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

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.

• The paper introduces a parity-constrained PINN method that accurately recovers the Morse index and nullity of the critical catenoid.
• It reduces the Jacobi–Steklov eigenvalue problem to one-dimensional Robin boundary-value problems using rotational symmetry and Fourier modes.
• The study tracks spectral flow along operator homotopy and validates the rigidity of the catenoid, opening paths for non-standard domain analysis.

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

Mathematical Framework and Motivation

The Morse index of a free boundary minimal surface (FBMS) quantifies instability, counting directions in which the area can be decreased via normal variations. For FBMS in the unit ball $\mathbb{B}^3$, classical results establish that the Morse index is encoded in the combined spectrum of fixed-boundary eigenvalue problems and the Jacobi–Steklov Dirichlet-to-Neumann map. The critical catenoid is the prototypical minimal annulus in $\mathbb{B}^3$, with a rigorously established Morse index of $4$ and nullity of $2$, as proved in Tran's spectral characterization (2606.21725). This work aims to numerically recover these spectral invariants using Physics-Informed Neural Networks (PINNs), leveraging their applicability to settings where analytic solutions are unavailable.

Jacobi–Steklov Eigenvalue Reduction and Spectral Characterization

To compute the Morse index numerically, the authors exploit rotational symmetry to reduce the Jacobi–Steklov eigenvalue problem to a sequence of one-dimensional Robin boundary-value problems, parameterized by Fourier mode $n$. For each mode, the physical boundary and eigenvalue threshold are precisely determined by geometric constraints and the Steklov framework, with the threshold $\delta = 1$ stemming from the mean curvature of the ambient boundary. The full spectrum, crucial for index recovery, is known analytically for the catenoid, providing a rigorous benchmark.

Figure 1: The critical catenoid embedded in $\mathbb{B}^3$ as the geometric setting for spectral analysis.

PINN Architecture and Parity Constraints

The PINN approach represents each eigenfunction as a neural network, explicitly enforcing mode parity via symmetrized network outputs ($f(t) = N(t) \pm N(-t)$ for even/odd modes). The eigenvalue $\delta$ is treated as a trainable parameter in the boundary loss, rather than pre-specified. The loss function incorporates the PDE residual, Robin boundary residual, and normalization to avoid trivial solutions. This parity constraint is essential, as unconstrained networks typically converge to spurious mixed-parity states that fail to recover accurate eigenvalues.

Numerical Results: Index and Nullity Recovery

The PINN solver trained on the critical catenoid recovers the Jacobi–Steklov eigenvalues below the threshold to within $10^{-6}$ to $\mathbb{B}^3$0 of analytic values. The fixed-boundary problem contributes one index via the Jacobi field, and PINN correctly finds the three $\mathbb{B}^3$1 modes—one $\mathbb{B}^3$2 odd and two $\mathbb{B}^3$3 even modes. Summing these yields the Morse index $\mathbb{B}^3$4 and nullity $\mathbb{B}^3$5 exactly. Notably, PINN achieves low PDE residuals ($\mathbb{B}^3$6), confirming accuracy for smooth, low-frequency eigenfunctions.

Figure 2: Top: analytic vs. PINN eigenfunctions for lowest Jacobi–Steklov modes; bottom: spectrum, with index and nullity annotated according to Theorem~\ref{thm:tran}.

The spectral gap is explicit: only three eigenvalues are strictly below the threshold, with no modes in $\mathbb{B}^3$7, echoing classical spectral separation in minimal surface theory.

Spectral Flow along Operator Homotopy

The eigenvalue spectrum is tracked along a one-parameter homotopy interpolating between a flat reference operator ($\mathbb{B}^3$8) and the critical catenoid Jacobi operator ($\mathbb{B}^3$9). At each $4$0, Jacobi–Steklov eigenvalues are computed, and index jumps are identified as eigenvalues cross $4$1. The index varies non-monotonically: it rises to $4$2 before settling at the catenoid value $4$3, indicating that only at the catenoid endpoint are geometric and spectral constraints simultaneously satisfied.

Figure 3: Spectral flow along operator homotopy, with eigenvalue crossings and index plateaus highlighted.

This flow counts the crossings and their Fourier types, providing an informative classification of the spectral events required to reach the catenoid's index.

Rigidity Analysis and Implications for Higher-Dimensional Settings

A formal rigidity result establishes that the catenoid is unique: the free-boundary condition, radius constraint, and minimality fix all parameters, precluding any genuine geometric deformation through FBMSs in $4$4. Varying the ambient radius is spectrally trivial, simply rescaling the threshold and spectrum without changing the index. The only nontrivial spectral flows arise in domains where curvature varies, such as ellipsoidal balls, for which neither analytic nor classical theory determines the Morse index. Here, PINN-based spectral pipelines—now validated—can provide genuinely geometric spectral flow and index computation.

Discussion and Theoretical/Practical Implications

The parity-constrained PINN scheme successfully calibrates against the critical catenoid, thus providing a robust numerical pipeline for spectral index computation where separation of variables is unavailable. This setup is poised to tackle the Jacobi–Steklov spectrum and Morse index for free boundary minimal surfaces in non-standard domains (e.g., ellipsoids), where analytic characterization fails due to non-constant boundary curvature. As such, future applications of PINN to two-dimensional spectral problems on parametrized geometric domains would enable the probing of open index problems and the tracking of genuinely geometric spectral flows.

On the theoretical side, this approach demonstrates the feasibility and reliability of PINN methods for spectral geometry on minimal surfaces, with spectral gap and index count numerical accuracy matching analytic benchmarks. Practically, it opens the door to robust index computation and spectral flow tracking in geometric configurations where classical methods, separation of variables, or spectral theory provide no closed formulas.

Conclusion

This paper validates a parity-constrained PINN pipeline for numerically recovering the Morse index and nullity of the critical catenoid in $4$5, matching analytic values to high accuracy. The spectral flow analysis along operator homotopy classifies index jumps and highlights the rigidity of the catenoid's geometric constraints. These results demonstrate both the necessity and future utility of neural spectral methods for investigating Morse index and spectral flow in genuinely geometric settings such as ellipsoidal balls, where analytic theory is absent.