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REVIEW 2 major objections 6 minor 6 references

An AI model solves the fastest-growing typhoon forecast-error patterns over five days, and the same perturbations grow fast when transplanted into a physics-based model.

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2026-08-02 17:17 UTC pith:4GVQHPB2

load-bearing objection Useful 5-day AI-based CNOP study with a WRF cross-check; the TDE proxy for track error is the main thing to fix before it can claim optimal track-error growth. the 2 major comments →

arxiv 2603.26165 v1 pith:4GVQHPB2 submitted 2026-03-27 physics.ao-ph

Error Growth Dynamic and Predictability of Tropical Cyclone in Machine Learning Weather Prediction Model

Jingchen Pu , Mu Mu , Jie Feng , Hao Li This is my paper
classification physics.ao-ph
keywords tropical cyclone predictabilityconditional nonlinear optimal perturbationmachine learning weather predictionerror growthFuXi modeltargeted observationensemble forecastingtyphoon track forecast
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the longstanding CNOP optimization, traditionally requiring adjoint models and short optimization windows, can be carried out for 5-day typhoon track forecasts using the FuXi machine-learning weather model. It reports that the perturbations found by maximizing total dry energy inside a box around the forecast typhoon position grow faster than random, random-plus-CNOP, and lagged-forecast perturbations, in both FuXi and the WRF physics-based model. Across twelve western North Pacific typhoons, the AI-derived CNOPs consistently produced the largest error growth, and for a case study the track deviation exceeded 1,400 km. The authors conclude that AI models can capture the most rapidly growing perturbation patterns and their nonlinear evolution, making AI useful for theoretical predictability studies, targeted observations, and ensemble design.

Core claim

On its own terms, the paper establishes that Conditional Nonlinear Optimal Perturbations (CNOPs) for 5-day typhoon track forecasts can be solved inside the FuXi AI model, and that these perturbations are optimal and physically consistent. The optimization maximizes total dry energy in a 10x10-degree box centered on the control forecast typhoon position at day 5, using FuXi's automatic differentiation and a spectral projected gradient algorithm. The resulting CNOPs, verified in both FuXi and WRF, grow faster than random and lagged-forecast perturbations, produce track deviations of over 1,400 km in the Chanthu case, and show strong spatial consistency between the two models. Sensitivity exper

What carries the argument

The central object is the Conditional Nonlinear Optimal Perturbation (CNOP): the initial perturbation of bounded amplitude that produces the largest forecast error at a specified time in a nonlinear model. The paper's implementation uses the FuXi AI model, whose automatic differentiation provides the gradient of the objective function with respect to initial conditions, replacing the adjoint model normally required in NWP-based CNOP computation. The objective is the total dry energy (TDE) norm integrated over a 10x10-degree box around the forecast typhoon position, and the constraint region is the western North Pacific domain (100-180E, 0-60N).

Load-bearing premise

The optimization maximizes total dry energy inside a box around the forecast typhoon position, not the typhoon track error itself, and the authors assert that larger energy differences 'generally' correspond to significant track displacement; if that proxy is weakly correlated with real track error in many cases, the perturbations found may not be the fastest-growing track-error patterns.

What would settle it

Re-run the same optimization with a direct, differentiable proxy for track error (for example, the distance between perturbed and control 500-hPa geopotential minima at day 5, or a smoothed vortex-center displacement) and compare the resulting CNOPs; if the TDE-optimal perturbation is not among the top growers of direct track error, or if the direct-optimized perturbation produces a much larger track deviation, the central claim fails. A second check is to test the TDE-CNOPs on a withheld set of typhoons beyond the twelve studied; if the perturbations consistently fail to produce large track d

