Related papers: Approximation and Generalization of DeepONets for …
We propose a novel convolutional neural network (CNN), called $\Psi$DONet, designed for learning pseudodifferential operators ($\Psi$DOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding…
Despite being effective in many application areas, Deep Neural Networks (DNNs) are vulnerable to being attacked. In object recognition, the attack takes the form of a small perturbation added to an image, that causes the DNN to misclassify,…
Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical…
Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering…
Deep neural networks (DNNs) are typically optimized using various forms of mini-batch gradient descent algorithm. A major motivation for mini-batch gradient descent is that with a suitably chosen batch size, available computing resources…
In this paper, by utilizing the theory of matched asymptotic expansions, an efficient and accurate neural network method, named as "MAE-TransNet", is developed for solving singular perturbation problems in general dimensions, whose…
This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized…
This paper focuses on the feasibility of Deep Neural Operator (DeepONet) as a robust surrogate modeling method within the context of digital twin (DT) for nuclear energy systems. Through benchmarking and evaluation, this study showcases the…
Neural network based data-driven operator learning schemes have shown tremendous potential in computational mechanics. DeepONet is one such neural network architecture which has gained widespread appreciation owing to its excellent…
The performance of deep neural networks (DNN) is very sensitive to the particular choice of hyper-parameters. To make it worse, the shape of the learning curve can be significantly affected when a technique like batchnorm is used. As a…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as…
We introduce a new general-purpose approach to deep learning on 3D surfaces, based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in…
We investigate deep morphological neural networks (DMNNs). We demonstrate that despite their inherent non-linearity, "linear" activations are essential for DMNNs. To preserve their inherent sparsity, we propose architectures that constraint…
In this paper we consider the limiting case of neural networks (NNs) architectures when the number of neurons in each hidden layer and the number of hidden layers tend to infinity thus forming a continuum, and we derive approximation errors…
Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for…
In certain practical engineering applications, there is an urgent need to perform repetitive solving of partial differential equations (PDEs) in a short period. This paper primarily considers three scenarios requiring extensive repetitive…
Despite prior advances in PINNs, significant challenges remain in localized solid mechanics problems because of the limitations of single network formulations in simultaneous resolution of smooth global responses and near-tip singularities,…
Modeling the recovery of interdependent critical infrastructure is a key component of quantifying and optimizing societal resilience to disruptive events. However, simulating the recovery of large-scale interdependent systems under random…
Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what…