Henrique N. Sá Earp

Elli Heyes, Edward Hirst, Henrique N. Sá Earp et al.

A numerical framework for approximating $\mathrm{G}_2$-structure 3-forms on contact Calabi-Yau manifolds is presented. The approach proceeds in three stages: first, existing neural network models are employed to compute an approximate Ricci-flat metric on a Calabi-Yau threefold. Second, using this metric and the explicit construction of a $\mathrm{G}_2$-structure on the associated 7-dimensional Calabi-Yau link in the 9-sphere, numerical approximations of the 3-form are generated on a large set of sampled points. Finally, a dedicated neural architecture is trained to learn the 3-form and its induced Riemannian metric directly from data, validating the learned structure and its torsion via a numerical implementation of the exterior derivative, which may be of independent interest.

Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva

The neural Willmore flow of a closed oriented $2$-surface in $\mathbb{R}^3$ is introduced as a natural evolution process to minimise the Willmore energy, which is the squared $L^2$-norm of mean curvature. Neural architectures are used to model maps from topological $2d$ domains to $3d$ Euclidean space, where the learning process minimises a PINN-style loss for the Willmore energy as a functional on the embedding. Training reproduces the expected round sphere for genus $0$ surfaces, and the Clifford torus for genus $1$ surfaces, respectively. Furthermore, the experiment in the genus $2$ case provides a novel approach to search for minimal Willmore surfaces in this open problem.

Daniel Fadel, Henrique N. Sá Earp, Tomás S. R. Silva

This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential $k$-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus -- without requiring full $C^1$ smoothness of the differential form. As a numerical application, we propose an algorithm for exterior differentiation in $\mathbb{R}^n$ that relies solely on black-box access to the differential form, offering a practical tool for computation without the need for mesh discretization or explicit symbolic expressions.

Renata Pelissari, Ricardo Suyama, Leonardo Tomazeli Duarte et al.

How can a National Artificial Intelligence Strategy be effectively monitored? To address this question, we propose a methodology consisting of two key components. First, it involves identifying relevant indicators within national AI strategies. Second, it assesses the alignment between these indicators and the strategic actions of a specific government's AI strategy, allowing for a critical evaluation of its monitoring measures. Moreover, identifying these indicators helps assess the overall quality of the strategy's structure. A lack of alignment between strategic actions and the identified indicators may reveal gaps or blind spots in the strategy. This methodology is demonstrated using the Brazilian AI strategy as a case study.