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CasEm: A Cascade Architecture for Long-Horizon Neural Emulation

Zhaoyi Li, Jingtao Ding, Shihua Li

arXiv:2609.34246v1cs.LG

TL;DR

Autoregressive neural emulators may accumulate error or diverge over long rollouts despite accurate short-term predictions. CasEm adds an independently evolving physical-aggregate model whose forecasts guide full-state corrections, and its error bound and diagnostics guide subsystem choice. Across ODE/PDE benchmarks and climate emulation, CasEm improves long-horizon accuracy, with regional total-water guidance reducing climate time-mean error for two frozen backbones.

  • Problem

    Autoregressive rollout can accumulate and amplify errors, so low one-step error alone does not guarantee reliable long-horizon emulation.

    Source: §1 · p. 1

  • Method

    CasEm uses an independently evolving model of physically specified aggregates to guide corrections to a full-state backbone, with an error bound and diagnostics guiding subsystem selection.

    Source: §1 · p. 2

  • Results

    Across four ODE/PDE benchmarks and climate emulation, CasEm improves long-horizon accuracy; regional total-water guidance reduces 10-year full-state time-mean error by 66.6% with frozen ACE and 46.3% with Spherical DYffusion.

    Source: §1 · p. 2

  • Takeaways & Limitations

    CasEm’s long-horizon gains are evaluated across diverse backbones, and the paper uses aggregate error coverage, modelability, and correctability to guide subsystem choice.

    Source: §1 · p. 2

    The approach focuses on physically specified aggregates for which prior knowledge suggests exact or approximate closure, with unresolved-variable influence small on encountered states.

    Source: §3.1 · p. 3

Abstract

from arXiv · show

Autoregressive neural emulators can drift or diverge over long rollouts despite accurate short-term predictions. We introduce Cascaded Emulation (CasEm), a one-way rollout architecture that augments an existing full-state backbone with an independently evolving model of physically specified aggregates. Its forecasts guide corrections to full-state predictions, without feedback from the backbone to the aggregate model. Effective guidance requires aggregates that cover substantial backbone error, remain accurately predictable, and support useful full-state corrections. We derive a finite-horizon error bound that clarifies these three factors and use empirical diagnostics to guide subsystem selection. Across four ODE/PDE benchmarks, CasEm reduces long-horizon rollout errors across diverse backbones and suppresses the trend toward error divergence in both diffusion tasks using Fourier neural operator backbones. In global climate emulation, CasEm with a regional total-water subsystem reduces 10-year full-state time-mean error by 66.6% and 46.3% for frozen ACE and Spherical DYffusion backbones, respectively, while adding less than 3% to inference time.

1 INTRODUCTION

Long autoregressive rollouts can accumulate error even when short-term predictions are accurate. CasEm adds an independently evolving aggregate model to guide full-state corrections, with design criteria formalized by an error bound and evaluated across dynamical systems.

1 INTRODUCTION

  • Autoregressive errors can accumulate and amplify, so low one-step error alone does not guarantee reliable long-horizon emulation.Long rollouts may distort long-term statistics, violate physical constraints, or diverge.Source: §1 · p. 1
  • CasEm combines a recurrent full-state backbone with an independently evolving model of physically specified aggregates whose forecasts guide full-state corrections.The backbone does not feed back into the aggregate model, so its accumulated errors cannot directly perturb subsequent aggregate forecasts.Source: §1 · p. 2
  • Effective guidance requires aggregates to cover backbone error, remain accurately predictable, and support useful full-state corrections.The finite-horizon error bound and empirical diagnostics guide subsystem selection and correction design.Source: §1 · p. 2
  • Across four ODE/PDE benchmarks and climate emulation, CasEm improves long-horizon accuracy; regional total-water guidance reduces 10-year full-state time-mean error by 66.6% with frozen ACE and 46.3% with Spherical DYffusion.The named backbones are ACE (Watt-Meyer et al., 2023) and Spherical DYffusion (R¨uhling Cachay et al., 2024).Source: §1 · p. 2

2 RELATED WORK

CasEm differs from reduced-order, multiscale, and climate-emulation approaches by retaining and recurrently evolving a full-state backbone alongside an independently evolving physical aggregate subsystem.

