Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid

• URL: http://arxiv.org/abs/2606.21725v1
• Date: Fri, 19 Jun 2026 20:23:17 GMT
• Title: Physics-Informed Neural Networks for Computing the Morse Index of the Critical Catenoid
• Abstract summary: A Morse index of a free boundary minimal surface is encoded in its Jacobi-Steklov spectrum.<n>We show how a physics-informed neural network reproduces that spectrum on a problem whose answer is already known in closed form.<n>We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 below the stability threshold to within $10^{-6}$ to $10^{-4}$ of their exact values, with PDE residuals of order $10^{-4}$; assembling them recovers the index $4$ and the nullity $2$. We then track the spectrum along a one-parameter homotopy joining a flat reference operator to the catenoid Jacobi operator and identify the crossings at which the index changes. Since the critical catenoid is rigid, a fact we prove, this homotopy deforms operators rather than surfaces. We close by explaining how the same pipeline, with its one-dimensional solver replaced by a two-dimensional one, is poised to address genuinely geometric families in ellipsoidal balls, where the boundary curvature is no longer constant, and the Morse index is not yet known.

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