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

This paper demonstrates that physics-informed neural networks can accurately compute the Morse index and nullity of the critical catenoid by reproducing its known Jacobi-Steklov spectrum and tracking eigenvalue crossings along a homotopy, thereby validating a pipeline ready for application to more complex geometric families with unknown indices.

Original authors: Miraj Samarakkody

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Miraj Samarakkody

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Counting "Wobbles" on a Soap Film

Imagine you have a soap film stretched inside a bubble. In math, this is called a minimal surface. Sometimes, these films are perfectly stable, but other times, they are like a wobbly Jell-O mold: if you push them in a certain way, they collapse or change shape to become smaller.

The Morse Index is just a number that counts how many different ways you can push the film to make it wobble and shrink.

Index 1: It's very stable; you can only push it one way to make it shrink.
Index 4: It's quite unstable; there are four distinct ways to push it to make it shrink.

The specific shape the authors are studying is called the Critical Catenoid. It looks like a hourglass or a soap film connecting two rings. Mathematicians already knew the answer for this specific shape: the index is 4.

The Problem: How to Solve It Without a Magic Formula

Usually, to find this number, you need a perfect mathematical formula (a "closed-form solution"). But for many complex shapes (like soap films in weirdly shaped bubbles), no such formula exists. You have to use a computer to guess and check.

The authors wanted to test a new type of computer program called a Physics-Informed Neural Network (PINN). Think of a PINN as a very smart student who doesn't just memorize answers but is forced to follow the laws of physics (like how soap films behave) while learning.

The Goal: They wanted to see if this "smart student" could correctly calculate the index of 4 for the Critical Catenoid, even though the answer was already known. If it works here, they can use it for shapes where the answer is unknown.

The Method: Teaching the Computer the Rules

To make the computer solve this, the authors broke the problem down into three steps:

The "Parity" Trick (The Mirror Rule):
The soap film is symmetrical. If you look at the left side, it's a mirror image of the right side.
- The Analogy: Imagine trying to teach a child to draw a butterfly. If you let them draw randomly, they might draw a lopsided mess. Instead, the authors told the computer: "Whatever you draw on the right, you must mirror on the left."
- Why it matters: Without this rule, the computer gets confused and draws "mixed-up" shapes that look okay but give the wrong answer. By forcing the symmetry, the computer finds the right answer much faster.
The "Trainable Number" (The Mystery Variable):
Usually, you have to guess the answer (the index) and then check if it's right.
- The Analogy: Imagine a detective trying to find a suspect's height. Instead of guessing, the detective has a "magic ruler" that changes its own length as it searches. The computer treats the answer (the eigenvalue) as a variable it can adjust, just like it adjusts its own internal settings. It keeps tweaking this number until the physics equations balance perfectly.
The "Homotopy" (The Slow Motion Movie):
To make sure the computer wasn't just lucky, they created a "movie" of the problem.
- The Analogy: Imagine starting with a flat, boring piece of paper (a simple shape) and slowly morphing it into the hourglass shape. As the shape changes, the computer watches the "wobble count" (the index) change in real-time. It counts every time a new wobble appears or disappears. This confirmed that the final count of 4 was reached through a logical path, not a fluke.

The Results

Did it work? Yes. The computer calculated the specific "wobble numbers" (eigenvalues) with extreme precision (correct to 4 or 5 decimal places).
The Count: It correctly identified that there are exactly 4 ways to make the film shrink and 2 ways to wiggle it without shrinking (nullity).
The Rigidity Check: The authors proved that the "movie" they made (morphing from flat to hourglass) wasn't actually morphing real soap films in between. It was just morphing the math equations. This is important because it means the computer is solving the math, not accidentally inventing new physical shapes that don't exist.

The Future: Why This Matters

The authors aren't just showing off a calculator for a shape we already understand. They are building a pipeline.

Current State: They tested the pipeline on the Critical Catenoid (where the answer is known).
Future Goal: They want to use this same pipeline on Ellipsoidal Balls (like squashed spheres). In these shapes, the boundary isn't a perfect circle, so the old math formulas don't work, and nobody knows the answer yet.
The Promise: Because the computer doesn't rely on a pre-written formula but instead learns the physics directly, this same method can be used to solve these new, unsolved problems.

Summary in One Sentence

The authors built a smart computer program that learns the laws of physics to count how unstable a specific soap film shape is, proving it works so they can use it later to solve similar problems where the answer is currently a mystery.

