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Pre-Strings Lectures on Artificial Intelligence

These lecture notes from the Pre-Strings 2026 school in Shanghai present a comprehensive overview of the bidirectional relationship between artificial intelligence and string theory, covering neural network fundamentals through a field-theoretic lens, the application of neural networks to define and solve field theories, and the use of AI techniques to advance research in string theory and related mathematical physics.

Original authors: James Halverson

Published 2026-07-07

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: James Halverson

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: A Three-Day Workshop

Imagine a physicist giving a three-day workshop to students who study the very fabric of the universe (string theory). The goal is to teach them how to use Artificial Intelligence (AI) not just as a calculator, but as a new way of thinking about physics.

The lectures are divided into three distinct days, moving from understanding the AI, to using AI to build physics, and finally using AI to solve physics problems.

Day 1: Understanding the AI Machine

The Analogy: The "Ensemble" of Dice Rollers

Usually, when we think of a neural network (the brain of modern AI), we think of one specific computer program. But the lecturer asks us to think differently. Imagine you don't just have one person rolling dice; you have a stadium full of people rolling dice.

Expressivity (How powerful is it?): The lecture explains that these networks are like a giant Lego set. With enough pieces (neurons) and the right connections, you can build almost any shape you can imagine. It mentions a famous math rule (Universal Approximation) that says even a simple network can mimic complex curves. It also introduces "Transformers" (the tech behind chatbots) as a way to read a story word-by-word, paying attention to how words relate to each other, rather than just looking at them in isolation.
Statistics (The Crowd's Opinion): Instead of asking one network what it thinks, we ask the whole stadium. If you start with random settings, the "average" answer of the whole crowd follows a very predictable pattern, like a bell curve. In physics terms, this is like a "free field theory"—a simple, calm state where everything is independent.
Dynamics (How it learns): When the network learns, it's like a hiker walking down a mountain to find the lowest valley (the best answer). The lecture explains that if the network is huge, the hiker barely moves the mountain; they just slide down a smooth, pre-determined path. However, if we tune the network just right, the hiker can actually reshape the mountain (learning new features), which is how deep learning really works.

Key Takeaway: AI isn't magic; it's a statistical system. If you look at a huge network as a crowd, its behavior follows the same math rules as gases or fluids in physics.

Day 2: Using AI to Build Physics

The Analogy: The "Reverse-Engineered Universe"

On the second day, the lecturer flips the script. Instead of using physics to understand AI, they use AI to create physics.

The Concept: Imagine you want to build a universe. Usually, you start with a rulebook (an equation) and see what happens. Here, you start with a neural network architecture and a set of random numbers (parameters). You treat the network's output as the "field" of your universe.
The Results:
- Free Theories: By setting up the network correctly, they can recreate simple, non-interacting particles (like a calm sea).
- Interacting Theories: By tweaking the rules of the network, they can create "interactions" (like waves crashing). They successfully recreated Liouville theory (a complex 2D universe) and the Bosonic String (a model of vibrating strings).
- The "Magic" Number 26: One of the biggest mysteries in string theory is why the universe needs 26 dimensions to work mathematically. Using this AI approach, the lecturer derived the number 26 simply by counting the "modes" or settings of the network. It's like discovering a secret code by counting the buttons on a remote control.
- Topological Defects: They even showed how to create "vortices" (swirls in a fluid) and phase transitions (like water freezing) by adding special "discrete" switches to the network.

Key Takeaway: You don't need a traditional equation to define a universe. You can define a universe by the architecture of a neural network and the probability of its settings. It turns out, AI is a valid way to construct the laws of physics.

Day 3: AI as a Research Assistant

The Analogy: The "Super-Intern"

The final day is about practical tools. The lecturer introduces AI Agents.

What is an Agent? Imagine a very smart intern who can read papers, write code, run simulations, check its own work, and ask for help if it gets stuck. It doesn't just answer a question; it does the work.
The Applications:
- Solving Geometry: String theory requires complex shapes called Calabi-Yau manifolds. We know these shapes exist, but we don't know their exact "metric" (how to measure distances on them). The AI uses a technique called "Physics-Informed Neural Networks" to guess the shape and then corrects itself until the math works out perfectly.
- Searching the Landscape: There are billions of possible universes (vacua) in string theory. Finding the one that looks like our world is like finding a needle in a haystack. The AI uses "Reinforcement Learning" (trial and error with rewards) to navigate this haystack much faster than a human could, even inventing new strategies to find the needle.
- Untying Knots: In math, deciding if a knot is actually just a loop (the "unknot") is hard. The AI learned to untie knots by learning a set of moves, effectively solving a math problem and providing a proof that a human can check.
- Conjecture Generation: The AI can look at data and spot patterns that humans miss. It can suggest a new mathematical rule (a conjecture), which a human mathematician can then try to prove.

