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Bounded Channel-Adaptive Spectral Learning for Forward-Consistent Inverse Flapping-Wing Aerodynamics

Haichuan Li

arXiv:2609.38730v1cs.RO

TL;DR

Recovering wing kinematics from aerodynamic histories is difficult because the inverse mapping can be non-unique and kinematic error alone does not ensure aerodynamic agreement. The paper proposes a recurrent-first model with controlled spectral correction and frozen-surrogate fine-tuning; it reports improved long-horizon prediction and surrogate-based consistency, with initialization sensitivity and a small tail-error trade-off.

  • Problem

    Recovering wing kinematics from desired aerodynamic trajectories is challenging because similar force histories can arise from different motions, and minimizing kinematic error alone does not ensure the requested aerodynamic response.

    Source: Introduction · p. 1

  • Method

    BCS-GRU keeps recurrent prediction primary, adds a bounded output-specific spectral correction, and fine-tunes with a causally aligned objective evaluated through a separately trained, frozen aerodynamic surrogate.

    Source: Introduction · p. 1

  • Results

    BCS-GRU improves long-horizon inverse prediction with modest computational overhead, while forward-consistent fine-tuning reduces aerodynamic discrepancy under a seed-matched frozen surrogate with a small kinematic tail-error trade-off.

    Source: Conclusion · p. 7

  • Takeaways & Limitations

    The findings support using spectral information to refine rather than replace the temporal representation in history-aware inverse modelling of flapping-wing aerodynamics.

    Source: Introduction · p. 1

    BCS-GRU's three-seed mean MSE improvement is initialization-sensitive: it improves over GRU for two seeds but reaches a weaker solution for the third.

    Source: Long-Horizon and Initialization… · p. 18

Abstract

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Flapping-wing vehicles regulate aerodynamic forces and moments through coordinated variations in stroke, deviation, and pitch motion. Because the resulting loads depend on both the instantaneous wing configuration and its preceding motion history, recovering suitable wing kinematics from a desired aerodynamic trajectory is a challenging inverse problem. Existing sequence models capture temporal dependencies, while spectral methods can exploit the periodic structure of flapping motion. However, unrestricted frequency-domain augmentation may interfere with temporal representations and produce inconsistent corrections across kinematic variables and prediction horizons. We propose the Bounded Channel-Adaptive Spectral Residual Gated Recurrent Unit (BCS-GRU), which retains recurrent temporal prediction as its primary representation and restricts spectral information to a controlled output-specific correction. We further introduce a causally aligned forward-consistency objective that evaluates predicted kinematics through a separately trained and frozen aerodynamic surrogate. Experiments under a unified episode-level protocol show that BCS-GRU improves inverse prediction over recurrent and adaptive-spectral baselines, with greater benefits at longer prediction horizons. Forward-consistent fine-tuning further improves surrogate-based aerodynamic consistency while maintaining mean kinematic accuracy. These results demonstrate that controlled spectral correction and forward-consistent learning provide an effective framework for history-aware inverse modelling of flapping-wing aerodynamics.

Introduction

The paper addresses history-dependent, potentially non-unique inverse mapping from aerodynamic trajectories to wing kinematics, where kinematic accuracy alone does not ensure aerodynamic consistency. It proposes a recurrent-first model with controlled spectral corrections and a frozen-surrogate consistency objective.

Problem Formulation

Introduction

  • BCS-GRU retains recurrent prediction as its primary representation and adds a bounded, channel- and horizon-adaptive spectral residual, while a frozen aerodynamic surrogate supports forward-consistent fine-tuning.The paper also introduces a unified episode-level evaluation protocol for forward and inverse learning.Source: Introduction · pp. 1–2

Temporal GRU Backbone

  • The GRU maps aerodynamic history to a multi-step kinematic prediction, while the spectral pathway extracts nonzero Fourier features from the input history.The recurrent pathway remains the primary predictor; the spectral pathway supplies a correction rather than replacing it.Source: Temporal GRU Backbone · p. 3, Spectral Feature Extraction · p. 3

