Related papers: Spectral methods for Neural Integral Equations
Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to…
We propose an empirical approach centered on the spectral dynamics of weights -- the behavior of singular values and vectors during optimization -- to unify and clarify several phenomena in deep learning. We identify a consistent bias in…
Speckles arise when coherent light interacts with biological tissues. Information retrieval from speckles is desired yet challenging, requiring understanding or mapping of the multiple scattering process, or reliable capability to reverse…
In this work we study integral equations defined on the whole real line. Using a suitable Banach space, we look for solutions which satisfy some certain kind of asymptotic behavior. We will consider spectral theory in order to find fixed…
Neural networks, whice have had a profound effect on how researchers study complex phenomena, do so through a complex, nonlinear mathematical structure which can be difficult for human researchers to interpret. This obstacle can be…
The Boltzmann equation, a fundamental model in kinetic theory, describes the evolution of particle distribution functions through a nonlinear, high-dimensional collision operator. However, its numerical solution remains computationally…
Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and…
In theoretical ML, the teacher-student paradigm is often employed as an effective metaphor for real-life tuition. The above scheme proves particularly relevant when the student network is overparameterized as compared to the teacher…
Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a $d$-dimensional system…
Existing methods for spectral reconstruction usually learn a discrete mapping from RGB images to a number of spectral bands. However, this modeling strategy ignores the continuous nature of spectral signature. In this paper, we propose…
Deep learning has been widely used for hyperspectral pixel classification due to its ability of generating deep feature representation. However, how to construct an efficient and powerful network suitable for hyperspectral data is still…
Despite their ability to represent highly expressive functions, deep learning models seem to find simple solutions that generalize surprisingly well. Spectral bias -- the tendency of neural networks to prioritize learning low frequency…
Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter…
Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this…
Vibrational spectroscopy is a key technique to elucidate microscopic structure and dynamics. Without the aid of theoretical approaches, it is however, often difficult to understand such spectra at a microscopic level. Ab initio molecular…
Vibrational spectroscopy is a cornerstone technique for molecular characterization and offers an ideal target for the computational investigation of molecular materials. Building on previous comprehensive assessments of efficient methods…
Data-driven models based on deep learning algorithms intend to overcome the limitations of traditional constitutive modelling by directly learning from data. However, the need for extensive data that collate the full state of the material…
Motivation: Tumor classification using Imaging Mass Spectrometry (IMS) data has a high potential for future applications in pathology. Due to the complexity and size of the data, automated feature extraction and classification steps are…
Integration is indispensable, not only in mathematics, but also in a wide range of other fields. A deep learning method has recently been developed and shown to be capable of integrating mathematical functions that could not previously be…
A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their…