Key Takeaway: AI agents are changing the workflow of physics. They handle the tedious "plumbing" (coding, checking, searching), allowing human physicists to focus on the big ideas and the "why."

Summary

This paper argues that the relationship between AI and Physics is a two-way street:

Physics explains AI: The math of neural networks is surprisingly similar to the math of quantum fields.
AI explains Physics: We can use neural networks to define new physical theories and solve problems that were previously too hard to calculate.
AI helps Physicists: By using "agents" that can think, act, and verify, the barrier to doing complex physics research is lowering, making it possible for researchers to tackle bigger questions faster.

The lecturer concludes that we are at the very beginning of a new era where AI is not just a tool, but a partner in the discovery of how the universe works.

Technical Summary: Pre-Strings Lectures on Artificial Intelligence

Problem Statement

The paper addresses the rapidly evolving intersection of theoretical physics (specifically string theory) and artificial intelligence. It identifies three primary gaps:

Theoretical Understanding: A lack of a unified field-theoretic framework to understand the expressivity, statistics, and dynamics of neural networks (NNs).
New Formulations of Physics: The need for a novel approach to define and compute field theories without relying on traditional path integrals or Lagrangians, potentially offering new insights into string theory and quantum field theory (QFT).
Practical Implementation: The high barrier to entry for applying ML techniques in string theory, often bogged down by "plumbing" (data wrangling, environment setup, debugging), and the need for interpretable methods that move beyond black-box prediction to conjecture generation and rigorous proof.

The lectures, delivered at the Pre-Strings 2026 school, aim to bridge these gaps by providing a field-theoretic lens on NNs, defining "Neural Network Field Theory" (NN-FT), and demonstrating how modern agentic AI workflows can accelerate research in string theory.

Methodology

The paper is structured as a three-day lecture series, progressing from foundational theory to advanced applications:

Day 1: Neural Network Essentials (Field-Theoretic Lens)

The author reframes standard ML concepts using physics terminology:

Expressivity: Analyzed via Universal Approximation Theorems (UAT) and the Kolmogorov-Arnold Representation Theorem. The latter motivates Kolmogorov-Arnold Networks (KANs), where learnable activation functions sit on edges rather than nodes. The role of Transformers and "chain-of-thought" reasoning is framed as an extension of expressivity through time/token depth.
Statistics: Neural network ensembles at initialization are treated as statistical field theories. The Neural Network / Gaussian Process (NNGP) correspondence is derived via the Central Limit Theorem (CLT) in the infinite-width limit ( $N \to \infty$ ), identifying the network as a generalized free field. Non-Gaussianities (interactions) arise from finite- $N$ corrections ( $1/N$ ) or breaking statistical independence. Symmetries are realized by absorbing transformations into the parameter density.
Dynamics: Gradient descent is analyzed via the Neural Tangent Kernel (NTK). In the infinite-width limit, the NTK freezes, leading to "lazy" training where features do not learn. A scaling analysis (following Pehlevan and Bordelon) identifies specific parameterizations (e.g., $\mu P$ ) that allow for non-trivial feature learning in finite time.

Day 2: Neural Network Field Theory (NN-FT)

The logic is reversed: a field theory is defined not by an action $S[\phi]$ , but by an architecture $\phi_\theta$ and a parameter density $P(\theta)$ .

Definition: An NN-FT is the data $(\phi_\theta, P(\theta))$ , with the partition function defined as an integral over parameters: $Z[J] = \int d\theta P(\theta) e^{\int J \phi_\theta}$ .
Engineering Theories:
- Free Theories: Engineered by satisfying CLT conditions and shaping the two-point function (spectrum) via architecture design (e.g., "Scalar-net" reproduces the free scalar propagator).
- Interactions: Introduced by deforming the parameter density $P(\theta)$ (e.g., inserting a potential $V(\phi)$ ) or breaking independence, rather than modifying an action.
- Quantumness: Defined by satisfying the Osterwalder-Schrader (OS) axioms (Euclidean invariance, reflection positivity, etc.) on the correlators.
Formalism: Ward identities and anomalies are derived via change-of-variables in parameter space, leading to a "Master Identity" involving the score function and Jacobian of the flow.

Day 3: Applied AI for Strings

Focuses on practical implementation using modern "agentic" workflows.

