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The manuscript’s central physical interpretation (recovery of inviscid Euler-like dynamics and a pressure-gradient mechanism) is not rigorously validated because the data-generating PDE(s), closure (equation of state / pressure model), physical parameters (e.g., viscosity), and numerical scheme are not explicitly specified or compared term-by-term against the discovered models (Sec. 1, Sec. 2.1, Sec. 3.2, Sec. 4.1). Without ground-truth disclosure or a quantitative comparison, it is difficult to distinguish true equation recovery from a qualitatively similar surrogate enabled by correlations in the dataset (e.g., $\nabla\rho$ acting as a proxy for $\nabla p$ under some regimes).
Recommendation: Add a dedicated description of the simulator in Sec. 2.1 (or a new subsection): governing PDE(s) (compressible Euler/Navier–Stokes? weakly compressible model?), pressure/closure (barotropic/isothermal? incompressible Poisson pressure?), viscosity/forcing, nondimensionalization/units, discretization (finite volume/spectral/etc.), and snapshot spacing in simulation time. In Sec. 3.2, provide a side-by-side table comparing the true PDE term structure and coefficients to the discovered model (relative coefficient errors where applicable). If the ground truth is unavailable, state that clearly in Sec. 1/Sec. 2.1 and reframe claims in Sec. 3.2/Sec. 4.1 as qualitative discovery/hypothesis generation rather than “recovering Euler.”
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Coefficient interpretability is currently unclear due to feature scaling/standardization and unknown nondimensionalization, yet coefficients are treated as directly physical—most critically in the inference of the physical time step $\Delta t_{\rm true}$ and sound speed from the learned advection/gradient coefficients (Sec. 2.3–2.4, Sec. 3.2, Sec. 4.1). If columns of $\Theta$ (and possibly targets) are scaled, then reported coefficients must be unscaled before interpreting them as PDE parameters; otherwise the $\Delta t_{\rm true}$ and $c_s$ estimates can be artifacts of preprocessing. Relatedly, setting $\Delta t_{\rm true}$ so that the advective coefficient becomes $1$ risks circular reasoning unless the nondimensional form is independently justified.
Recommendation: In Sec. 2.3–2.4, specify exactly how $\Theta$ columns were scaled ($z$-score, $L_2$ norm, max norm, etc.), whether $\partial_t X$ targets were scaled, and how coefficients were transformed back to the original units before being reported/used in Sec. 3.2. In Sec. 3.2/Sec. 4.1, explicitly list the assumptions required for the mapping “learned coefficient $\rightarrow \Delta t_{\rm true} \rightarrow$ inferred $c_s$” (unit advective coefficient in the chosen nondimensionalization; correct derivative stencil scaling; negligible viscosity; barotropic relation; $\rho \approx 1$), and—if possible—compare inferred $\Delta t_{\rm true}$ to simulation metadata. If metadata are unavailable, present $\Delta t_{\rm true}$ and $c_s$ as hypothesis-consistent effective parameters and soften wording accordingly.
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Key SINDy configuration details are underspecified and robustness is not established: the library definition, ridge/threshold schedules, convergence/refit procedure, and the precise BIC computation are not fully documented; nor are there ablations/sensitivity analyses to show that the Euler-like structure is stable rather than an artifact of a particular library/penalty (Sec. 2.3–2.4, Sec. 3.2, Sec. 4.2). This is especially important because the library is likely highly collinear in 3D, and the paper already observes “canceling” derivative combinations—suggesting identifiability/stability issues under correlated features.
Recommendation: Expand Sec. 2.3 with a compact but explicit enumeration of library terms (categories + maximum polynomial degree + whether mixed derivatives are included) and report the library size (\#features) per target; if lengthy, include the full list in an Appendix while keeping counts in the main text. Expand Sec. 2.4 with: ridge $\lambda$ grid, hard-threshold rule (absolute/relative), number of iterations/stopping criterion, and whether coefficients are refit on the active set without ridge. Provide the exact BIC formula with definitions of $N$ and $k$ and discuss how spatial correlation affects the effective sample size. In Sec. 3.2 (or new Sec. 3.5), add robustness checks: (i) ablate second-derivative terms, (ii) ablate pure density polynomials, (iii) vary scaling/threshold/BIC penalty strength, (iv) subsample spatial points and/or hold out time slices, and report term-support stability and coefficient variability (e.g., bootstrap intervals).