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • AI models can generate optimal growing perturbations at 5-day lead times without the costly adjoint development that limits NWP-based CNOP applications.
  • Sensitive areas for targeted typhoon observations can be identified inexpensively from AI-derived CNOPs, potentially improving where dropsondes or extra soundings are deployed.
  • The case-dependent mix of inner-core versus far-environment sensitivity implies that ensemble perturbation design should be tailored to each typhoon's surrounding flow.
  • Because the AI-derived perturbations grow rapidly and consistently when transplanted into WRF, the paper suggests the patterns are physically meaningful features of the atmosphere, not artifacts of the machine-learning model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the total-dry-energy proxy generalizes, the same optimization recipe could be applied to other high-impact weather events whose forecast-error measures are non-differentiable, such as heavy-rainfall location or heatwave extent.
  • The 5-day CNOP's outward shift to midlatitude systems suggests that medium-range typhoon track uncertainty is dominated by errors in the large-scale steering flow, which could guide a two-tier observation strategy: dense inner-core sampling for short lead times, broader midlatitude sampling for longer lead times.
  • A direct, testable extension would be to optimize a smoothed track-displacement objective (for example, the distance between the perturbed and control vortex centers at day 5) and compare the resulting CNOPs; agreement would directly validate the TDE proxy, while disagreement would identify cases where the proxy fails.
  • The method's restriction to wind and temperature perturbations above 925 hPa leaves moisture and boundary processes unperturbed; extending the perturbation variables as AI models improve their moist physics could reveal additional error-growth channels for typhoon intensity as well as track.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper develops a method to compute Conditional Nonlinear Optimal Perturbations (CNOPs) for 5-day tropical cyclone (TC) track forecasts using the FuXi machine-learning weather model, exploiting automatic differentiation to make the optimization tractable. The authors validate the resulting perturbations by (i) comparing their growth in FuXi against random, hybrid, and lagged forecast perturbations, and (ii) transplanting the CNOP into the WRF model and showing similar rapid growth and structural evolution. Twelve western North Pacific TCs are analyzed, with a detailed case study for Typhoon Chanthu, including sensitivity to perturbation amplitude and optimization window.

Significance. If the central claim holds, this is a valuable contribution: it demonstrates that AI-based weather models can support long-window CNOP computations, and that the resulting perturbations exhibit physical consistency in an independent NWP model. The manuscript includes multiple sensitivity experiments, a 12-case statistical comparison, and cross-model verification — these are substantial strengths. The paper also provides open data/model links, and the optimization framework is clearly described. However, the headline claim about optimal growth of TC track errors is weakened by the use of a total dry energy (TDE) proxy as the optimization target, without quantitative validation of the TDE–track displacement link.

major comments (2)
  1. [Section 3.2, Eq. (2)] The objective function J is the total dry energy (TDE) in a 10°×10° box centered on the control TC position, not the TC track error. The paper asserts (Section 3.2) that larger TDE differences are 'generally related' to significant TC displacement, but no quantitative support is provided. All optimality comparisons (Figs. 5c-d, 7c/f, 9c/f, 13a-b) are in TDE units. The central claim that the CNOP achieves the fastest nonlinear development of TC track errors is therefore not established. Please validate the TDE–track displacement relationship (e.g., correlation across many random/optimal perturbations), or reframe the conclusions to refer to TDE-optimal perturbations.
  2. [Section 4, Figs. 5c-d] The FuXi comparison of J(CNOP) vs J(RP) is partially circular: the CNOP is the maximizer of J in FuXi by construction. The independent WRF transfer confirms physical consistency of the same perturbation, but it does not verify track-error optimality, since J is still TDE. The phrase 'optimality verified' in the abstract and Section 5.1.3 should be tempered, or the authors should test whether the TDE-optimal perturbation also maximizes actual track error relative to a broader set of perturbations.
minor comments (6)
  1. [Section 2.2] WRF boundary conditions, initialization, and the interpolation of FuXi perturbations to the WRF grid are not described. This information is needed for reproducibility of the cross-model verification.
  2. [Section 3.2] Typo: 'westly' should be 'westerly'; also 'predictivity' appears in Section 3.2.
  3. [Figure 8 caption] Only CNOPs for 24h and 72h optimization windows are displayed, although the text reports 12, 48, 96, and 120h. Please indicate where these are shown or add them.
  4. [Section 4] Please specify the number of random perturbations and how CNOP+RP is constructed.
  5. [Section 5.1.3] The statement 'track deviations exceeding 1400 km' should be accompanied by the track-error plot with clear units (already in Fig. 5e).
  6. [Section 3.3] The gradient spectral cutoff of 2° is a free parameter; no sensitivity analysis is shown. A brief justification or test would help.

