PINNs for Geometric Analysis: Lessons from the Asymptotic Plateau Problem
A contribution to the proceedings, stemming from a presentation at the 2026 workshop 'DANGER: Data, Numbers, and Geometry', details a machine learning approach utilizing physics-informed neural networks (PINNs) to develop near-minimal discs in hyperbolic space that are asymptotic to a specified knot at infinity. This research, conducted in collaboration with Marco Usula, offers numerical support for a conjecture proposed by Joel Fine, which connects minimal surfaces in H^4 to the coefficients of the HOMFLY polynomial. The paper emphasizes two critical elements for the method's effectiveness: integrating the problem's geometry into the model's structure and addressing boundary conditions. Comprehensive findings are available in the preprint arXiv:2605.26234v2.
Key facts
- The paper is a proceedings contribution based on a presentation at the 2026 workshop 'DANGER: Data, Numbers, and Geometry'.
- It elaborates on findings from arXiv:2605.26234v2, a joint work with Marco Usula.
- The framework uses physics-informed neural networks (PINNs) to construct near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity.
- The method provides numerical evidence for a conjecture by Joel Fine relating minimal surfaces in H^4 to coefficients of the HOMFLY polynomial.
- The paper discusses two aspects that determine whether the method works: encoding geometry in the model's architecture and boundary conditions.
- The full results are presented in the associated preprint arXiv:2605.26234v2.
- The workshop 'DANGER: Data, Numbers, and Geometry' took place in 2026.
- The work involves hyperbolic space and knots at infinity.
Entities
Artists
- Marco Usula
- Joel Fine
Institutions
- arXiv