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A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem

This paper serves as a methodological companion to a study on minimal surfaces in hyperbolic space, detailing how encoding geometric constraints directly into the neural network architecture and optimizing PDE residual evaluation through second-order jets and graph compilation can reduce training costs by a factor of forty to fifty, thereby enabling efficient Physics-Informed Neural Network applications in geometric analysis.

Original authors: Tancredi Schettini Gherardini

Published 2026-08-03

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Tancredi Schettini Gherardini

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 Shape of Knots and the Art of Teaching Computers

Imagine you are a master sculptor, but instead of clay, you are working with invisible, stretchy soap films. In the world of mathematics, there is a famous puzzle called the "Plateau problem." It asks a simple question: if you dip a wire loop shaped like a knot into soapy water, what shape will the soap film take? Usually, nature is smart; it wants to use the least amount of energy possible, so the film settles into a "minimal surface"—the smoothest, flattest shape it can be while still holding onto the wire.

Now, imagine taking this idea to the extreme. Instead of a wire in a bathtub, imagine a knot floating in a strange, curved universe called "hyperbolic space," where the rules of geometry are warped. In this universe, the knot isn't just a loop in the middle of the room; it's a shape drawn on the very edge of the world, infinitely far away. The question becomes: can we find a perfect, minimal soap film that stretches out from the center of this universe to meet that distant knot? Mathematicians have known for a long time that such a film exists, but actually finding its exact shape is like trying to solve a maze that keeps changing its walls. This is where the paper steps in, using a special kind of artificial intelligence to do the heavy lifting.

The Paper: Teaching AI to Be a Geometry Genius

This paper is essentially a "user's guide" for mathematicians who want to use a specific type of AI called Physics-Informed Neural Networks (PINNs) to solve these tricky geometry problems. The author, Tancredi Schettini Gherardini and Marco Usula, didn't just use AI to guess the answer; they built a super-smart, highly efficient system that provided numerical evidence for a deep connection between the shape of knots and a complex algebraic formula called the HOMFLY polynomial.

Think of the HOMFLY polynomial as a secret code hidden inside every knot. If you twist a knot a certain way, the code changes. The author wanted to see if the number of times their AI-generated soap films crossed over themselves (self-intersections) matched the numbers in this secret code. They tested this on many different knots, from simple loops to complex "figure-eight" shapes, and found that the AI's results matched the code perfectly every time. This gives strong numerical evidence for a conjecture by mathematician Joel Fine, suggesting that the geometry of these infinite soap films is deeply tied to the algebra of knots.

How They Made the AI Work (The Two Big Secrets)

The paper admits that if you just throw a standard AI at this problem, it will likely fail or take forever to learn. The author discovered that the difference between a successful AI and a useless one comes down to two clever tricks they used to "hard-code" the rules of the universe into the AI's brain.

1. Don't Ask the AI to Learn the Rules; Build Them In
Usually, when you train an AI, you give it a goal (like "minimize energy") and a penalty if it breaks the rules (like "if you don't touch the knot, add a penalty"). The author realized this approach is too messy. Instead, they built the AI so that it is physically impossible for it to break the rules.

Imagine you are teaching a robot to draw a circle. A normal approach is to say, "Draw a line, and if it's not round, I'll give you a bad grade." The AI might draw a square, get a bad grade, and try again, but it might never get it perfect. The author's approach was different: they built the robot's arm so that it only moves in a circle. No matter how the robot tries to move, it can only draw a circle. In the paper, they designed the AI's structure so that it always touches the knot at infinity and always behaves correctly at the edge of the universe. This meant the AI didn't have to waste time learning the basics; it only had to figure out the tricky details in the middle. This removed the need for a "balancing act" between different penalties and made the training much more reliable.

2. Stop Re-inventing the Wheel Every Time
The second trick was about speed. To train the AI, the computer has to calculate how the shape changes at every single point, which involves complex math called "derivatives." The standard way of doing this is like asking a student to re-derive a math formula from scratch every single time they take a test. It's accurate, but incredibly slow.

The author realized they could do this much faster by using a technique called "forward propagation of second-order jets." In plain English, instead of asking the computer to re-calculate the slope and the curvature of the shape from scratch every time, they taught the computer to carry the "slope" and "curvature" along with the shape as it moves through the network, like a backpack. They also built a "compiler" that took the entire set of instructions and turned them into a single, super-fast machine code program, rather than rebuilding the instructions every time.

The result? These two changes made the training process 40 to 50 times faster. On a regular laptop, a task that would have taken days or required a massive supercomputer was finished in about 40 minutes. Without these tricks, the author says, their entire study of dozens of knots would have required a cluster, or would not have been done on ordinary computers.

What This Means for the Future

The paper doesn't claim to have solved every knot problem in the universe, but it has provided a powerful new tool. By showing that they could get the AI to produce results that match deep mathematical theories, they have opened the door for other mathematicians to use similar methods. They suggest that with these efficient techniques, we might soon be able to use computers to help prove the existence of complex shapes in geometry, turning numerical experiments into solid mathematical proofs.

In short, the author didn't just find a way to make soap films in a computer; they figured out how to teach the computer to understand the rules of the game so well that it can play perfectly, fast enough to help us solve some of the oldest puzzles in geometry.