Circularity Check

1 steps flagged

FuXi-side optimality comparison is tautological, but WRF transfer provides independent support; TDE proxy is a stated assumption, not hidden circularity.

specific steps
  1. fitted input called prediction [Section 5.1.3 (also Section 4 Experimental Design; Eq. 1 and Eq. 2)]
    "In both models, the CNOP (red stars) consistently yields higher objective function values than either RP (dark green dots), CNOP+RP (light blue dots), or LFPs (orange points), confirming its superiority in capturing the mode of maximal nonlinear error growth."

    The CNOP is computed by maximizing the same objective function J (Eq. 1, with J defined by the TDE norm in Eq. 2) in FuXi using SPG2. Section 3.3 states the optimization is iterated 200 times and runs until the increment of J is below 0.5. Therefore, for the FuXi model, J(CNOP) >= J(any admissible perturbation), including RPs, CNOP+RPs, and LFPs, is a property of the optimization/convergence, not an independent empirical finding. Reporting this FuXi comparison as 'confirming its superiority' verifies convergence, not physical optimality. The same comparison in WRF is not circular, because J is evaluated with a different nonlinear model that did not participate in the optimization; the paper's independent evidence therefore rests on the WRF transfer, not on the FuXi comparison.

full rationale

The derivation chain is not globally circular. The CNOP is defined as the maximum of J (Eq. 1) under the TDE norm (Eq. 2), so any statement that the CNOP maximizes J in the same model is definitional. The paper uses such FuXi J-comparisons as evidence of optimality, which is the one genuinely tautological component. This is partially mitigated because the FuXi-optimized perturbation is transplanted into WRF, a physics-based model that did not participate in the optimization; the WRF comparisons provide independent, non-construction evidence that the perturbation grows faster than the random/lagged comparators in a second system. The TDE proxy for TC track error is an explicit assumption ('we adopt the TDE as an alternative to approximately measure the TC forecast error'), not a hidden redefinition, so it is a validity/correctness risk rather than a circular step. No load-bearing self-citation or imported uniqueness theorem is present; citations to Mu et al. (2003), Qin et al. (2024), and Li et al. (2025) are antecedent work rather than the justification of the present computation. Therefore the circularity is partial and localized, giving a score of 4.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest on several modeling choices the paper makes but does not independently justify: the TDE-as-track-error proxy, the restriction to dry wind/temperature perturbations, the gradient scale cutoff, and the unstated FuXi-to-WRF perturbation transfer. None of these is a fitted physical constant, but δ and the cutoff are hand-selected and directly shape the computed CNOP.

free parameters (3)
  • Initial perturbation norm constraint δ = 0.4 m^2 s^-2
    Chosen from sensitivity tests (0.05, 0.1, 0.2, 0.4, 0.8, 1.6 m^2 s^-2) as "suitable perturbation strength"; not derived from an independent measurement.
  • Gradient spectral cutoff = 2 degrees (~200 km)
    Scale components below 2 degrees are removed from the gradient for stable CNOP computation (§3.3 Step 3); a hand-set threshold based on effective model resolution.
  • Optimization iterations and stopping tolerance = 200 iterations; ΔJ < 0.5
    Numerical termination settings; no convergence proof to a global optimum (SPG2 on a nonconvex objective).
axioms (5)
  • domain assumption TDE difference inside the target box is a valid proxy for TC track forecast error
    Introduced in §3.2 because TC track is not directly differentiable; the relationship is asserted but not quantified against actual track errors.
  • domain assumption Perturbing only wind and temperature above 925 hPa, excluding surface and moisture variables, is sufficient for reliable TC track uncertainty analysis
    §3.2 justifies exclusion by AI model limitations, but moisture and surface processes can affect TC track; the assumption is untested here.
  • domain assumption Removing perturbation scales below 2 degrees from the gradient does not compromise the relevant optimality
    §3.3 Step 3 justifies this with effective resolution, but its effect on the final CNOP is not sensitivity-tested.
  • domain assumption The FuXi-derived perturbation can be transplanted into WRF without structural degradation
    The WRF verification is the main independent evidence, yet the interpolation and variable mapping between FuXi and WRF are not described (§4, §5).
  • domain assumption FuXi's representation of large-scale atmospheric fields is physically reliable enough for CNOP optimization
    Supported by track-error comparisons and pattern correlations in §5.1.1, but the physical consistency of AI weather models remains an active debate (the paper cites Bonavita 2024, Selz and Craig 2023).

