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Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

Michael P. Brenner, Vincent Cohen-Addad, David Woodruff

TL;DR

A neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings is presented.

Abstract

This paper demonstrates that artificial intelligence can accelerate mathematical discovery by autonomously solving an open problem in theoretical physics. We present a neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search (TS) framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings. Specifically, the agent evaluated the core integral $I(N,α)$ for arbitrary loop geometries, directly improving upon recent AI-assisted attempts \cite{BCE+25} that only yielded partial asymptotic solutions. To substantiate our methodological claims regarding AI-accelerated discovery and to ensure transparency, we detail system prompts, search constraints, and intermittent feedback loops that guided the model. The agent identified a suite of 6 different analytical methods, the most elegant of which expands the kernel in Gegenbauer polynomials $C_l^{(3/2)}$ to naturally absorb the integrand's singularities. The methods lead to an asymptotic result for $I(N,α)$ at large $N$ that both agrees with numerical results and also connects to the continuous Feynman parameterization of Quantum Field Theory. We detail both the algorithmic methodology that enabled this discovery and the resulting mathematical derivations.

Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

TL;DR

A neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings is presented.

Abstract

This paper demonstrates that artificial intelligence can accelerate mathematical discovery by autonomously solving an open problem in theoretical physics. We present a neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search (TS) framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings. Specifically, the agent evaluated the core integral for arbitrary loop geometries, directly improving upon recent AI-assisted attempts \cite{BCE+25} that only yielded partial asymptotic solutions. To substantiate our methodological claims regarding AI-accelerated discovery and to ensure transparency, we detail system prompts, search constraints, and intermittent feedback loops that guided the model. The agent identified a suite of 6 different analytical methods, the most elegant of which expands the kernel in Gegenbauer polynomials to naturally absorb the integrand's singularities. The methods lead to an asymptotic result for at large that both agrees with numerical results and also connects to the continuous Feynman parameterization of Quantum Field Theory. We detail both the algorithmic methodology that enabled this discovery and the resulting mathematical derivations.
Paper Structure (30 sections, 77 equations, 3 figures, 1 table)

This paper contains 30 sections, 77 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology: AI-Accelerated Discovery
  3. Gemini Deep Think
  4. Tree Search (TS)
  5. Prompts and Intermittent Feedback
  6. Methodological Transparency
  7. General Structure
  8. Class I: Monomial Basis Approaches
  9. Preliminaries: The Taylor Coefficients
  10. Method 1: Generating Function Approach
  11. Method 2: Gaussian Integral Lifting
  12. Method 3: Hybrid Coordinate Transformation
  13. Class II: Spectral Basis Approaches
  14. Method 4: Spectral Galerkin (Matrix Method)
  15. Method 5: Spectral Volterra (Recurrence Method)
  16. ...and 15 more sections

Figures (3)

  • Figure 1: Verification of the Method 6 analytical solution. The solid curves represent the closed-form expression derived for the integral $I(N, \alpha)$. Open circles represent reference values obtained via direct numerical integration. The excellent agreement across a range of parameters $N$ and $\alpha$ validates the derivation. We note that for this range of $N$ all of the methods achieve similar level of accuracy.
  • Figure 2: Comparison of methods: absolute error and speed for $N=20$. The top panel shows the absolute error $|I_{\text{method}} - I_{\text{numerical}}|$ of the different methods for $N=20$ as a function of $\alpha$ on a logarithmic scale. The stable spectral methods hug the numerical noise floor, while Method 2 diverges and fails due to numerical instability. The bottom panel compares the speed of the methods, with the spectral methods evaluating orders of magnitude faster. We also note a distinct spike in computation time for Method 5 around $\alpha \approx 1.05$; this corresponds to a transient matrix conditioning issue near a root of the associated Legendre polynomials.
  • Figure 3: Convergence of the asymptotic models toward the exact spectral ground truth for $N=10,100,1000$