2 RELATED WORK

  • Unlike reduced-order and latent-dynamics methods that reconstruct full-state forecasts from evolved coordinates, CasEm corrects predictions from a separately recurrent full-state backbone using physical aggregates.Multiscale-LED alternates latent rollout with full-system simulation and feeds re-encoded states back into the latent recurrence.Source: §2 · p. 2
  • Unlike coarse-to-fine methods that evolve only coarse states or HINE’s hierarchy of predicted coarse futures, CasEm recurrently evolves both subsystem and full state with one-way guidance.ReLIFT uses restrict–evolve–lift propagation; HINE conditions full-state prediction on predicted coarse future states (Jiang et al., 2025).Source: §2 · p. 2
  • CasEm builds on pretrained climate emulators by keeping their full-state backbones frozen and adding an independently evolving regional-water subsystem.The stated aim is to reduce 10-year mean-field errors for ACE and S-DY.Source: §2 · pp. 2–3

3 CASEM: MACROSCOPIC TREND FIRST, FULL-STATE EMULATION SECOND

CasEm retains a full-state backbone and adds an independently evolving model of physically specified aggregate variables, whose forecasts are coordinated into full-state predictions. Its finite-horizon error analysis separates aggregate error coverage, subsystem modelability, and correction effectiveness.

3.1 PROBLEM SETUP AND CASCADED ROLLOUT

  • Physical knowledge specifies a low-dimensional aggregate state, typically representing large-scale or collective features and potentially defined by nonlinear readouts.The formulation focuses on aggregates with exact or approximate closure, where unresolved-variable influence is small on encountered states.Source: §3.1 · p. 3
  • CasEm retains the full-state backbone and adds a learned subsystem model that evolves independently, while a prescribed or learned coordination map combines its forecast with the backbone proposal.The cascade uses an upstream-first discretization.Source: §3.1 · p. 3

3.2 MECHANISTIC ANALYSIS

  • The analysis characterizes aggregate-direction error coverage (ρ), subsystem modelability (q), and full-state correctability (γ); q < 1 indicates more accurate independent subsystem predictions.Positive γ indicates beneficial correction, while γ > 1 can reflect gains in complementary directions through coupled full-state dynamics.Source: §3.2 · p. 4
  • The finite-horizon error bound relates CasEm’s normalized accumulated full-state error to ρ, q, and γ under an orthonormal aggregate readout and a minimum-change correction satisfying the predicted aggregate constraint.The bound also assumes a well-defined corrected recurrence with finite response to subsystem-target perturbations, summarized by B.Source: §3.2 · p. 4
  • The bound motivates screening aggregates for substantial error coverage, good independent modelability, and high correctability.The passage points to sequential empirical diagnostics for subsystem selection and correction design.Source: §3.2 · p. 4

4 ODE AND PDE BENCHMARKS

Across four ODE/PDE benchmarks, CasEm improves long-horizon predictions across tested systems and backbones, while subsystem-size experiments show that broader coverage can be offset by harder-to-model dynamics.

4 ODE AND PDE BENCHMARKS

  • The benchmarks span Heat and Fisher–KPP diffusion, spherical barotropic vorticity, and Alanine dipeptide dynamics, using Fourier, spherical-harmonic, and center-of-mass subsystems.The evaluation covers four ODE/PDE systems and multiple backbone architectures.Source: §4 · pp. 4–5

4.1 OVERALL PERFORMANCE

  • CasEm reduces long-horizon errors across tested systems and backbones, and suppresses the trend toward error divergence in both diffusion tasks with FNO backbones.With ResCNN backbones, CasEm slows this trend even when it does not fully prevent it; its sample standard deviations are generally smaller than the baselines, especially over longer rollouts.Source: §4.1 · pp. 5–6, Figure 3 · p. 5
  • Subsystem-only rollouts can be stable over long horizons but cannot represent detailed dynamics, whereas full-state emulators retain short-horizon fidelity and may develop explosive error growth.The comparison supports combining full-state detail with subsystem stability; HINE does not consistently outperform Base on these PDE benchmarks.Source: §4.1 · p. 6

