Physics-Informed Neural Networks for Modeling Galactic Gravitational Potentials

TL;DR

This work tackles modeling galactic gravitational potentials by marrying analytic baseline potentials with neural residuals under physics-informed constraints. It introduces a six-component design that couples a physics loss, radial scaling, analytic fusion, Bayesian uncertainty, and a neural ODE for time evolution to learn static and time-dependent potentials. On mock Milky Way–LMC systems, it achieves sub-percent acceleration errors, stable energy evolution, and accurate recovery of LMC parameters, outperforming purely analytic baselines. The approach yields interpretable, uncertainty-quantified, time-resolved potential models that are well-suited for integration with observational data and complex dynamical analyses.

Abstract

We introduce a physics-informed neural framework for modeling static and time-dependent galactic gravitational potentials. The method combines data-driven learning with embedded physical constraints to capture complex, small-scale features while preserving global physical consistency. We quantify predictive uncertainty through a Bayesian framework, and model time evolution using a neural ODE approach. Applied to mock systems of varying complexity, the model achieves reconstruction errors at the sub-percent level ($0.14\%$ mean acceleration error) and improves dynamical consistency compared to analytic baselines. This method complements existing analytic methods, enabling physics-informed baseline potentials to be combined with neural residual fields to achieve both interpretable and accurate potential models.

Figures (4)

• Figure 1: Impact of model design choices on reconstructing the MW--LMC system.Left: Radial profile of relative acceleration percent error ($100 \cdot \tfrac{\|\mathbf{a}_\text{pred} - \mathbf{a}_\text{true}\|}{\|\mathbf{a}_\text{true}\|}$). Black dashes mark the training point locations, and black crosses represent the performance of an analytic MW model without NN correction. Right: Relative residual of the reconstructed potential field in the $x$–$y$ plane.
• Figure 2: Model performance on the MW--LMC test system(a) Orbit initialized at the LMC center with the local circular velocity and integrated for $1\,\mathrm{Gyr}$ under four models: the true potential, an analytic MW--only baseline without the LMC, an MW-LMC model with misspecified parameters (LMC center offset by $1\,\mathrm{kpc}$ and scale radius $r_s$ misestimated by $2\%$), and the BNN reconstruction. (b) Relative deviation from true energy, where $\Delta E(t) = E_{\text{pred}} - E_{\text{true}}$. (c) Relative energy drift along the predicted orbit. (d) Posterior distributions of the inferred LMC parameters (mass, scale radius, and Galactocentric distance). Each posterior sample is obtained by reconstructing the full learned potential and fitting it to a MW-LMC model with the MW parameters held fixed to their true values. Uncertainty envelopes in all panels correspond to the 16--84th ($1\sigma$) and 2.5--97.5th ($2\sigma$) percentiles across 1000 posterior draws.
• Figure 3: Relative percent error of the time-dependent MW--LMC potential reconstruction, binned by radius and time relative to the present ($t=0$; $t<0$ earlier). The dashed vertical lines mark training snapshot times; the solid horizontal line indicates the maximum training radius.
• Figure 4: Effect of network size on performance. Each square corresponds to one network configuration, where the diagonal separates the two metrics: the lower-left shows the mean acceleration error (MAE, $\%$), evaluated on 32,768 testing points; the upper-right shows the mean orbit deviation (MOD, kpc), computed on a set of 100 orbits initialized at randomly sampled positions and integrated for 500 Myr. Training time is encoded by the color of each cell border. The star marks the configuration selected for follow-up tests.