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Variational autoencoders understand knot topology

Anna Braghetto, Sumanta Kundu, Marco Baiesi, Enzo Orlandini

TL;DR

The paper develops a variational autoencoder augmented with a knot-type classifier (VAEC) to identify and generate knotted polymer configurations under spherical confinement. By training on 3D coordinates, the model learns a topology-aware latent space where knots cluster by family and complexity, and chirality can be distinguished even for knots not seen during training. The approach yields high classification accuracy and enables generation of topologically faithful configurations, with latent axes correlating to unknotting number and braid index. It also demonstrates robust generalization to longer chains and enhanced chirality recognition beyond traditional polynomial invariants, highlighting a promising route for topology-aware generative ML in polymer physics.

Abstract

Supervised machine learning (ML) methods are emerging as valid alternatives to standard mathematical methods for identifying knots in long, collapsed polymers. Here, we introduce a hybrid supervised/unsupervised ML approach for knot classification based on a variational autoencoder enhanced with a knot type classifier (VAEC). The neat organization of knots in its latent representation suggests that the VAEC, only based on an arbitrary labeling of three-dimensional configurations, has grasped complex topological concepts such as chirality, unknotting number, braid index, and the grouping in families such as achiral, torus, and twist knots. The understanding of topological concepts is confirmed by the ability of the VAEC to distinguish the chirality of knots $9_{42}$ and $10_{71}$ not used for its training and with a notoriously undetected chirality to standard tools. The well-organized latent space is also key for generating configurations with the decoder that reliably preserves the topology of the input ones. Our findings demonstrate the ability of a hybrid supervised-generative ML algorithm to capture different topological features of entangled filaments and to exploit this knowledge to faithfully reconstruct or produce new knotted configurations without simulations.

Variational autoencoders understand knot topology

TL;DR

The paper develops a variational autoencoder augmented with a knot-type classifier (VAEC) to identify and generate knotted polymer configurations under spherical confinement. By training on 3D coordinates, the model learns a topology-aware latent space where knots cluster by family and complexity, and chirality can be distinguished even for knots not seen during training. The approach yields high classification accuracy and enables generation of topologically faithful configurations, with latent axes correlating to unknotting number and braid index. It also demonstrates robust generalization to longer chains and enhanced chirality recognition beyond traditional polynomial invariants, highlighting a promising route for topology-aware generative ML in polymer physics.

Abstract

Supervised machine learning (ML) methods are emerging as valid alternatives to standard mathematical methods for identifying knots in long, collapsed polymers. Here, we introduce a hybrid supervised/unsupervised ML approach for knot classification based on a variational autoencoder enhanced with a knot type classifier (VAEC). The neat organization of knots in its latent representation suggests that the VAEC, only based on an arbitrary labeling of three-dimensional configurations, has grasped complex topological concepts such as chirality, unknotting number, braid index, and the grouping in families such as achiral, torus, and twist knots. The understanding of topological concepts is confirmed by the ability of the VAEC to distinguish the chirality of knots and not used for its training and with a notoriously undetected chirality to standard tools. The well-organized latent space is also key for generating configurations with the decoder that reliably preserves the topology of the input ones. Our findings demonstrate the ability of a hybrid supervised-generative ML algorithm to capture different topological features of entangled filaments and to exploit this knowledge to faithfully reconstruct or produce new knotted configurations without simulations.

Paper Structure

This paper contains 14 sections, 2 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Simulations and Neural Networks
  3. Simulations
  4. Variational Autoencoders with Classifier
  5. Architecture
  6. Validation
  7. Simple knots
  8. Generalization on longer chains
  9. Knots up to seven crossings
  10. Including chirality
  11. Knot families
  12. Knot complexity
  13. Distinguishing among $9_{42}^\pm$ and $10_{71}^\pm$
  14. Conclusions

Figures (13)

  • Figure 1: Sketches (a) of an achiral knot, (b) a torus knot, and (c) a twist knot.
  • Figure 2: Two complex knots whose chirality cannot be computed with any known polynomial invariant.
  • Figure 3: Table of knots used for the training of the VAEC.
  • Figure 4: Sketches (a) of the standard VAE architecture and (b) of the VAEC implemented in this study. The input to the encoder consists of a set of coordinates $\bm{x} = (x_i,y_i,z_i)$ for $i\le N$, representing a polymer configuration. These coordinates are mapped into a latent space by the encoder. Such a low-dimensional $\bm{z}$ representation is re-expanded to three-dimensional ring chains by the decoder. In addition, in the VAEC, the latent representation of the polymer serves as an input for the classifier designed to predict the topology of the polymer.
  • Figure 5: Kernel manipulation to make the model suitable for chains longer than $N$. Each filter is doubled for chains of length $2N$. Iterating this step, we deal with chains of length $4N$.
  • ...and 8 more figures