4.2 SUBSYSTEM CHOICE AND MECHANISM ANALYSIS

  • Larger subsystems do not always improve predictions: increasing Fisher–KPP modes from 16 to 32 worsens CasEm despite improving Oracle results, while Heat gains diminish beyond K = 4.Across most of the rollout, aggregate directions capture a growing share of Base error and γ is approximately 1; Heat’s exactly closed subsystem has negligible normalized error, unlike the visible actual–ideal gap for approximately closed Fisher–KPP.Source: Figure 4 · p. 5, §4.2 · p. 6

5 CLIMATE EMULATION

CasEm improves long-horizon climate emulation with regional water-subsystem guidance, including lower full-state errors at low inference-time overhead. Regional results and ablations show gains across anomaly and transport diagnostics, with guidance sensitive to temporal and vertical structure.

5 CLIMATE EMULATION

  • The climate evaluation uses 11 ten-year FV3GFS simulations and compares frozen ACE and S-DY backbones with CasEm variants on 34 prognostic fields.The water subsystem models regional averages of total water at each vertical level; TM NRMSE measures persistent bias in 10-year mean fields, while ST NRMSE measures area-weighted trajectory errors.Source: §5 · p. 7

5.1 OVERALL PERFORMANCE

  • Water-subsystem guidance reduces 10-year full-state TM errors by 66.6% for ACE and 46.3% for S-DY, and S-DY’s full-state fair CRPS by 62.2%.CasEm also reduces total-water TM errors by 82.1% and 69.4%, respectively, and improves S-DY’s non-water predictions overall.Source: Table 2 · p. 7, §5.1 · p. 7
  • CasEm adds 2.6% and 1.4% to ACE and S-DY inference times, respectively, while ACE with CasEm beats uncorrected S-DY on full-state ST and TM errors.ACE with CasEm requires about one-third of S-DY’s inference time; all 25 S-DY members improve across all four rollout metrics.Source: Table 3 · p. 7, §5.1 · p. 7
  • Across four initialization times, annual mean full-state and total-water TM and ST errors are lower in every year, with generally smaller variation across starts.Both backbones improve all four metrics on each of the other 10 training trajectories, reducing mean full-state TM errors by 62.4% for ACE and 43.2% for S-DY.Source: §5.1 · p. 7, Figure 5 · p. 8

5.2 REGIONAL CASE STUDY

  • In the South Asian regional case, CasEm reduces amplitude error by 84.5%, P95 exceedance-frequency bias by 94.9%, and slow-anomaly RMSE by 41.3%.Daily total-water transport and convergence errors decrease, and regional-mean wind time series move closer to the reference.Source: Table 4 · p. 8, §5.2 · p. 8, Figure 6 · p. 8

5.3 ABLATION STUDIES

  • Ablations show guidance depends on subannual variability, seasonal timing, year-specific evolution, target-grid resolution, and layerwise correction allocation.Annual averaging, shifted or repeated targets, coarser grids, reduced vertical resolution, and fixed-profile redistribution degrade performance; refining the grid from 12 × 24 to 18 × 36 adds little improvement.Source: Table 5 · p. 9, §5.3 · p. 9

6 CONCLUSION AND FUTURE WORK

CasEm combines independently evolving low-dimensional subsystems with full-state backbones to improve long-horizon emulation, and the paper develops theoretical guidance for its rollout architecture.