Bounded Channel-Adaptive Spectral Residual

  • The spectral correction is bounded and modulated by forecast position, output channel, and input history before being added to the temporal prediction.The learned scale varies across forecast positions and kinematic variables, while the sample-dependent gate adjusts the correction for each history.Source: Bounded Channel-Adaptive Spectral… · p. 3

BCS-GRU Pretraining Objective

  • Pretraining uses normalized kinematic MSE as its primary objective, with an additional MAE term and a penalty on spectral-correction magnitude.The MAE term reduces dominance of large squared residuals, while the penalty discourages unnecessarily large corrections.Source: BCS-GRU Pretraining Objective · p. 4

Forward Aerodynamic Surrogate

  • A separately trained forward GRU evaluates predicted kinematics and remains frozen, although gradients can pass through it to update the inverse model.Freezing prevents the surrogate from adapting to inverse predictions and preserves it as a fixed evaluator.Source: Forward Aerodynamic Surrogate · p. 4

Causal Forward-Consistency Alignment

  • For causal alignment, the predicted kinematic state enters shifted context, and the surrogate target is the subsequent aerodynamic state at s+L+1.The one-step shift follows because the predicted state at s+L becomes the latest context element used to predict the response at s+L+1.Source: Causal Forward-Consistency Alignment · p. 4

Staged Fine-Tuning

  • Fine-tuning combines kinematic and force-consistency losses, initially updates only the spectral pathway, then jointly optimizes the inverse model while the forward surrogate stays frozen.The principal experiments set the optional smoothness weight to zero; the schedule first adjusts the spectral correction without immediately perturbing the temporal representation.Source: Forward-Consistent Fine-Tuning Objective · p. 4, Staged Fine-Tuning · p. 4

Experiments

The experiments use episode-level partitions and matched temporal windows to evaluate inverse prediction and forward consistency. Model comparisons use validation-based selection, seed-matched frozen surrogates, and separately reported kinematic and force-consistency errors.

Dataset and Experimental Protocol

  • The dataset is partitioned by episode before window generation, keeping overlapping windows from one physical trajectory within the same partition.Kinematic and aerodynamic variables are standardized separately using training-episode statistics, and primary losses and metrics use normalized target space.Source: Dataset and Experimental Protocol · p. 5

Causal Forward-Consistency Evaluation

  • Forward-consistency experiments use single-step inverse prediction, insert the predicted state into causally shifted kinematic context, and compare the reconstructed response with f at s+257.The measured subsequent kinematic state is excluded to prevent future-kinematic leakage.Source: Causal Forward-Consistency Evaluation · p. 5

Models and Training Protocol

  • The comparison includes a matched GRU, attention-based Seq2Seq, and Seq2Seq+ASL, with checkpoints and hyperparameters selected on validation data and consistency runs paired with seed-matched frozen forward checkpoints.Within each seed, the zero- and nonzero-consistency conditions hold the other listed training and evaluation choices fixed.Source: Models and Training Protocol · p. 5

Evaluation Metrics

  • Inverse metrics aggregate the three kinematic channels, while forward and force-consistency metrics aggregate five aerodynamic channels and are reported separately.Force-consistency errors measure agreement under the frozen surrogate, not direct experimental error.Source: Evaluation Metrics · p. 5

Results

BCS-GRU improves long-horizon inverse prediction, though its advantage over GRU varies across seeds. Bounded, output-specific spectral corrections support the prediction results, while forward-consistent fine-tuning reduces surrogate-based force errors with a small kinematic tail-error trade-off.

Inverse Prediction

  • At H = 1, 16, and 32, seed-42 BCS-GRU reduces MSE over GRU by 3.7%, 7.3%, and 21.5%, respectively.At H = 32, its MAE and P95 reductions are 5.1% and 13.2%; the H = 16 P95 is slightly higher.Source: Inverse Prediction · p. 5
  • At H = 32, BCS-GRU obtains the lowest mean MSE, MAE, and P95 among evaluated architectures: 0.072±0.011, 0.184±0.009, and 0.568±0.029.Relative to GRU, mean MSE, MAE, and P95 fall by 10.59%, 2.84%, and 6.48%; the reported 32-step comparison spans three seeds.Source: Inverse Prediction · p. 6, Table 3 · p. 6

