Physics-guided weak-form discovery of reduced-order models for trapped ultracold hydrodynamics

TL;DR

The paper addresses the challenge of modeling relaxation dynamics in a strongly collisional, ultracold dipolar gas confined in a harmonic trap, where first-order hydrodynamics is insufficient, by focusing on the width observables $\boldsymbol{\sigma}=(\sigma_\perp,\sigma_z)$. It employs a physics-guided weak-form equation learning (WFEL) approach, via WSINDy, to identify a sparse, interpretable closed set of ODEs for the widths from high-fidelity DSMC simulations, starting from a fluctuating Gaussian Ansatz. Three learning strategies—Naïve Discovery (ND), Ansatz Verification (AV), and Physics-Informed (PI)—progressively incorporate physics, revealing anisotropic corrections to the hydrodynamic volume $\mathcal{V}_{\rm hy}$ and higher-order viscous/forcing terms that improve accuracy across trap aspect ratios $\lambda$. The resulting WSINDy-PI models achieve errors at the few-percent level, outperform the previous Gaussian Ansatz, and provide interpretable mechanisms such as velocity-field corrections and potential vortical forcing that connect to higher-order hydrodynamics, with implications for experimental thermometry and evaporation protocols. The framework offers a general, data-driven path to discovering reduced-order descriptions in complex, multi-phase hydrodynamic systems constrained by known physics.

Abstract

We study the relaxation of a highly collisional, ultracold but nondegenerate gas of polar molecules. Confined within a harmonic trap, the gas is subject to fluid-gaseous coupled dynamics that lead to a breakdown of first-order hydrodynamics. An attempt to treat these higher-order hydrodynamic effects was previously made with a Gaussian ansatz and coarse-graining model parameter [R. R. W. Wang & J. L. Bohn, Phys. Rev. A 108, 013322 (2023)], leading to an approximate set of equations for a few collective observables accessible to experiments. Here we present substantially improved reduced-order models for these same observables, admissible beyond previous parameter regimes, discovered directly from particle simulations using the WSINDy algorithm (Weak-form Sparse Identification of Nonlinear Dynamics). The interpretable nature of the learning algorithm enables estimation of previously unknown physical quantities and discovery of model terms with candidate physical mechanisms, revealing new physics in mixed collisional regimes. Our approach constitutes a general framework for data-driven model identification leveraging known physics.

Figures (4)

• Figure 1: Learned ${\cal V}_{{\rm hy}, ij}$ (Eq. \ref{['eq:Vhydij']}) using the WSINDy-AV models. The empirical functional form for ${\cal V}_{\rm hy}$Wang23_PRA2 is independent of $\lambda$ and given by the black dashed line (SI Eq. 1.11). Red and blue curves represent values extracted from the learned equations for $\ddot{\sigma}_\perp$ and $\ddot{\sigma}_z$, showing anisotropies in the hydrodynamic volume at different trap ratios $\lambda$. Modest gains in accuracy indicate that such anisotropy is favorable (Fig. \ref{['fig:PIres']}).
• Figure 2: Fidelity of the ansatz (blue), WSINDy-ND (red), WSINDy-AV (yellow), and WSINDy-PI (purple) models with respect to DSMC data. WSINDy-PI captures the dynamics to within $5\%$ across the spectrum of trap aspect ratios $\lambda$, reducing errors by an order of magnitude from the ansatz.
• Figure 3: Comparison between DSMC, Ansatz, and WSINDy-PI for $\lambda=8$ (pancake-shaped trap). The relative errors of the WSINDy (WS) and Ansatz (Ans) time traces with respect to DSMC results are listed in the titles.
• Figure 4: Comparison of the specific forcing time traces along $\sigma_z$ between the Ansatz (dot-dashed red curve), and WSINDy-PI (dashed blue curve) for $\lambda=8$ (pancake-shaped trap), for both the viscous (left) and forcing (right) components. The difference between the WSINDy and Ansatz forcing terms $\Delta\boldsymbol{F}_z = \boldsymbol{F}^{WS}_z - \boldsymbol{F}^{Ans}_z$ is plotted in solid green.