6 CONCLUSION AND FUTURE WORK

  • Experiments across ODE/PDE benchmarks and large-scale climate emulation demonstrate gains in long-horizon accuracy and rollout stability.Future work will explore additional climate subsystems capturing cross-variable dynamics, organized into parallel branches or multilevel cascades, and uncertainty-aware coordination.Source: §6 · p. 9

A.1 CONTINUOUS-TIME DEFINITION

  • The cascade has triangular dependence: downstream dynamics may depend on upstream state, but upstream dynamics do not depend on downstream state.This structural restriction prevents downstream disturbances or modeling errors from feeding back into the upstream state.Source: §A.1 · p. 14

A.2 UPSTREAM-FIRST DISCRETIZATION

  • CasEm uses an upstream-first update so the downstream full state can be coordinated with the upstream subsystem at the same time.The upstream subsystem advances through gψ, while fθ and Cϕ jointly update the downstream full state.Source: §A.2 · p. 14

A.3 APPLICATIONS AND STRUCTURAL ADVANTAGES

  • The cascade’s modular structure places an independently evolving low-dimensional subsystem upstream of the full-state emulator.Its stages can have specialized roles, use different models, and interact through explicit interfaces.Source: §A.3 · p. 15

B.1 THEORETICAL SETUP

  • Theorem 1 assumes subsystem-target error is within a radius r and that rollout sensitivity B is finite.B bounds amplification of subsystem-target errors into complementary directions during rollout.Source: §B.2 · p. 16, §B.1 · p. 16

B.2 PROOF OF THEOREM 1

  • The proof separates aggregate-direction correction from complementary rollout error, with learned-target errors able to affect the latter.Although the correction preserves the complementary component of each proposal, the corrected state enters subsequent backbone predictions.Source: §B.1 · p. 16, §B.2 · p. 17

B.3 INTERPRETATION OF THEOREM 1

The error-bound interpretation weighs the benefit of exact aggregate guidance against subsystem prediction error and its propagation. Approximate closure, dynamics-learning error, and autonomous amplification govern subsystem rollout error.

B.3 INTERPRETATION OF THEOREM 1

  • A sufficient condition for improvement is that ideal correction gain ργ exceeds direct subsystem error and its propagation cost.With no target-error propagation, the condition reduces to q < γ for ρ > 0; when γ = 1, it reduces further to q < 1.Source: §B.3 · p. 18
  • With nonzero propagation, greater subsystem accuracy is needed, but sufficiently accurate targets can limit propagated error even when downstream response is appreciable.The Heat/FNO diagnostic supports this mechanism interpretation but does not establish a uniform upper bound on B.Source: §B.3 · p. 18

B.4 APPROXIMATE CLOSURE AND SUBSYSTEM ROLLOUT ERROR

  • Approximate-closure residual depends jointly on unresolved-state spread within an aggregate fiber and its coupling into aggregate dynamics.The residual is bounded by κ_cl,t(z, u)d_Q,t(z), rather than by a state-independent absolute residual.Source: §B.4 · p. 18
  • Subsystem rollout error is controlled by closure, dynamics-learning error, and autonomous amplification; Theorem 2 provides an optional analytical bound, not a premise of Theorem 1.Exact closure removes closure forcing, but not dynamics-learning error or amplification through Γ_z.Source: §B.4 · p. 19

C SUPPLEMENTARY METHODOLOGICAL DETAILS

The appendix gives practical subsystem-selection diagnostics, contrasts CasEm’s temporal recurrences with other prediction structures, and details benchmark subsystems, corrections, reconstructions, and simulation settings.

C.1 SUBSYSTEM SELECTION

  • Subsystem selection balances error coverage, modelability, and correctability; compare candidate designs under matched development data, then test shortlisted designs in actual CasEm rollouts.Coverage identifies aggregate directions explaining accumulated error; modelability is assessed with autonomous multistep forecasts, and correction maps with Oracle rollouts.Source: §C.1 · p. 20

C.2 COMPARISON OF PREDICTION STRUCTURES

  • CasEm recurrently evolves both subsystem and full state, with guidance only from subsystem to full state; the corrected state feeds the next backbone step.Unlike coarse-to-fine rollout, the subsystem is not the complete system at reduced resolution; unlike joint sequence refinement, its updates represent physical-time recurrence.Source: §C.2 · pp. 20–21

