Related papers: Fractional Concepts in Neural Networks: Enhancing …
Fourier Neural Operators (FNOs) have emerged as leading surrogates for solver operators for various functional problems, yet their stability, generalization and frequency behavior lack a principled explanation. We present a systematic…
Current research suggests that the key factors in designing neural network architectures involve choosing number of filters for every convolution layer, number of hidden neurons for every fully connected layer, dropout and pruning. The…
In recent years, as fractional calculus becomes more and more broadly used in research across different academic disciplines, there are increasing demands for the numerical tools for the computation of fractional…
Diffractive neural network (DNN), which can perform machine learning tasks based on the light propagation and diffraction, has recently emerged as a promising optical computing paradigm due to its high parallel processing speed and low…
Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…
The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In…
Our study focuses on fractional order compartment models derived from underlying physical stochastic processes, providing a more physically grounded approach compared to models that use the dynamical system approach by simply replacing…
While machine learning models are typically trained to solve prediction problems, we might often want to use them for optimization problems. For example, given a dataset of proteins and their corresponding fluorescence levels, we might want…
The study of systems with memory requires methods which are different from the methods used in regular dynamics. Systems with power-law memory in many cases can be described by fractional differential equations, which are…
Functional connectivity (FC) refers to the investigation of interactions between brain regions to understand integration of neural activity in several regions. FC is often estimated using functional magnetic resonance images (fMRI). There…
Functional Data Analysis (FDA) is an extension of traditional data analysis to functional data, for example spectra, temporal series, spatio-temporal images, gesture recognition data, etc. Functional data are rarely known in practice;…
Activation functions in neural networks are typically selected from a set of empirically validated, commonly used static functions such as ReLU, tanh, or sigmoid. However, by optimizing the shapes of a network's activation functions, we can…
The fractional order system, which is described by the fractional order derivative and integral, has been studied in many engineering areas. Recently, the concept of fractional order has been generalized to the distributed order concept,…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds…
Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more…
Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the…
Effective sequence modeling fundamentally requires balancing the retention of unbounded history with the high-resolution detection of abrupt short-term variations common in real-world phenomena. However, existing state space models (SSMs)…
In this work, we propose activation functions for neuronal networks that are refinable and sum the identity. This new class of activation functions allows the insertion of new layers between existing ones and/or the increase of neurons in a…
Fractional calculus is an effective tool in incorporating the effects of non-locality and memory into physical models. In this regard, successful applications exist rang- ing from signal processing to anomalous diffusion and quantum…