Architectural Ablation

  • Unrestricted fusion performs worse than GRU, while an unbounded residual approaches GRU-level MSE but does not match complete BCS-GRU.The learned correction varies by channel and forecast position; at h = 32, pitch has a mean absolute correction of approximately 0.31 versus approximately 0.17 for stroke and deviation, and the GRU retains the dominant trajectory shape.Source: Architectural Ablation · pp. 6–7, Figure 2 · p. 6

Forward-Consistent Fine-Tuning

  • Across three seeds, λF = 0.05 reduces mean force-consistency MSE, MAE, and P95 by 6.12%, 2.74%, and 5.10%, respectively.Force-consistency MSE decreases for every seed, while kinematic P95 increases by 1.43%; these are frozen-surrogate metrics, not direct physical validation.Source: Forward-Consistent Fine-Tuning · p. 7, Table 7 · p. 7, Figure 4 · p. 7
  • Median batch-one overhead remains between 6.3% and 7.2%, while the seed-42 MSE reduction reaches 21.5% at H = 32.The accuracy–latency comparison is reported for batch-one GPU latency on an AMD Instinct MI250X.Source: Forward-Consistent Fine-Tuning · p. 7, Table 8 · p. 7

Conclusion

The study presents BCS-GRU for history-aware inverse modelling of flapping-wing aerodynamics. It reports improved long-horizon prediction and surrogate-based aerodynamic consistency, while identifying validation and robustness priorities for future work.

Conclusion

  • BCS-GRU retains the GRU as primary predictor and uses spectral information as a bounded, channel-adaptive residual to improve long-horizon prediction with modest computational overhead.Forward-consistent fine-tuning further reduces aerodynamic discrepancy under a seed-matched frozen surrogate, with a small kinematic tail-error trade-off.Source: Conclusion · p. 7
  • The reported aerodynamic-discrepancy reduction is under a seed-matched frozen surrogate, alongside a small kinematic tail-error trade-off.Source: Conclusion · p. 7
  • Future work should validate reconstructed motions using higher-fidelity simulations or experiments and reduce initialization sensitivity and tail error.Source: Conclusion · p. 7

Additional Experimental Details

The additional experimental details describe a fixed episode-level protocol, data processing and normalization, causally aligned consistency evaluation, and an oracle check of that alignment.

Unified Experimental Protocol

  • The experimental protocol keeps the episode partition fixed across models, horizons, and task directions; the five-seed extension applies to the matched 32-step GRU and BCS-GRU comparison.Split seed 42 assigns episodes, while model seeds control initialization, mini-batch ordering, and active dropout sampling.Source: Unified Experimental Protocol · p. 9, Table 9 · p. 9

Dataset Processing and Window Construction

  • The dataset contains 548 episodes with three-channel kinematic and five-channel aerodynamic trajectories, partitioned into 438 training, 55 validation, and 55 test episodes before window generation.Overlapping windows from a physical trajectory therefore remain in the same partition.Source: Dataset Processing and Window… · p. 9

Training-Derived Normalization

  • Separate training-derived standardizers are fitted for kinematic and aerodynamic channels, then applied unchanged to validation and test data; losses and primary metrics use normalized target space.Source: Training-Derived Normalization · p. 9

Causal Alignment for Forward Consistency

Oracle Validation of Causal Alignment

  • Oracle alignment yields MSE 0.018 and MAE 0.102, close to standalone forward-GRU results of 0.019 ± 0.001 and 0.103 ± 0.003 across three seeds.The oracle check validates pipeline alignment and is not treated as an inverse-prediction result.Source: Oracle Validation of Causal Alignment · p. 10

Additional Multi-Seed Evaluation

The five-seed comparison finds the clearest average BCS-GRU benefit at H = 1, close results at H = 16, and nearly identical mean MSE at H = 32.

Five-Seed GRU and BCS-GRU Comparison Across Horizons

Model and Training Configurations Principal Models

The principal configurations pair recurrent and sequence-to-sequence inverse models with a bounded spectral-residual design and a separately trained forward surrogate. Comparisons use shared partitions and matched controls for architectural, training, and forward-consistency experiments.

