Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid
The Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum, and we test how faithfully a physics-informed neural network (PINN) reproduces that spectrum on a problem whose answer is already known in closed form. The benchmark is the critical catenoid in the unit ball $\mathbb{B}^3$, where it is well known that the Morse index equals $4$ and the nullity equals $2$. Separating the angular variable reduces the eigenvalue problem to a family of one-dimensional Robin problems on $[-T,T]$, one for each Fourier mode. A network that enforces the parity of each mode by construction, and carries the eigenvalue as a trainable parameter, returns the three eigenvalues...
| Authors | Miraj Samarakkody |
|---|---|
| Primary category | math.DG |
| Published | 19 June 2026 |
| Last updated at source | 19 June 2026 |
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