A new functional S vanishes precisely on free line arrangements and enables discovery of verified free examples for every admissible exponent pair with up to 20 lines.
Minimising Willmore Energy via Neural Flow
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abstract
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.
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PINNs can address differential geometry problems by training neural networks to minimize functionals that encode geometric conditions, as shown through summaries of three related studies.
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A semicontinuous relaxation of Saito's criterion and freeness as angular minimization
A new functional S vanishes precisely on free line arrangements and enables discovery of verified free examples for every admissible exponent pair with up to 20 lines.
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PINNs in More General Geometry
PINNs can address differential geometry problems by training neural networks to minimize functionals that encode geometric conditions, as shown through summaries of three related studies.