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Derivative estimation is a critical bottleneck given only $10$ time slices and very small temporal derivatives (especially for $\rho$), but the manuscript provides mainly qualitative discussion and no quantitative sensitivity analysis; second-order finite differences may be suboptimal for periodic domains where spectral derivatives are natural (Sec. 2.2, Sec. 3.1–3.2). As a result, it is unclear whether failures (density) and even key recovered coefficients (velocity) are robust to more accurate/regularized differentiation.
Recommendation: In Sec. 2.2 and Sec. 3.1–3.2, add a quantitative derivative-sensitivity study: compare current second-order central differences to (i) spectral spatial derivatives (FFT-based) given periodicity, and (ii) at least one temporal denoising/differentiation approach (e.g., Savitzky–Golay in time, smoothing splines/Tikhonov). Report how $R^2$/MSE and the main coefficients (advective and gradient terms) change. Also include an empirical estimate of derivative noise/error (e.g., forward vs. backward vs. central disagreement; magnitude of $\partial_t\rho$ relative to estimated noise floor).
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The density equation identification and its interpretation as continuity-consistent are not yet technically convincing: the discovered form reported (e.g., $-0.0648\nabla\cdot \mathbf{v} + 0.0638\rho\nabla\cdot \mathbf{v}$) does not directly match $\partial_t\rho = -\mathbf{v}\cdot\nabla\rho - \rho\nabla\cdot\mathbf{v}$, and it is unclear whether $v\cdot\nabla\rho$ terms were present in the library and rejected, or absent altogether (Sec. 3.2; also affects claims in Sec. 4.1). Given the extremely low signal-to-noise for $\partial_t\rho$, relying on $R^2$ is also problematic.
Recommendation: In Sec. 3.2, write the full learned density equation explicitly (all selected terms and coefficients) and state explicitly whether $v_i\partial_i\rho$ terms were included in $\Theta$ and what coefficients they received along the regularization path (even if thresholded out). Add alternative evaluation metrics for the density target that are meaningful in low-variance settings (e.g., correlation, normalized residual norm, or relative error vs. a baseline). Consider learning/enforcing continuity structure directly (e.g., include $\nabla\cdot(\rho\mathbf{v})$ as a single composite feature, or impose a constraint) and report whether this improves identifiability.
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Model evaluation is too limited to establish dynamical fidelity: results rely on in-sample fit and a single one-step forward Euler prediction between consecutive snapshots, which can appear strong even for imperfect models when the system evolves slowly (Sec. 2.5, Sec. 3.3–3.4). There are no longer-horizon rollouts, stability checks, out-of-sample tests, or baseline comparisons (e.g., persistence), making it hard to assess whether the learned PDE truly captures the dynamics.
Recommendation: Extend Sec. 3.4 to include multi-step rollouts over as many steps as the data allow (e.g., 3–5 steps), using a more stable integrator (RK2/RK4) and clearly specifying step size and any substepping. Report error growth curves (NRMSE/relative $L_2$) and include at least a persistence baseline $X(t+\Delta t) = X(t)$. Where relevant, compare distributional/structural statistics (e.g., kinetic energy, divergence norms, spectra) between predicted and true fields. If longer rollouts are infeasible, state that limitation explicitly and qualify conclusions to “one-step/short-horizon only.”
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Generality/novelty is not sufficiently supported: the experimental evidence is based on a single dataset with a very short temporal window, and there is no comparison to alternative PDE-discovery approaches or to prior SINDy/PDE-FIND results in fluids to clarify what is new beyond a case study (Sec. 1–4).
Recommendation: In Sec. 1 and Sec. 4.2, strengthen positioning relative to existing PDE-discovery work in fluids (representative SINDy/PDE-FIND Navier–Stokes/Euler studies, compressible/weakly compressible cases) and state clearly the novelty (e.g., 3D $128^3$ scale, limited snapshots, time-step inference). If feasible, add at least one additional scenario (different initial condition/parameter regime) or a controlled robustness test (synthetic noise injection; alternative derivative schemes) and/or a comparison to another identification baseline. Otherwise, frame the paper explicitly as a single-case feasibility/diagnostic study.