D ODE AND PDE BENCHMARKS

  • The four benchmarks use Fourier modes for Heat and Fisher–KPP, degree-1 spherical harmonics for barotropic vorticity, and center-of-mass variables for Alanine.Table 6 summarizes the benchmark systems and specified subsystems.Source: Table 6 · p. 23, §D.1 · p. 23

D.1 DYNAMICAL SYSTEMS AND SUBSYSTEM VARIABLES

  • Heat retains the mean and low Fourier modes in an exactly closed 2K + 1-dimensional subsystem; Fisher–KPP retains corresponding modes but has approximate closure because nonlinear reactions couple modes.The main configurations use K = 4 for Heat and K = 16, or 33 real variables, for Fisher–KPP.Source: §D.1 · p. 23
  • The barotropic-vorticity subsystem is the exactly closed 3-dimensional degree-1 rotational component, while Alanine’s exactly closed translational subsystem comprises center-of-mass position and velocity.The rotational coefficients are associated with angular momentum; Alanine’s six variables describe overall translation rather than internal conformation.Source: §D.1 · p. 24

D.2.1 FULL-STATE CORRECTION

  • Corrections impose subsystem forecasts while preserving unconstrained degrees of freedom or learning residual adjustments; Alanine instead shifts all atoms uniformly to match center-of-mass forecasts.For Heat and Fisher–KPP, spectral replacement preserves other coefficients for FNO, while ResCNN adds a residual with selected modes removed; BVE also uses a learned residual correction.Source: §D.2.1 · pp. 24–25

D.2.2 SUBSYSTEM-ONLY RECONSTRUCTION

  • Subsystem-only diagnostics reconstruct spectral benchmarks with unretained coefficients set to zero, while an Alanine decoder predicts internal positions and velocities from center-of-mass variables.The molecular decoder is fitted on training trajectories and mass-centers its outputs before adding the predicted center-of-mass components.Source: §D.2.2 · p. 25

D.3 NUMERICAL CONFIGURATIONS

  • The appendix reports benchmark discretizations, emulator prediction intervals, and training and development trajectory counts; normalization uses training data only.Comparisons use the same development trajectories and initial states.Source: §D.3 · p. 25, Table 7 · p. 25, Table 8 · p. 25

D.4.1 PREDICTION PERFORMANCE

CasEm improves long-horizon performance in the Heat and Fisher–KPP diffusion benchmarks, including with Fourier neural operator backbones. Low-frequency Fourier subsystems are especially effective for Fisher–KPP.

D.4.1 PREDICTION PERFORMANCE

  • At Heat/FNO h = 16000, CasEm’s full-state endpoint NRMSE is (1.32 ± .594)×10−4, compared with .0874 ± .0820 for Base and .0794 ± .0802 for HINE (mean ± sample SD).Source: §D.4.1 · p. 26, Table 9 · p. 26
  • At h = 1000, Fisher–KPP/ResCNN CasEm reaches full-state NRMSE .443 ± .374, versus Base (8.31 ± 17.3)× 10^3 and HINE (3.38 ± 7.63)× 10^3.CasEm also has lower endpoint NRMSE than Base and HINE at h = 100 and h = 500.Source: §D.4.1 · p. 26, Table 9 · p. 26
  • At h = 1000, Fisher–KPP/FNO CasEm reaches full-state NRMSE .0181 ± .00628, versus Base 744 ± 241 and HINE (2.52 ± .793)× 10^3.CasEm also has lower endpoint NRMSE than Base and HINE at h = 100 and h = 500.Source: §D.4.1 · p. 26, Table 9 · p. 26

D.4.2 SUBSYSTEM SELECTION

  • The appendix also evaluates subsystem size for Heat/FNO and Fisher–KPP/FNO at 1000 steps, with the FNO backbone fixed across subsystem sizes.Tables 10 and 11 contain the numerical results; no cell values are provided in the supplied passages.Source: §D.4.2 · p. 26, Table 10 · p. 26, Table 11 · p. 27
  • For Fisher–KPP/FNO, the low-frequency band [0, 8) outperforms equal-width and wider high-frequency Fourier bands at 1000 steps.Base rollout error increasingly concentrates in low-frequency components, consistent with the effectiveness of correcting those modes.Source: §D.4.2 · p. 27, Table 12 · p. 27, Figure 10 · p. 27

D.5 ADDITIONAL EXPERIMENTAL EXAMPLE

The central-force example tests CasEm with nonlinear aggregate readouts and MLP and Hamiltonian neural network backbones. CasEm substantially outperforms baselines, though the subsystem’s retained nonlinear dynamics leave a gap to Oracle.

