closure problem. For achievement of particular solutions, there are variety of numerical simulations including Direct Numerical Simulation…

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…

Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. The…