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

The paper formulates a weak-form equation-learning approach (WSINDy/WFEL) for the reduced dynamics of σ⊥ and σz, starting from the ansatz equation Ei(σ,σ̇,σ̈) − 2kBT0/(mσi) = RHS (Eq. 1), and builds linear weak-form systems (G,b) using compactly supported test functions (Eqs. 2–3). It then introduces three libraries (ND, AV, PI) and shows that the physics-informed (PI) library—with Lf monomials up to |p|+|q|≤3 and Lµ terms involving σ̇j/(σjσk), (σ̇iσj)/σi, and (σ̇iσ̇j2)/(σkσj)—achieves ≤5% fidelity across all trap aspect ratios λ and supports the linearized matrices δ̈σ + 2ΓPI δ̇σ + OPI δσ ≈ 0 with OPI, ΓPI given by Eq. (10) (bracketed ansatz contributions plus learned Jacobians). All of these appear explicitly in the PDF (Eq. 1; WFEL construction; Table I libraries; PI model Eq. 7; linearization Eqs. 9–10; Fig. 2 reporting ≤5% errors) . The candidate solution reproduces this methodology nearly verbatim: it constructs the same weak-form linear system, employs the same Lf and Lµ choices, matches the error metric Δ2 and the forward validation criterion (≤5%), and derives OPI and ΓPI via the same Jacobian formulas around σ(∞). Minor implementation differences (e.g., STRidge vs MSTLS) do not affect conceptual correctness. The candidate is incomplete only in that it withholds numerical coefficients pending data, but its method is consistent with the paper’s approach and claims.