D.5 ADDITIONAL EXPERIMENTAL EXAMPLE

  • The subsystem state (r, pr, ℓ) captures radial orbital evolution without spatial orientation and is exactly closed, but its aggregation map from (q, v) is nonlinear.It retains radial distance, radial velocity, and specific angular-momentum magnitude.Source: §D.5 · p. 28
  • The experiment uses randomly oriented initial orbits, time-dependent radial input, 128 training trajectories, and 24 development trajectories.Methods are compared on the same development trajectories and initial states.Source: §D.5 · p. 28
  • For MLP, a learned correction network adjusts the reconstructed proposal; for HNN, learned phase-correction and fusion modules adjust the angular increment before reconstruction.The full-state correction matches radius, radial velocity, and specific angular momentum while retaining candidate radial and tangential directions.Source: §D.5 · p. 28
  • CasEm substantially outperforms baselines with both MLP and HNN backbones, suppressing explosive MLP error growth and improving HNN prediction accuracy.The central-force subsystem offers only modest simplification: it reduces the state from six dimensions to three while retaining nonlinear radial dynamics, which may contribute to the larger gap from Oracle.Source: §D.5 · p. 28, Table 13 · p. 29

D.6 ABLATION OF UNIDIRECTIONAL COUPLING

The appendix finds that reverse coupling does not improve long-horizon accuracy in the tested Heat and Fisher–KPP configurations. It also specifies the climate water and energy subsystems, their corrections, and the evaluation metrics.

D.6 ABLATION OF UNIDIRECTIONAL COUPLING

  • The ablation adds a learned full-state residual to the subsystem update; Heat modes are exactly closed, whereas Fisher–KPP’s nonlinear reaction couples retained and discarded modes.Only the residual network is fine-tuned, using aggregate-prediction losses over progressively longer rollout windows.Source: §D.6 · p. 29, Figure 11 · p. 29
  • Bidirectional coupling does not improve long-horizon accuracy in any of four Heat and Fisher–KPP configurations; errors can exceed Base, while aggregate-component accuracy deteriorates.The authors interpret this as consistent with backbone errors contaminating aggregate forecasts and then affecting later full-state predictions.Source: §D.6 · p. 29, Figure 12 · p. 30

E.1.1 WATER SUBSYSTEM

  • The climate water subsystem predicts regional total-water aggregates independently from background-subtracted, low-mode DCT coefficients, then restores the seasonal–trend background.Its inputs are exogenous rather than taken from the full-state rollout; reconstructed water is clipped to be nonnegative.Source: §E.1.1 · pp. 30–31, Figure 13 · p. 31
  • Water targets correct only full-state water fields every 12 days, using layer-specific regional mixing-ratio increments with gain 0.5 capped at ±5 × 10−4 kg kg−1.Three subsystem updates of four days correspond to eight S-DY rounds; no non-water field is directly changed.Source: §E.1.1 · p. 31, Figure 13 · p. 31

E.1.2 ENERGY SUBSYSTEM

  • The energy subsystem is global-mean column-integrated moist energy, excluding kinetic energy, and evolves independently through a fitted stable forced affine model.A spatially conditioned diffusion U-Net generates temperature increments from energy mismatch, forcing, and backbone temperature and pressure fields.Source: §E.1.2 · p. 32
  • Energy correction uses a spatially conditioned diffusion model for temperature increments, followed by a uniform temperature offset that enforces the desired energy readout.No other fields are directly changed, and the final corrected frame is fed back to the backbone.Source: §E.1.2 · p. 32

