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An end-to-end strategy for recovering a free-form potential from a snapshot of stellar coordinates

Wassim Tenachi, Rodrigo Ibata, Foivos I. Diakogiannis

TL;DR

The paper tackles the problem of recovering a free-form gravitational potential from snapshot phase-space data of stars. It develops an end-to-end differentiable pipeline that learns a neural-network potential $\Phi_\alpha$ from phase-space data, maps to actions $\vec{J}$, and uses a phase-space density $f(\vec{J})$ estimated by normalizing flows or diffusion models to maximize a likelihood. It then distills $\Phi_\alpha$ into an interpretable analytical form via the physics-aware symbolic regression framework $\Phi$-SO, which searches the function space for physically plausible expressions using reinforcement learning. A toy isochrone test demonstrates the method with a mean relative error of about $0.1\%$, illustrating that gradient-based backpropagation can propagate through orbit integration and density estimation to recover the gravitational field from observations, with extensions discussed for Gaia-scale data and stellar streams; code is available on GitHub.

Abstract

New large observational surveys such as Gaia are leading us into an era of data abundance, offering unprecedented opportunities to discover new physical laws through the power of machine learning. Here we present an end-to-end strategy for recovering a free-form analytical potential from a mere snapshot of stellar positions and velocities. First we show how auto-differentiation can be used to capture an agnostic map of the gravitational potential and its underlying dark matter distribution in the form of a neural network. However, in the context of physics, neural networks are both a plague and a blessing as they are extremely flexible for modeling physical systems but largely consist in non-interpretable black boxes. Therefore, in addition, we show how a complementary symbolic regression approach can be used to open up this neural network into a physically meaningful expression. We demonstrate our strategy by recovering the potential of a toy isochrone system.

An end-to-end strategy for recovering a free-form potential from a snapshot of stellar coordinates

TL;DR

The paper tackles the problem of recovering a free-form gravitational potential from snapshot phase-space data of stars. It develops an end-to-end differentiable pipeline that learns a neural-network potential from phase-space data, maps to actions , and uses a phase-space density estimated by normalizing flows or diffusion models to maximize a likelihood. It then distills into an interpretable analytical form via the physics-aware symbolic regression framework -SO, which searches the function space for physically plausible expressions using reinforcement learning. A toy isochrone test demonstrates the method with a mean relative error of about , illustrating that gradient-based backpropagation can propagate through orbit integration and density estimation to recover the gravitational field from observations, with extensions discussed for Gaia-scale data and stellar streams; code is available on GitHub.

Abstract

New large observational surveys such as Gaia are leading us into an era of data abundance, offering unprecedented opportunities to discover new physical laws through the power of machine learning. Here we present an end-to-end strategy for recovering a free-form analytical potential from a mere snapshot of stellar positions and velocities. First we show how auto-differentiation can be used to capture an agnostic map of the gravitational potential and its underlying dark matter distribution in the form of a neural network. However, in the context of physics, neural networks are both a plague and a blessing as they are extremely flexible for modeling physical systems but largely consist in non-interpretable black boxes. Therefore, in addition, we show how a complementary symbolic regression approach can be used to open up this neural network into a physically meaningful expression. We demonstrate our strategy by recovering the potential of a toy isochrone system.
Paper Structure (3 sections, 2 figures)

This paper contains 3 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Learning a potential from a snapshot positions and velocities
  3. Distilling a neural network into an analytical function

Figures (2)

  • Figure 1: Proposed strategy for recovering a free form neural network potential that stabilizes a distribution of stars $\{(\vec{x}, \vec{v})_i\}_{i<n_*}$ using auto-differentiation and gradient descent. See § 1 of Section \ref{['sec:MassFinder']} for a full description of the workflow.
  • Figure 2: Distilling a neural network into a interpretable analytical expression with $\Phi$-SO physo. An RNN generates trial expressions, their ability to reproduce the neural network $\Phi_\alpha$ is assessed and the best ones are reinforced. This process is repeated iteratively until convergence of the RNN and the extraction a set of high quality expressions.