Model and Training Configurations Principal Models

  • Seq2Seq and Seq2Seq+ASL were retrained under a common partition, normalization, model-selection, and evaluation protocol rather than reproducing every original training component.A separately trained forward GRU serves as the frozen aerodynamic surrogate.Source: Model and Training Configurations… · p. 10

BCS-GRU Spectral Configuration

  • BCS-GRU uses a 64-dimensional single-layer GRU and a spectral pathway that projects the first 16 nonzero Fourier bins into an additive prediction correction.The spectral projector, prediction head, and sample-dependent gate use dropout 0.10; recurrent dropout is disabled.Source: BCS-GRU Spectral Configuration · p. 10

Architectural Variants

  • The architectural variants compare unrestricted fusion, an unbounded residual, multi-band residuals, a two-layer BCS-GRU, and the bounded single-layer design.All variants use the same episode partition, normalization, checkpoint selection, and seed.Source: Architectural Variants · pp. 10–11

Published Forecasting Baselines

  • Six published forecasting architectures are evaluated for inverse and forward prediction at horizons 1, 16, and 32 across three model seeds.They share the principal models’ episode split, standardization, direct horizons, validation-based selection, and test metrics.Source: Published Forecasting Baselines · p. 11

Optimization and Model Selection

  • Model settings are selected using validation performance, while the held-out test partition is excluded from model and checkpoint selection.BCS-GRU pretraining allows up to 2,000 epochs with early stopping after 100 epochs without validation improvement.Source: Optimization and Model Selection · p. 11

Random-Seed and Matched-Control Protocol

  • The episode split is fixed across experiments, while model seeds vary initialization, minibatch ordering, and active dropout sampling without changing the episodes.Architectural and loss-weight ablations use seed 42; selected configurations are subsequently evaluated across three seeds.Source: Random-Seed and Matched-Control Protocol · p. 12

Forward-Consistency Conditions

  • Forward-consistency conditions compare nonzero consistency weights with a matched λF = 0 control that holds training and evaluation procedures fixed.The control accounts for additional fine-tuning and staged freezing; the smoothness term is disabled in reported experiments.Source: Forward-Consistency Conditions · pp. 12–13

Evaluation and Statistical Aggregation Metric Definitions

The evaluation aggregates inverse-prediction errors across kinematic channels and forward or consistency errors across aerodynamic channels. It reports sample standard deviations and paired comparisons, then contrasts the strongest added models for inverse and forward prediction.

Evaluation and Statistical Aggregation Metric Definitions

  • Inverse metrics aggregate across three kinematic channels, while forward and force-consistency metrics aggregate across five aerodynamic channels.The two error families are reported separately because the weighted training objective changes scale with λF.Source: Evaluation and Statistical Aggregation… · p. 13

Multi-Seed and Paired Statistics

  • Reported uncertainty is sample standard deviation, not standard error or a confidence interval, and rankings use unrounded values.Paired control–FCI relative error reduction is positive when FCI has lower error.Source: Multi-Seed and Paired Statistics · p. 13

Inverse Prediction

  • TCN has the lowest MSE and P95 among the six added inverse models at every horizon; TimesNet is second in MSE and has the lowest MAE at H = 32.Linear and patch-based models perform substantially worse on the inverse mapping.Source: Inverse Prediction · p. 13

Forward Prediction

  • The single-step forward GRU used for FCI obtains MSE of 0.019 ± 0.001, below TCN’s 0.026 ± 0.001; TCN is the strongest added forward model at every horizon.TCN is retained as an independent candidate for subsequent cross-surrogate evaluation.Source: Forward Prediction · p. 13

Additional Long-Horizon Results

At 32 steps, BCS-GRU leads the evaluated architectures on all three reported inverse metrics, although its improvement over GRU varies by initialization. The section also reports paired episode behavior, output-dependent corrections, and forward-consistency gains with a kinematic tail-error trade-off.

