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Using "AI Poincare" to analyze non-linear integrable optics

Lazare Osmanov, Nilanjan Banerjee

TL;DR

The paper addresses automated discovery of conserved quantities in non-linear accelerator dynamics using AI Poincaré applied to the McMillan non-linear integrable optics system. It introduces a three-step workflow—pre-whitening, Monte-Carlo perturbations, and a neural network pullback to the manifold—coupled with PCA to infer local invariants and manifold dimensionality, validated on the McMillan map which preserves two invariants. The study identifies an optimal perturbation walk-length range $L \,\in\ [0.1,0.3]$ for reliable manifold learning, demonstrates improved network designs that enhance invariant recovery, and shows that AI Poincaré can extract invariants from real IOTA experimental data despite decoherence. These results suggest practical applicability for characterizing integrable optics and guiding accelerator design, with potential to reveal time-local invariants under realistic measurement conditions.

Abstract

This study dives into the applicability of using automated discovery of conserved quantities in dynamical systems relevant to accelerator physics. Specifically, we explore the performance of AI Poincaré in analyzing numerical trajectory data obtained using the McMillan system of non-linear integrable optics. A comprehensive evaluation of the algorithm's performance is conducted through diverse methodologies. These include the analysis of the estimated number of conserved quantities embedded in a dataset and the deviation of interpolated points on the inferred manifold with respect to points in actually in the dataset. the investigation identifies an optimal range of perturbation distances where the underlying manifold extraction algorithm inside AI Poincaré exhibits optimal performance. Additionally, an improved neural network architecture is proposed based on the observed results. Finally, we apply the algorithm to preliminary experimental data from the Integrable Optics Test Accelerator at Fermilab to successfully infer the number of conserved quantities even in the presence of fast decoherence of the measured signal.

Using "AI Poincare" to analyze non-linear integrable optics

TL;DR

The paper addresses automated discovery of conserved quantities in non-linear accelerator dynamics using AI Poincaré applied to the McMillan non-linear integrable optics system. It introduces a three-step workflow—pre-whitening, Monte-Carlo perturbations, and a neural network pullback to the manifold—coupled with PCA to infer local invariants and manifold dimensionality, validated on the McMillan map which preserves two invariants. The study identifies an optimal perturbation walk-length range for reliable manifold learning, demonstrates improved network designs that enhance invariant recovery, and shows that AI Poincaré can extract invariants from real IOTA experimental data despite decoherence. These results suggest practical applicability for characterizing integrable optics and guiding accelerator design, with potential to reveal time-local invariants under realistic measurement conditions.

Abstract

This study dives into the applicability of using automated discovery of conserved quantities in dynamical systems relevant to accelerator physics. Specifically, we explore the performance of AI Poincaré in analyzing numerical trajectory data obtained using the McMillan system of non-linear integrable optics. A comprehensive evaluation of the algorithm's performance is conducted through diverse methodologies. These include the analysis of the estimated number of conserved quantities embedded in a dataset and the deviation of interpolated points on the inferred manifold with respect to points in actually in the dataset. the investigation identifies an optimal range of perturbation distances where the underlying manifold extraction algorithm inside AI Poincaré exhibits optimal performance. Additionally, an improved neural network architecture is proposed based on the observed results. Finally, we apply the algorithm to preliminary experimental data from the Integrable Optics Test Accelerator at Fermilab to successfully infer the number of conserved quantities even in the presence of fast decoherence of the measured signal.
Paper Structure (7 sections, 3 equations, 9 figures)

This paper contains 7 sections, 3 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Description of AI Poincare
  3. Validating the fitted manifold
  4. Testing AI Poincare on real experimental data
  5. Optimum Network design for NEB
  6. Appendix
  7. Conclusions

Figures (9)

  • Figure 1: Schematics of FermiLab integrable optics acceleration "IOTA" c4
  • Figure 2: Graphical representation of the IOTA (Integrable Optics Test Accelerator) ring showcasing its structural layout. The Danilov-Nagaitsev non-linear element is placed in section BR, while the McMillan kick will be realized as an electron lens incorporated in section DR.
  • Figure 3: Explained ratio diagram obtained by applying AI Poincaré to a typical turn-by-turn phase-space dataset generated numerically for McMillan lens system
  • Figure 4: Relative deviation of invariants computed from points $\mathbf{x}' = P_\theta{\mathbf{y}}$ inferred from the pull network. Panels (a) and (b) depict relative error of the invariants $I_1$ and $I_2$ with respect to the original values, respectively, as a function of the rms walk length scale used during inference. Panels (c) and (d) show the uncertainty of the relative error predictions for invariants $I_1$ and $I_2$ respectively.
  • Figure 5: Representation of the displacement between the original point and corresponding pulled-back points for two distinct Neural Networks. The networks were individually trained for different perturbation lengths, on panel (a) the perturbation length $L_1=1.65$ and on panel (b) $L_2=0.13$.
  • ...and 4 more figures