E.2 EVALUATION PROTOCOL AND METRICS

  • Climate evaluation fits the subsystem on IC1–10 and uses IC11 for the main evaluation; global rollout and energy-readout errors are normalized by reference variability.ST and TM errors use cosine-latitude area weighting; S-DY ensemble scores use 25 members and average equally over variables within each group.Source: §E.2 · p. 32, §E.2.1 · pp. 32–33

E.2.2 SPATIAL DIAGNOSTICS AND RELATIVE CHANGES

  • Spatial diagnostics report grouped 10-year mean errors and relative reductions, with positive ablation changes indicating degradation.The spatial weighting uses cosine latitude, and group reductions are calculated from weighted group errors rather than averages of grid-cell percentages.Source: §E.2.2 · pp. 33–34

E.2.3 REGIONAL CASE-STUDY EVALUATION

The appendix evaluates South Asian regional diagnostics and tests robustness, energy guidance, and variable-wise and spatial effects. Results show improvements across many fields, while energy-guidance benefits are less consistent across trajectories.

E.2.3 REGIONAL CASE-STUDY EVALUATION

  • The South Asian regional case study uses daily TWP and slow anomalies formed after fitting an intercept, linear trend, and four annual harmonics; filtering precedes June–August selection.The selected June–August date set contains 920 days across 10 summers.Source: §E.2.3 · p. 34, Figure 14 · p. 34
  • Regional RMSE measures errors after spatial averaging, whereas native-grid RMSE retains spatial variation; transport, convergence, and wind diagnostics use unfiltered daily means and regional averages.Source: §E.2.3 · p. 35

E.3.1 ROBUSTNESS ACROSS TRAJECTORIES AND INITIALIZATION TIMES

E.3.2 ENERGY-SUBSYSTEM RESULTS

  • Energy-subsystem guidance improves energy readout and full-state predictions on IC11 and all six reported errors averaged across IC1–11, but benefits are less consistent across trajectories than water guidance.For S-DY, Base already predicts energy accurately, leaving limited room for further gains; the analysis therefore focuses on ACE.Source: §E.3.2 · p. 36, Table 15 · p. 36

E.3.3 VARIABLE-WISE AND SPATIAL IMPROVEMENTS

  • Member-mean RMSE decreases in 31/34 fields, including 23/26 non-water fields; across non-water fields, mean member-RMSE and fair-CRPS reductions are 5.97% and 5.66%.Source: §E.3.3 · p. 37
  • Area-averaged local error decreases by 73.94% for water and 50.54% for the full state, while non-water gains are smaller and spatially heterogeneous.Source: §E.3.3 · p. 37, Figure 16 · p. 37

E.3.4 SUBSYSTEM ABLATIONS

The ablations test how target resolution, vertical correction allocation, and horizontal correction distribution relate to CasEm performance. Horizontally uniform corrections worsen errors despite preserving each layer’s net correction.

E.3.4 SUBSYSTEM ABLATIONS

  • The horizontal-grid and vertical-group ablations vary subsystem configurations, including retrained target grids from 3×6 to the 12×24 reference.The 1×1 subsystem retains one arithmetic mean per water layer, for a dynamic dimension of 8 rather than 32.Source: §E.3.4 · pp. 37–38, Table 17 · p. 37, Table 19 · p. 38
  • Temporal target interventions compare ST/TM NRMSE changes against full CasEm, with 10-year averaging replacing time-varying targets by a single 10-year-mean target.Source: §E.3.4 · p. 38, Table 18 · p. 38
  • Vertical-group tests use pressure-thickness-weighted layer means and share mixing-ratio increments across group layers; layer-mean redistribution spreads the summed eight-layer correction equally before constraints.Source: §E.3.4 · p. 38, Table 20 · p. 38
  • Horizontally uniform corrections increase ST and TM errors despite preserving each layer’s net correction; longitude averaging remains comparable, while latitude averaging raises TM errors.The contrast suggests latitude-dependent correction distribution matters more than detailed longitude dependence in this setting.Source: §E.3.4 · p. 39, Table 21 · p. 39
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