Minimising Willmore Energy via Neural Flow

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.

A semicontinuous relaxation of Saito's criterion and freeness as angular minimization

We introduce a nonnegative functional on the space of line arrangements in $\mathbb{P}^2$ that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion for freeness. Given an arrangement $\mathcal{A}$ of $n$ lines with candidate exponents $(d_1, d_2)$, we parameterize the spaces of logarithmic derivations of degrees $d_1$ and $d_2$ via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree $n$ polynomials. The functional then admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial $Q(\mathcal{A})$ in coefficient space, and equals zero if and only if its image contains the line spanned by $Q(\mathcal{A})$. This provides a computable measure of how far a given arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. Using this functional as a reward signal, we develop a sequential construction procedure in which lines are added one at a time so as to minimize the angular distance to freeness, implemented via reinforcement learning with an adaptive curriculum over arrangement sizes and exponent types. Our results suggest that semicontinuous relaxation techniques, grounded in the geometry of polynomial coefficient spaces, offer a viable approach to the computational exploration of freeness in the theory of line arrangements.

Neural and numerical methods for $\mathrm{G}_2$-structures on contact Calabi-Yau 7-manifolds

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.