Expanded 32-Step Comparison

  • BCS-GRU has the lowest mean MSE, MAE, and P95 across evaluated architectures, reducing these metrics relative to TCN by approximately 42.5%, 26.5%, and 22.5%.TCN is the strongest additional model in MSE, while TimesNet has lower MAE than TCN.Source: Expanded 32-Step Comparison · p. 13, Table 15 · p. 15

Paired GRU and BCS-GRU Results

  • BCS-GRU improves MSE and MAE for seeds 42 and 2048 but yields a weaker solution for seed 1024, indicating initialization sensitivity rather than uniform improvement.The lower three-seed mean represents a positive average effect.Source: Paired GRU and BCS-GRU Results · p. 13

Paired Comparison with Adaptive Spectrum Learning

  • BCS-GRU outperforms Seq2Seq+ASL for every matched initialization, a more consistent paired comparison than its comparison with GRU.The paired results are summarized for 32-step MSE.Source: Paired Comparison with Adaptive… · p. 13

Episode-Level Paired Analysis

  • For seed 42, BCS-GRU has lower MSE in 25 of 55 held-out episodes; positive mean and negative median reductions indicate gains concentrated in some high-error trajectories.The paired-bootstrap probability of positive mean reduction is 98.13% for MSE and 88.50% for MAE; the MSE interval excludes zero, while the MAE interval includes zero.Source: Episode-Level Paired Analysis · p. 14

Output- and Horizon-Dependent Correction

  • At h = 32, pitch scale and mean gate reach approximately 0.86 and 0.80, while mean absolute pitch correction reaches approximately 0.31, versus approximately 0.17 for stroke and deviation.The spectral contribution varies across channels, input histories, and forecast positions.Source: Output- and Horizon-Dependent Correction · p. 14

Residual Bounding and Temporal–Spectral Balance

  • In the single-step experiment, the two-layer BCS-GRU has normalized test MSE 0.211 versus 0.060 for the single-layer model, showing that residual bounding alone does not ensure effective integration.The results emphasize the compact temporal backbone alongside controlled residual formulation.Source: Residual Bounding and… · p. 14

Matched-Control Interpretation

  • The matched λF = 0 fine-tuning control improves kinematic reconstruction over the pretrained checkpoint but only slightly reduces reconstructed aerodynamic discrepancy.It controls for additional training, staged freezing, and validation-based checkpoint selection.Source: Matched-Control Interpretation · p. 14

Three-Seed Consistency Results

  • Across three matched seeds, λF = 0.05 lowers mean force-consistency MSE, MAE, and P95 by 6.12%, 2.74%, and 5.10%, respectively, alongside a 1.43% increase in kinematic P95.The absolute closed-loop force-consistency error remains above oracle forward error because evaluation uses an inverse-predicted state rather than the measured subsequent state.Source: Three-Seed Consistency Results · p. 15

Inverse BCS-GRU Decomposition Diagnostics

The decomposition examines the temporal base and spectral correction from one trained BCS-GRU checkpoint, isolating the correction used in the final prediction. It reports a lower final MSE, generally beneficial correction alignment, and adaptive modulation that varies by horizon, channel, and sample.

Inverse BCS-GRU Decomposition Diagnostics

Additive decomposition

  • The temporal pathway produces the base prediction, and the bounded spectral residual is scaled and gated before being added to it.Exported components reconstruct the model output within a tolerance of 10^-5.Source: Additive decomposition · p. 16

Correction-to-base ratio

  • Relative spectral contribution generally increases with forecast horizon, with the largest ratios for later-horizon pitch, smaller ratios for surge, and intermediate contributions for heave.The ratio is not itself an error-reduction measure and can be large when the temporal base magnitude is near zero.Source: Correction-to-base ratio · p. 17

Correction alignment

  • Correction alignment is positive across evaluated horizons and channels, strongest for surge and pitch and weaker but still positive for heave.The correction is generally directed toward reducing temporal-base error.Source: Correction alignment · p. 17

Adaptive modulation statistics

  • The sample-dependent gate ranges from 0.007 to approximately 1.000, allowing near-suppression for some sample-horizon-channel combinations and larger corrections for others.The bounded final correction remains approximately within [−0.852, 0.856] in normalized target space.Source: Adaptive modulation statistics · p. 17

Visualization

  • Figure 6 shows a representative trajectory, horizon-dependent correction ratios, positive alignment, and heatmaps of learned scale, mean gate, and correction magnitude.The representative sample is selected by temporal-base MSE closest to the test-set median.Source: Visualization · pp. 17–18

Spectral contribution to inverse prediction

Interpretation and scope

  • The spectral correction reduces normalized MSE without merely increasing prediction magnitude, while the learned scale, gate, and bounded residual limit unrestricted correction magnitudes.The temporal GRU remains the base trajectory predictor rather than being replaced by the spectral pathway.Source: Interpretation and scope · p. 18

Additional Efficiency Results

BCS-GRU adds a modest GPU inference cost, while the reported CPU measurements at two horizons are not a reliable basis for claiming an inherent latency advantage.