Agents: Reasoning models (LLMs with reinforcement learning) that can plan, execute code, search literature, and verify results. These agents lower the implementation barrier for ML-for-physics.
Physics-Informed Neural Networks (PINNs): Used to solve the Monge-Ampère equation to approximate Ricci-flat Calabi-Yau metrics, a problem previously intractable analytically. The method uses a surrogate loss based on the Monge-Ampère equation rather than direct Ricci tensor minimization.
Reinforcement Learning (RL): Applied to search the string landscape (e.g., finding consistent D-brane configurations satisfying tadpole cancellation and supersymmetry) and to solve topological problems (e.g., unknotting knots via braid word simplification).
Conjecture Generation: Using interpretable models (decision trees, KANs, symbolic regression) to identify patterns in data, formulate conjectures, and guide rigorous proofs (e.g., in knot theory and F-theory gauge groups).

Key Contributions and Results

1. Theoretical Unification (NN-FT)

The paper establishes that Neural Network Field Theory is a valid and powerful framework for defining field theories.

Universality: It proves that any probability distribution over tempered distributions (including Euclidean QFTs) admits a NN-FT description via Borel isomorphism.
Liouville Theory: The authors construct Liouville theory as an NN-FT where all interactions are carried by a conditioned zero-mode. Numerical Monte Carlo in parameter space reproduces the DOZZ structure constants to within a few percent.
Bosonic String: The gauge-fixed worldsheet theory of the bosonic string is realized as an infinite-width NN ensemble. The authors analytically compute the Virasoro-Shapiro and Veneziano amplitudes directly in parameter space, recovering the standard results without a worldsheet action.
Critical Dimension: A new derivation of the critical dimension $D=26$ for the bosonic string is presented. It arises from a mode-counting mismatch in the Jacobian of the parameter-space flow under Weyl rescaling, rather than normal ordering or central charge cancellation.
Topological Transitions: The Berezinskii-Kosterlitz-Thouless (BKT) transition is realized by introducing discrete latent variables (vortices) into the NN ensemble, successfully reproducing the universal jump in stiffness and vortex proliferation.

2. Dynamics and Scaling

The paper clarifies the conditions for feature learning in deep networks, identifying the $\mu P$ parameterization as the unique solution that allows features to evolve in finite time, contrasting with the "lazy" NTK regime.
It establishes a hierarchy of metric flows, showing that Ricci flow is a special case of a neural network metric flow in the infinite-width, local, Perelman-loss limit.

3. Applied AI and Agentic Workflows

Calabi-Yau Metrics: The paper demonstrates state-of-the-art numerical computation of Calabi-Yau metrics using PINNs, enabling the calculation of previously inaccessible quantities like Yukawa couplings and Kaluza-Klein modes.
RL in String Landscapes: An RL agent is shown to discover consistent string vacua up to 200 times faster than random search and to invent novel strategies (e.g., "filler branes") not previously codified by humans.
Rigorous Math: RL agents are used to produce human-verifiable certificates for unknotting knots and to eliminate candidate counterexamples to the smooth 4D Poincaré conjecture.
Conjecture Generation: The paper highlights successful cases where ML identified new invariants (e.g., the "natural slope" in knot theory) and proved conjectures regarding F-theory gauge groups.

Significance and Claims

The paper claims that the integration of AI and physics is entering a new phase driven by agentic reasoning models.

Lowered Barriers: The primary significance of the "Day 3" results is that agentic workflows have reduced the implementation cost of ML-for-physics techniques by an order of magnitude, allowing researchers to focus on physics rather than engineering.
New Derivations: The NN-FT framework provides a "parameter-space" perspective that yields new derivations of classic results (e.g., $D=26$ , string amplitudes) and offers a rigorous probabilistic definition of field theories without an action.
From Prediction to Understanding: The paper argues that the future of ML in physics lies not just in prediction, but in conjecture generation and interpretability. By using agents to search for patterns and humans to verify and prove them, the field can move toward discovering new mathematical structures and physical laws.
Modest Scope: The author explicitly notes that while agents lower the cost of implementation, they do not replace human judgment. The responsibility for correctness, taste, and the final verification of results remains with the physicist. The results presented are often "proofs of concept" or specific case studies (e.g., Liouville theory, specific string amplitudes) rather than a complete overhaul of all string theory methods.

In summary, the paper posits that neural networks are not just tools for data analysis but are themselves a new language for defining field theories, and that agentic AI is the catalyst that will make these advanced techniques accessible and productive for the broader string theory community.