Related papers: Understanding Deep Learning using Topological Dyna…

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs)…

The representations learned by deep neural networks are difficult to interpret in part due to their large parameter space and the complexities introduced by their multi-layer structure. We introduce a method for computing persistent…

Empirically, Deep Learning (DL) has demonstrated unprecedented success in practical applications. However, DL remains by and large a mysterious "black-box", spurring recent theoretical research to build its mathematical foundations. In this…

Probabilistic deep learning is deep learning that accounts for uncertainty, both model uncertainty and data uncertainty. It is based on the use of probabilistic models and deep neural networks. We distinguish two approaches to probabilistic…

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of…

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…

A neural network works as an associative memory device if it has large storage capacity and the quality of the retrieval is good enough. The learning and attractor abilities of the network both can be measured by the mutual information…

The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is…

Due to the nonlinearity of artificial neural networks, designing topologies for deep convolutional neural networks (CNN) is a challenging task and often only heuristic approach, such as trial and error, can be applied. An evolutionary…

Graph embeddings play a critical role in graph representation learning, allowing machine learning models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form $\dot x = \sum_{i=1}^lF_i(x)u_i$, with linear dependence in the controls, and we use…

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large…

Machine learning methods are being actively considered as a new tool of describing many body physics. However, so far, their capabilities has been only demonstrated in previously studied models, such as e.g. Ising model. Here, we consider a…

This paper introduces a novel network topology that seamlessly integrates dynamic inference cost with a top-down attention mechanism, addressing two significant gaps in traditional deep learning models. Drawing inspiration from human…

Dense prediction tasks such as depth perception and semantic segmentation are important applications in computer vision that have a concrete topological description in terms of partitioning an image into connected components or estimating a…

We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…

We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented…

Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an…