We study the relationship between the frequency of a function and the speed at which a neural network learns it. We build on recent results that show that the dynamics of overparameterized neural networks trained with gradient descent can bewell approximated byalinear system. When normalized training data is uniformly distributed on ahypersphere, the eigenfunctions of this linear system are spherical harmonic functions.

We use machine learning to search for examples of Z/2 eigenfunctions on the 2-sphere. For this we created a multivalued version of a feedforward deep neural network, and we implemented it using the JAX library. We found Z/2 eigenfunctions for three cases: In the first two cases we fixed the branch points at the vertices of a tetrahedron and at a cube respectively. In a third case, we allowed the AI to move the branch points around and, in the end, it positioned the branch points at the vertices of a squashed tetrahedron.

From characterizing the speed of a thermal system's response to computing natural modes of vibration, eigenvalue analysis is ubiquitous in engineering. In spite of this, eigenvalue problems have received relatively little treatment compared to standard forward and inverse problems in the physics-informed machine learning literature. In particular, neural network discretizations of solutions to eigenvalue problems have seen only a handful of studies. Owing to their nonlinearity, neural network discretizations prevent the conversion of the continuous eigenvalue differential equation into a standard discrete eigenvalue problem. In this setting, eigenvalue analysis requires more specialized techniques. Using a neural network discretization of the eigenfunction, we show that a variational form of the eigenvalue problem called the "Rayleigh quotient" in tandem with a Gram-Schmidt orthogonalization procedure is a particularly simple and robust approach to find the eigenvalues and their corresponding eigenfunctions. This method is shown to be useful for finding sets of harmonic functions on irregular domains, parametric and nonlinear eigenproblems, and high-dimensional eigenanalysis. We also discuss the utility of harmonic functions as a spectral basis for approximating solutions to partial differential equations. Through various examples from engineering mechanics, the combination of the Rayleigh quotient objective, Gram-Schmidt procedure, and the neural network discretization of the eigenfunction is shown to offer unique advantages for handling continuous eigenvalue problems.

We analyze the capacity of neural networks to approximate solutions to an elliptic PDE assuming that the boundary condition can be approximated efficiently. Our focus is on the Laplace operator with Dirichlet boundary condition on a half space and on neural networks with a single hidden layer and an activation function that is a power of the popular ReLU activation function.

We introduce Harmonic Robustness, a powerful and intuitive method to test the robustness of any machine-learning model either during training or in black-box real-time inference monitoring without ground-truth labels. It is based on functional deviation from the harmonic mean value property, indicating instability and lack of explainability. We show implementation examples in low-dimensional trees and feedforward NNs, where the method reliably identifies overfitting, as well as in more complex high-dimensional models such as ResNet-50 and Vision Transformer where it efficiently measures adversarial vulnerability across image classes.

Harmonic functions are abundant in nature, appearing in limiting cases of Maxwell's, Navier-Stokes equations, the heat and the wave equation. Consequently, there are many applications of harmonic functions from industrial process optimisation to robotic path planning and the calculation of first exit times of random walks. Despite their ubiquity and relevance, there have been few attempts to incorporate inductive biases towards harmonic functions in machine learning contexts. In this work, we demonstrate effective means of representing harmonic functions in neural networks and extend such results also to quantum neural networks to demonstrate the generality of our approach. We benchmark our approaches against (quantum) physics-informed neural networks, where we show favourable performance.

Due to the advent of the expressions of data other than tabular formats, the topological compositions which make samples interrelated came into prominence. Analogically, those networks can be interpreted as social connections, dataflow maps, citation influence graphs, protein bindings, etc. However, in the case of social networks, it is highly crucial to evaluate the labels of discrete communities. The reason underneath for such a study is the non-negligible importance of analyzing graph networks to partition the vertices by using the topological features of network graphs, solely. For each of these interaction-based entities, a social graph, a mailing dataset, and two citation sets are selected as the testbench repositories. This paper, it was not only assessed the most valuable method but also determined how graph neural networks work and the need to improve against non-neural network approaches which are faster and computationally cost-effective. Also, this paper showed a limit to be excesses by prospective graph neural network variations by using the topological features of networks trialed.

Movie scenes with robots and vehicles autonomously moving around and interacting with their environment are no longer science fiction but slowly becoming part of our daily lives. On Mars, robotic vehicles already travel around freely and perform tasks. However, on Earth, the setup is quite different: robots have to navigate through crowded places and plan their route numerous times per second to safely avoid pedestrians and property. Currently, the development of rapidly responsive robot path planning is a hot topic of research. In computational methods, a robot's ability to pivot heavily depends on continuously solving differential equations.