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

Problem Statement
The paper addresses the computation of the Morse index for free boundary minimal surfaces (FBMS), specifically the critical catenoid in the unit ball $B^3$ . The Morse index, which counts the independent directions in which a surface can be perturbed to decrease area to second order, is encoded in the Jacobi–Steklov spectrum. While the Morse index of the critical catenoid is known analytically to be 4 (with nullity 2), the authors aim to develop a numerical pipeline that does not rely on closed-form spectral solutions. The ultimate goal is to validate a method capable of handling geometric families where the boundary curvature is non-constant and the index is unknown, such as FBMS in ellipsoidal balls.

Methodology
The authors employ a Physics-Informed Neural Network (PINN) to solve the Jacobi–Steklov eigenvalue problem. The approach involves the following steps:

Mathematical Reduction: Leveraging the rotational symmetry of the critical catenoid, the problem is reduced from a two-dimensional surface to a family of one-dimensional Robin eigenvalue problems on the interval $[-T, T]$ , one for each Fourier mode $n$ . The governing equation is $f''(t) + (2/\cosh^2 t - n^2)f(t) = 0$ , subject to Robin boundary conditions involving the eigenvalue $\delta$ .
PINN Architecture: A fully connected neural network with four hidden layers (width 64, tanh activation) approximates the eigenfunction $f(t)$ $f (t)$ .
- Parity Constraint: A critical architectural modification enforces the parity of the solution by construction ( $f(t) = N(t) \pm N(-t)$ for even/odd modes). This prevents the network from converging to mixed-parity spurious minima that corrupt eigenvalue estimates.
- Trainable Eigenvalue: The eigenvalue $\delta$ is treated as a trainable scalar parameter, optimized via gradient descent alongside the network weights.
Loss Function: The network is trained to minimize a composite loss function comprising:
- The interior ODE residual ( $L_{pde}$ ).
- The Robin boundary residuals ( $L_{bc}$ ).
- A normalization constraint ( $L_{norm}$ ) to prevent the trivial solution.
Spectral Flow Analysis: To understand how the index changes, the authors construct a one-parameter operator homotopy from a flat reference operator to the catenoid Jacobi operator. They track the movement of eigenvalues across the stability threshold ( $\delta = 1$ ) using high-accuracy ODE shooting, validated by PINN spot-checks.

Key Contributions

Parity-Constrained PINN Recovery: The authors successfully recover the three Jacobi–Steklov eigenvalues below the stability threshold ( $\delta < 1$ ) with errors between $10^{-6}$ and $10^{-4}$ relative to exact values. This allows for the correct numerical computation of the Morse index (4) and nullity (2).
Spectral Gap Identification: The study establishes a spectral gap statement, confirming that no eigenvalues exist in the open interval $(0.4392, 1)$ , isolating the eigenvalues responsible for the index.
Operator Homotopy and Spectral Flow: The paper computes the spectral flow of a homotopy connecting a flat strip to the catenoid. It identifies specific eigenvalue crossings (particularly the $n=0$ even mode and the $n=2$ even pair) that drive the non-monotonic changes in the index (rising to 7 before settling at 4).
Rigidity Proof: The authors prove that the critical catenoid is rigid; the parameters defining the homotopy do not correspond to actual free boundary minimal surfaces for intermediate values. This clarifies that the homotopy deforms operators, not surfaces.

Results

Accuracy: The PINN recovers the exact Morse index of 4 and nullity of 2. The eigenfunctions match analytic solutions to plotting accuracy, and PDE residuals are of order $10^{-4}$ .
Spectral Flow Dynamics: The spectral flow reveals that the index is piecewise constant along the homotopy. The index rises to 7 in the intermediate regime (due to transient eigenvalues crossing the threshold) and drops to 4 only at the endpoint $s=1$ , where the free-boundary condition is strictly satisfied.
Rigidity Confirmation: The proof confirms that the intermediate operators in the homotopy do not correspond to any FBMS, validating the interpretation of the spectral flow as an operator-theoretic phenomenon rather than a geometric deformation of surfaces.

Significance and Claims
The paper explicitly states that the PINN is not competitive with classical eigensolvers for this specific one-dimensional problem. Its significance lies in the transferability of the pipeline. Unlike separation of variables, which fails for non-symmetric domains, the PINN construction (parity constraints, eigenvalue-as-parameter, loss structure) remains structurally unchanged when moving to two-dimensional problems.

The authors position this work as a calibration step for a future application to ellipsoidal balls. In such geometries, the boundary curvature is non-constant, the separation of variables is impossible, and the Morse index is not known. The paper demonstrates that the proposed pipeline can successfully recover known indices in the symmetric case, thereby establishing a foundation for addressing genuinely geometric families where analytic solutions are unavailable. The work does not claim to solve the ellipsoidal problem yet but provides the validated methodology required to attempt it.