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MultiSTOP: Solving Functional Equations with Reinforcement Learning

Alessandro Trenta, Davide Bacciu, Andrea Cossu, Pietro Ferrero

TL;DR

MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics, is developed by adding multiple constraints derived from domain-specific knowledge, even in integral form, to improve the accuracy of the solution.

Abstract

We develop MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics. This new methodology produces actual numerical solutions instead of bounds on them. We extend the original BootSTOP algorithm by adding multiple constraints derived from domain-specific knowledge, even in integral form, to improve the accuracy of the solution. We investigate a particular equation in a one-dimensional Conformal Field Theory.

MultiSTOP: Solving Functional Equations with Reinforcement Learning

TL;DR

MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics, is developed by adding multiple constraints derived from domain-specific knowledge, even in integral form, to improve the accuracy of the solution.

Abstract

We develop MultiSTOP, a Reinforcement Learning framework for solving functional equations in physics. This new methodology produces actual numerical solutions instead of bounds on them. We extend the original BootSTOP algorithm by adding multiple constraints derived from domain-specific knowledge, even in integral form, to improve the accuracy of the solution. We investigate a particular equation in a one-dimensional Conformal Field Theory.
Paper Structure (21 sections, 18 equations, 7 figures, 1 algorithm)

This paper contains 21 sections, 18 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. MultiSTOP
  3. The Conformal Bootstrap Equation
  4. BootSTOP
  5. MultiSTOP: enforcing additional physical constraints
  6. Results
  7. Weak coupling and the problem of degeneracy
  8. Strong coupling
  9. Conclusion
  10. Details on the model and the algorithm
  11. Details on the model's equations and constraints
  12. A short RL introduction
  13. Algorithm details
  14. Implementation remarks
  15. Speed-up techniques
  16. ...and 6 more sections

Figures (7)

  • Figure 1: Results on $C_2^2, C_3^2$ at weak coupling. Green regions represent the bounds from 1DCFTConstraints+smallg. The blue dots are the results for the best $25$ runs based on reward for some values of $g$. Results have high precision, with a standard error of around $0.1\%$.
  • Figure 2: Strong coupling results for $C_4^2, C_5^2$ and $C_4^2 +C_5^2 + C_6^2 + C_8^2$ as a function of the coupling constant $g$. Blue points represent the individual values for the best $25$ runs based on reward, while red points, lines and bars represent their means with standard deviation. Green lines represent theoretical expectations. Standard error on $C_4^2, C_5^2$ is around $10-50\%$ or worse and increases with $g$ due to the degeneracy problem. Standard error on $C_4^2 +C_5^2 + C_6^2 + C_8^2$ is calculated on the values of the sum for each run. Standard error is always below $0.1\%$, making the vertical bars invisible in figure \ref{['fig:OPEstrongsum_results']}.
  • Figure 3: Values for the first $10$ scaling dimensions $\Delta_n$ in the $1D$ CFT of interest, from 1DCFTConstraints+smallg.
  • Figure 4: Experimental results on the values of the rewards as a function of the coupling constant $g$. The green line represents reward of the best performing run only, blue line and region represent the average of the best $25$ runs and the standard deviation. The green line and blue region are almost invisible, showing the stability of the method.
  • Figure 5: Strong coupling results for $C_6^2$ as a function of the coupling constant $g$. Blue points represent the individual values for the best $25$ runs based on reward, while red points and bars represent their means with standard deviation. Standard error is around $10-50\%$ or worse and increases with $g$ due to the degeneracy problem.
  • ...and 2 more figures