Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid (opens in new tab)

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 ei...