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Graph Neural Networks (GNNs) have gained popularity in various learning tasks, with successful applications in fields like molecular biology, transportation systems, and electrical grids. These fields naturally use graph data, benefiting…

Deep neural networks learn feature representations via complex geometric transformations of the input data manifold. Despite the models' empirical success across domains, our understanding of neural feature representations is still…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

Transitional accounts of evolution emphasise a few changes that shape what is evolvable, with dramatic consequences for derived lineages. More recently it has been proposed that cognition might also have evolved via a series of major…

Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within…

We propose a learning paradigm for the numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework…

Geometry is a ubiquitous tool in computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative…

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances…

In this paper we propose a novel augmentation technique that improves not only the performance of deep neural networks on clean test data, but also significantly increases their robustness to random transformations, both affine and…

The estimation of continuous parameters from measured data plays a central role in many fields of physics. A key tool in understanding and improving such estimation processes is the concept of Fisher information, which quantifies how…

Existing adaptive bias techniques, which seek to estimate free energies and physical properties from molecular simulations, are limited by their reliance on fixed kernels or basis sets which hinder their ability to efficiently conform to…

We propose an approach based on Artificial Neural Networks (ANNs) to evaluate geometric constants relevant to the analysis and design of numerical schemes for partial differential equations. These constants play a central role,…

This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks…

Biological neural networks have evolved to maintain performance despite significant circuit damage. To survive damage, biological network architectures have both intrinsic resilience to component loss and also activate recovery programs…

Understanding how neural networks learn remains one of the central challenges in machine learning research. From random at the start of training, the weights of a neural network evolve in such a way as to be able to perform a variety of…

Complex models in physics, biology, economics, and engineering are often sloppy, meaning that the model parameters are not well determined by the model predictions for collective behavior. Many parameter combinations can vary over decades…

This work provides a computable, direct, and mathematically rigorous approximation to the differential geometry of class manifolds for high-dimensional data, along with nonlinear projections from input space onto these class manifolds. The…

Transformer models have consistently achieved remarkable results in various domains such as natural language processing and computer vision. However, despite ongoing research efforts to better understand these models, the field still lacks…

Modelling the complex physics of the Interstellar Medium (ISM) in the context of large-scale numerical simulations is a challenging task. A number of methods have been proposed to embed a description of the ISM into different codes. We…