GPU Inference

  • BCS-GRU increases median batch-one latency by 7.2%, 6.3%, and 6.6% at horizons H = 1, 16, and 32, respectively; batch-512 throughput decreases approximately 4.5%–4.9%, while peak allocated memory remains below 243 MiB.Source: GPU Inference · p. 18
  • Batch-512 throughput decreases by approximately 4.5%–4.9%, while peak allocated memory remains below 243 MiB.Source: GPU Inference · p. 18

Single-Threaded CPU Inference

  • Elevated upper-tail latency in the horizon-1 and horizon-32 GRU measurements means lower measured BCS-GRU CPU latency at those horizons is not an inherent computational advantage.The GPU benchmark provides the primary efficiency comparison.Source: Single-Threaded CPU Inference · p. 18

Reproducibility and Artifact Tracking

The results compare BCS-GRU with published inverse baselines, examine its spectral residual and initialization behavior, and report forward-consistency outcomes. The paper also describes artifact tracking and notes limits to component attribution and surrogate-based evaluation.

Reproducibility and Artifact Tracking

  • Runs record configurations, split manifests, seeds, selected epochs, checkpoints, and metrics; retained artifacts include normalization statistics and efficiency measurements.A read-only registry with SHA-256 checksums protects evaluation checkpoints, and the published-baseline campaign contains 108 validated metric files.Source: Reproducibility and Artifact Tracking · p. 18

Controlled Spectral Residual Learning

  • Unrestricted fusion, an unbounded residual, multi-band expansion, and added recurrent depth do not match complete BCS-GRU, and the current ablation does not isolate its individual components.Component-level ablations are needed before assigning the improvement to residual bounding, channel adaptation, horizon adaptation, or sample-dependent gating.Source: Controlled Spectral Residual Learning · p. 18
  • BCS-GRU outperforms the evaluated TCN, TimesNet, iTransformer, PatchTST, DLinear, NLinear, and Seq2Seq+ASL models in the three-seed 32-step comparison.Source: Controlled Spectral Residual Learning · p. 18

Long-Horizon and Initialization Behaviour

  • 3.7%, 7.3%, and 21.5% are the seed-42 MSE reductions over GRU at H = 1, 16, and 32, respectively.The three-seed comparison shows initialization sensitivity: BCS-GRU improves over GRU for seeds 42 and 2048 but converges to a weaker solution for seed 1024.Source: Long-Horizon and Initialization… · p. 18
  • BCS-GRU obtains lower MSE on 25 of 55 test episodes; large gains on a subset of high-error trajectories produce a positive mean reduction.GRU retains smaller advantages on more episodes.Source: Long-Horizon and Initialization… · p. 18

Implications of the Published Baseline Comparison

  • TCN has the lowest MSE and P95 among the six additional models at every evaluated horizon, while TimesNet is second-best in MSE and has the lowest H = 32 MAE.DLinear, NLinear, and PatchTST perform weakly on the inverse mapping under the evaluated protocol; iTransformer remains less accurate than the compact recurrent and convolutional models.Source: Implications of the Published Baseline… · pp. 18–19

Forward-Surrogate Consistency

  • Forward-consistent fine-tuning reduces mean force-consistency MSE by 6.12% relative to the matched λF = 0 control, with improvements for all three seeds.Mean kinematic MSE also decreases slightly, while kinematic P95 increases by 1.43%.Source: Forward-Surrogate Consistency · p. 19
  • These consistency results measure agreement under a learned, frozen forward surrogate rather than direct aerodynamic performance, and freezing does not rule out surrogate-error exploitation or distribution-shift degradation.Source: Forward-Surrogate Consistency · p. 19

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