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Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…
We consider arbitrary bounded discrete time series originating from dynamical system with recursivity. More precisely, we provide an explicit construction of recurrent neural networks which effectively approximate the corresponding discrete…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
Data-driven algorithms, in particular neural networks, can emulate the effect of sub-grid scale processes in coarse-resolution climate models if trained on high-resolution climate simulations. However, they may violate key physical…
Data-enabled predictive control (DeePC) for linear systems utilizes data matrices of recorded trajectories to directly predict new system trajectories, which is very appealing for real-life applications. In this paper we leverage the…
While deep feature learning has revolutionized techniques for static-image understanding, the same does not quite hold for video processing. Architectures and optimization techniques used for video are largely based off those for static…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
CoVariance Neural Networks (VNNs) perform convolutions on the graph determined by the covariance matrix of the data, which enables expressive and stable covariance-based learning. However, covariance matrices are typically dense, fail to…
Deep neural networks have proven to be particularly effective in visual and audio recognition tasks. Existing models tend to be computationally expensive and memory intensive, however, and so methods for hardware-oriented approximation have…
Visual error metrics play a fundamental role in the quantification of perceived image similarity. Most recently, use cases for them in real-time applications have emerged, such as content-adaptive shading and shading reuse to increase…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Deep learning based on deep neural networks has been very successful in many practical applications, but it lacks enough theoretical understanding due to the network architectures and structures. In this paper we establish some analysis for…
Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed…
Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…
In computer vision, superpixels have been widely used as an effective way to reduce the number of image primitives for subsequent processing. But only a few attempts have been made to incorporate them into deep neural networks. One main…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
We introduce a novel model architecture that incorporates network effects into discrete choice problems, achieving higher predictive performance than standard discrete choice models while offering greater interpretability than…
Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and…
In recent years, the researches about solving partial differential equations (PDEs) based on artificial neural network have attracted considerable attention. In these researches, the neural network models are usually designed depend on…