[Paper Review] Machine Learning Algebraic Geometry for Physics

Jiakang Bao, Yang‐Hui He+2 more|arXiv (Cornell University)|Apr 21, 2022

Geological Modeling and Analysis4 citations

TL;DR

This paper reviews the integration of machine learning (ML) with algebraic geometry in theoretical physics, focusing on how ML techniques—particularly supervised and unsupervised learning—can map tensor-based geometric structures in string theory, such as Calabi-Yau manifolds, polytopes, and Hilbert series. The key contribution is a conceptual and computational dictionary linking Kähler geometry, Hessian manifolds, and optimal transport to deep neural networks, especially generative models like GANs, via the Brenier potential and Monge-Ampère equations.

ABSTRACT

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

Motivation & Objective

• To investigate how machine learning can be used to analyze and classify complex algebraic geometric structures in theoretical physics, particularly in string theory.
• To establish a conceptual and computational correspondence between differential geometry (e.g., Kähler potentials, Hessian manifolds) and deep neural network architectures (e.g., GANs, autoencoders).
• To demonstrate that ML techniques such as principal component analysis (PCA), clustering, and neural networks can uncover hidden patterns and symmetries in high-dimensional geometric data from the string landscape.
• To explore the role of optimal transport theory in unifying geometric structures in algebraic geometry with generative models in machine learning.
• To propose a dictionary mapping geometric objects (e.g., Kähler potentials, metrics) to neural network components (e.g., discriminators, generators, potentials), enabling cross-domain insight.

Proposed method

• Uses supervised learning to map tensor representations of algebraic varieties (e.g., hypersurfaces, polytopes, Hilbert series) to physical or geometric invariants.
• Applies unsupervised techniques like PCA and clustering to identify low-dimensional structures and degrees of freedom in high-dimensional geometric data.
• Employs generative adversarial networks (GANs) to model the distribution of geometric objects, with the generator learning to produce synthetic Calabi-Yau manifolds or related structures.
• Establishes a correspondence between the Monge-Ampère equation in optimal transport and the Kähler potential in complex geometry, using the Brenier potential as a bridge.
• Maps the discriminator function in GANs to the Kantorovich potential in optimal transport, and identifies the generator’s output with the optimal transport map.
• Uses the Hessian manifold structure to relate the dual geometry of the Kähler potential to the loss landscape of neural networks, particularly via the Hessian of the generator’s objective function.

Experimental results

Research questions

• RQ1 How can machine learning be systematically applied to classify and predict invariants of Calabi-Yau threefolds and related algebraic varieties in the string landscape?
• RQ2 What is the mathematical correspondence between the Kähler potential in complex geometry and the Brenier potential in optimal transport theory as realized in GANs?
• RQ3 In what ways can unsupervised ML techniques such as PCA and clustering reveal the intrinsic dimensionality and symmetry structure of geometric data in algebraic geometry?
• RQ4 How can the generative process in GANs be interpreted as solving a geometric optimal transport problem on a manifold of algebraic varieties?
• RQ5 Can the duality between a manifold and its Hessian dual be mirrored in the loss landscape and optimization dynamics of a neural network?

Key findings

• The paper establishes a formal dictionary between Kähler geometry and deep generative models, identifying the Kähler potential with the Brenier potential in optimal transport.
• The Monge-Ampère equation governing optimal transport in GANs is shown to correspond to the complex Hessian equation in Kähler geometry, linking the two frameworks mathematically.
• The generator of a GAN can be interpreted as the optimal transport map from a latent distribution to the data manifold, with the discriminator function corresponding to the Kantorovich potential.
• The Hessian structure of the Kähler potential is found to mirror the Hessian of the generator’s loss function, suggesting a duality between geometric curvature and optimization landscape.
• The correspondence allows for the use of neural networks to sample from complex algebraic varieties such as Calabi-Yau manifolds, with the generated samples indistinguishable from real geometric data under the GAN’s discriminator.
• The framework enables the use of ML to explore the string landscape by learning to generate and classify geometric objects like brane webs, quiver mutations, and dessins d’enfants via tensor-based representations.