pith-pipeline@v1.3.0-alltime-deepseek · 21034 in / 11283 out tokens · 114056 ms · 2026-08-02T17:17:09.147156+00:00 · methodology

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Cite this review

Pith. "Pith review of Error Growth Dynamic and Predictability of Tropical Cyclone in Machine Learning Weather Prediction Model." pith.science (2026). https://pith.science/paper/4GVQHPB2

@misc{pith2026260326165,
  author       = {Pith},
  title        = {Pith review of: Error Growth Dynamic and Predictability of Tropical Cyclone in Machine Learning Weather Prediction Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GVQHPB2}},
  note         = {Machine review of arXiv:2603.26165}
}
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read the original abstract

Predictability analysis, which focuses on perturbation growth dynamic, is a key problem in both weather and climate prediction. Among all perturbations, the conditional nonlinear optimal perturbation (CNOP) leads to maximum uncertainties in forecasts, which is fundamentally important for theoretical studies and applications. Traditionally, CNOPs are solved through iterative optimization of numerical weather prediction (NWP) systems. Their large computational demands pose significant challenges to long-term predictability analysis. In our study, using a fast and accurate Artificial intelligence (AI) model, i.e. FuXi, a low-cost optimization framework for solving 5-day tropical cyclone (TC) CNOP is developed. For the first time, CNOPs that achieve the optimal (i.e., fastest) nonlinear development of long-term TC forecast errors are solved, with their optimality and physical explainability verified. Results demonstrate that perturbations with specific spatial structures undergo significant development. In both AI and NWP models, AI-based CNOPs exhibit rapid and physically consistent error growth across diverse TC cases, faster compared to random and lagged forecast perturbations. Furthermore, sensitivity analysis reveals that far-environment systems and processes are more crucial for long-term TC forecasts. Structural analyses of the CNOP emphasizes the interactions between TC internal and external processes for rapid perturbation growth. The success derivation of AI-based CNOPs, with their rapid growth and physical explainability verified in both AI and NWP models, suggests that AI models can capture the most rapidly growing perturbation patterns and their subsequent nonlinear evolution. Thus, potential of AI models is highlighted for advancing atmospheric predictability researches, including theoretical analysis, targeting observations and ensemble forecasts.

Figures

Figures reproduced from arXiv: 2603.26165 by Hao Li, Jie Feng, Jingchen Pu, Mu Mu.