Technical Summary: A User's Guide to PINNs in Geometric Analysis

Problem Statement
The paper addresses the asymptotic Plateau problem in hyperbolic space $\mathbb{H}^{n+1}$ : determining whether a closed submanifold $M^k$ at the sphere at infinity bounds a complete, properly immersed minimal submanifold. Specifically, the author focuses on surfaces in $\mathbb{H}^4$ bounding knots $K \subset S^3$ . While Anderson's theorem guarantees the existence of area-minimizing solutions, the structure of the full solution space remains complex. The author aims to numerically construct near-minimal discs asymptotic to prescribed knots to test a conjecture by Joel Fine, which posits a relationship between the signed count of self-intersections of these minimal surfaces and the coefficients of the HOMFLY polynomial.

Methodology: A Physics-Informed Neural Network (PINN) Framework
Rather than reviewing the geometric results (detailed in the companion preprint arXiv:2605.26234v2), this note focuses on the methodological framework required to make the PINN approach viable for this specific geometric analysis problem. The author identifies two critical pillars that determine the success of the method:

Hard-Constrained Architecture (Encoding Geometry):
- The Challenge: Standard PINNs minimize a weighted sum of interior PDE residuals and boundary penalties. This approach fails here because the boundary is at infinity (where the metric degenerates), and balancing the weights ( $\lambda$ ) between boundary and interior accuracy is fragile. Furthermore, generic networks must "learn" known regularity properties (e.g., orthogonal intersection with the boundary at infinity), wasting capacity.
- The Solution: The model architecture is designed to satisfy boundary conditions and asymptotics exactly for all learnable parameters $\theta$ $θ$ .
  - The map $u_\theta$ is constructed as a composite: $u_\theta = (\rho e^{NN_X}, \text{ext}(\gamma) + \rho^k NN_Y)$ .
  - $\rho$ is a boundary-defining function (vanishing at $\partial D^2$ ), ensuring the image lies in $\mathbb{H}^4$ and reaches the boundary at infinity precisely over the knot.
  - $\text{ext}(\gamma)$ is a stereobiharmonic extension of the knot $\gamma$ . This choice ensures the extension is smooth and satisfies the Neumann condition (orthogonality) required for asymptotic minimality.
  - Result: The boundary term is removed entirely from the loss function. The network only learns the interior perturbation, and the loss function becomes a single-component term encoding only the minimality condition (vanishing tension field).
Engineered Residual Evaluation (Computational Efficiency):
- The Challenge: Evaluating the loss for a second-order PDE requires first and second derivatives of the model. The implementation uses nested reverse-mode automatic differentiation (AD), rebuilding the computational graph at every optimization step. This is computationally prohibitive, especially on CPUs where the overhead of graph construction dominates arithmetic.
- The Solution: The author replaces standard AD with a custom, compiled pipeline:
  - Forward Propagation of 2-Jets: Instead of using AD for spatial derivatives, the author explicitly propagates second-order jets (value, Jacobian, Hessian) through the network layers using forward-mode differentiation rules (Taylor-mode). This avoids building a "graph of graphs" for spatial derivatives.
  - Static Compilation: The entire residual evaluation pipeline (jet propagation, metric algebra, residual calculation) is compiled once into a static computational graph (using torch.compile). The training loop is designed to be shape-static (fixed batch sizes, fixed collocation points) to prevent recompilation.
- Result: These changes reduce the cost of a training step by a factor of 40–50 on identical hardware (laptop CPUs), making the training of dozens of models feasible without cluster resources.

Key Results and Performance

Accuracy: The framework successfully produces near-minimal discs for various knots (unknot, torus knots, figure-eight, etc.). The signed counts of self-intersections computed numerically agree with the coefficients of the HOMFLY polynomial, corroborating Fine's conjecture.
Efficiency:
- Baseline: Nested reverse-mode AD takes ~427 ms per step (Adam) on a laptop CPU.
- Optimized: Forward 2-jets with static compilation takes ~8.6 ms per step.
- Precision: The author emphasizes the necessity of double precision (float64). Single precision causes rounding errors in second-derivative computations that saturate residuals near $10^{-4}$ , preventing convergence to the precision required for topological analysis.
Robustness: The hard-constrained design ensures that the residual vanishes along the boundary even before training begins, allowing the optimizer to focus solely on the interior elliptic region.

Significance and Claims
The paper positions itself as a methodological companion to the geometric results. Its primary contribution is not a new geometric theorem, but a set of practical guidelines for applying PINNs to geometric variational problems where:

Constraints can be parametrized: Boundary conditions and asymptotics should be built into the model architecture rather than penalized in the loss.
Regularity is known: Prior knowledge of solution behavior (e.g., polyhomogeneous expansions) should be encoded in the ansatz.
Differentiation is engineered: For low-dimensional domains, explicit forward-mode jet propagation combined with static compilation is vastly superior to nested reverse-mode AD.

The author claims that without these specific methodological choices, the study of [18] (the companion paper) would have been infeasible on standard hardware, requiring a cluster or failing entirely. The note aims to lower the "activation energy" for other researchers in differential geometry to deploy PINNs on similar problems, emphasizing that the difference between a successful and failed PINN application often lies in these undocumented implementation choices rather than the network architecture itself.

The Shape of Knots and the Art of Teaching Computers

The Paper: Teaching AI to Be a Geometry Genius

How They Made the AI Work (The Two Big Secrets)

What This Means for the Future

Technical Summary: A User's Guide to PINNs in Geometric Analysis

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