Figure 1
Figure 1. Figure 1: Tracks and forecast errors of selected TC cases. Figure (a) gives TC tracks from IBTrACS dataset. Figure (b) gives mean track errors from FuXi, WRF and IFS [PITH_FULL_IMAGE:figures/full_fig_p037_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The procedures for solving CNOP in data driven model (blue arrows) and for verifying CNOP in NWP models [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The 120h forecast and analysis of TC Chanthu initialized at 0000 UTC Sep 9, 2021. Figures (a1-a4) give the 0h, 24h, 72h, 120h forecast of 500hPa geopotential field (contours, interval at 40gpm, thickened at 5880gpm) and temperature field (shadings). Green lines for TC track. Figures (b1-b4) and (c1-c4) are the same as (a1-a4), but for results from WRF and reanalysis from ERA5. Figure (d) compares the TC tr… view at source ↗
Figure 4
Figure 4. Figure 4: CNOP structures of TC Chanthu. Figures (a1-a4) gives initial temperature perturbations (shadings) on isobaric surface of 925hPa, 850hPa, 500hPa and 300hPa, respectively. Figures (b1-b4) are similar as (a1-a4), but for initial wind perturbations (vectors) and its corresponding vorticity field (shadings). Contours represent the geopotential field. Green lines and red lines give TC tracks from control forecas… view at source ↗
Figure 5
Figure 5. Figure 5: The evolution and verification of CNOP for TC Chanthu. Figures (a1-a3) gives the 24h, 72h, 120h evolution of 500hPa geopotential perturbation (shading) and wind perturbation (vectors) in FuXi. Contours give the geopotential height from control forecast (interval at 80gpm and thickened at 5880gpm). Green lines and red lines represent TC tracks from control forecast and perturbed forecast. Figures (b1- b3) a… view at source ↗
Figure 6
Figure 6. Figure 6: CNOP structures of TC Chanthu with different initial norms. Figure (a1, a2) gives initial perturbations of temperature (shading) and wind (vectors) on 850hPa and 500hPa with an initial norm of 0.1. Contours represent geopotential heights. Green lines and red lines represent TC tracks from control forecast and perturbed forecast in FuXi. Figures (b1, b2) and (c1, c2) are similar as Figures (a1, a2), but for… view at source ↗
Figure 7
Figure 7. Figure 7: Forecasts of TC Chanthu perturbed by CNOPs with different norms. Figure (a) gives TC tracks of control and CNOP-perturbed forecasts from FuXi. Figure (b) gives track errors caused by different CNOPs in FuXi. Figure (c) gives target values of CNOPs with different norms (red star), random perturbation (RP, green dots) and CNOPs combined with RPs (blue dots) evaluated from FuXi. Figures (d-f) are similar as F… view at source ↗
Figure 8
Figure 8. Figure 8: CNOP structures of TC Chanthu with different optimization times. Figures (a1, a2) gives initial perturbations of temperature (shading) and wind (vectors) on 850hPa and 500hPa with an optimization time of 24h. Contours represent geopotential heights. Green lines and red lines represent TC tracks from control forecast and perturbed forecast in FuXi. Figures (b1, b2) are similar as Figures (a1, a2), but for o… view at source ↗
Figure 9
Figure 9. Figure 9: Forecasts of TC Chanthu perturbed by CNOPs with different optimization times. Figure (a) gives TC tracks of control and CNOP-perturbed forecasts from FuXi. Figure (b) gives track errors caused by different CNOPs in FuXi. Figure (c) gives target values of CNOPs with different optimization times (red star), random perturbation (RP, green dots) and CNOPs combined with RPs (blue dots) evaluated from FuXi. Figu… view at source ↗
Figure 10
Figure 10. Figure 10: CNOPs of 12 TC cases in 850hPa temperature (shading) and wind (vector) perturbations. Contours represent geopotential heights. Green lines and red lines represent TC tracks from control forecast and perturbed forecast [PITH_FULL_IMAGE:figures/full_fig_p046_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: CNOP evolution of TC Maysak in FuXi and WRF. Figures (a1-a3) give the CNOP-caused 500hPa geopotential (shading) and wind (vectors) perturbations at 24h, 72h and 120h. Contours represent geopotential height from control forecast (interval at 80gpm, thickened at 5880gpm). Figures (b1-b3) are similar as Figures (a1-a3), but for results from WRF [PITH_FULL_IMAGE:figures/full_fig_p047_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Same as [PITH_FULL_IMAGE:figures/full_fig_p048_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Target values and track error growth of different TC cases in FuXi and WRF. Figure (a) gives target values of CNOPs (red star), random perturbation (RP, green dots), lagged forecast perturbations (LFP, orange dots) and CNOPs combined with RPs (CNOP+RP, blue dots) evaluated from FuXi in different TC cases. Figure (b) is similar as Figure (a), but for results from WRF. Figure (c) gives CNOP￾caused TC track … view at source ↗
Figure 14
Figure 14. Figure 14: TC tracks from control and perturbed forecasts and their track errors. Figures (a, b, c) give the tracks of TC Maysak, Chanhom and Chanthu from control forecast (black) and forecasts perturbed by center (green), middle (blue), outlying (purple) and all (red) parts of CNOP. Figures (d, e, f) give the errors of those perturbed forecasts from control forecasts. Green (blue) squares in a-c show the boundary o… view at source ↗

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Reference graph

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