ML-Based Gust Load Prediction for Wing Structures

Implement a reduced-order aeroelastic model for time-domain gust response. Model the wing as a cantilever beam with 3–5 structural modes (first and second bending, first torsion, and optionally higher modes). Use Roger's rational function approximation or a quasi-steady strip theory approach for unsteady aerodynamics. The state-space form is: dx/dt = A*x + B*w_g(t), where x is the state vector and w_g(t) is the gust velocity input.

Implement both 1-minus-cosine discrete gusts (FAR 25.341(a): w_g = U_ds/2 * (1 - cos(2*pi*s/2H)) for 0 <= s <= 2H, with gust gradient H from 30 to 350 feet) and von Karman continuous turbulence (filter white noise through the von Karman power spectrum to generate realistic turbulence time histories). Validate against published gust response results.

Create a comprehensive dataset by systematically varying: flight condition (Mach number, altitude, dynamic pressure), gust parameters (gradient length H for discrete gusts, intensity sigma for continuous turbulence), aircraft mass/CG (light/heavy, forward/aft center of gravity), and structural damping. For each combination, run a 5-second time-domain simulation and extract the peak wing root bending moment.

Generate at least 50,000 samples. Store each as a pair: the input gust velocity time history (resampled to 200 time steps) and the scalar output (peak bending moment, normalized by the static 1g bending moment). Also store the full bending moment time history for models that predict the complete response. Use HDF5 format via h5py for efficient storage and random access during training.

Build a bidirectional LSTM in PyTorch that takes the gust velocity time series plus flight condition parameters as input and predicts peak bending moment. The architecture should have: an input projection layer (mapping each time step's features to a 64-dimensional embedding), 2 bidirectional LSTM layers with 128 hidden units, and a fully connected output head (128 -> 64 -> 1) with ReLU activations.

Concatenate the flight condition parameters (Mach, altitude, mass, CG) as additional features at each time step — this is a standard technique for conditioning sequence models on static context. Alternatively, inject them only at the output head. Experiment with both approaches and measure performance differences. Use MSE loss with an additional asymmetric penalty term that penalizes underestimation more heavily than overestimation — for structural loads, underestimating the peak is the dangerous failure mode.

Train the LSTM with Adam optimizer (lr=1e-3), batch size 256, for up to 100 epochs with early stopping on validation loss (patience=10). Use a cosine annealing learning rate schedule for stable convergence. Monitor both MSE and peak load error at the 99th percentile — this extreme-value metric matters more than average accuracy for structural design.

Implement and train two alternative architectures for comparison: a 1D CNN (4 convolutional layers with kernel size 5, followed by global average pooling and FC head) and a Transformer encoder (4 layers, 4 attention heads, positional encoding). Compare all three on the test set. The Transformer may capture long-range temporal dependencies better, but the LSTM typically handles the causal nature of gust response more naturally.

For structural loads, the tails of the distribution matter most. Plot the cumulative distribution function (CDF) of predicted vs. actual peak loads. Overlay both on a Gumbel probability plot — if the model captures the extreme value statistics correctly, the points should follow a straight line. Compute the exceedance probability: at the design load level (e.g., 2.5g limit load), how accurately does the model predict whether a given gust encounter exceeds the limit?

Analyze the model's errors conditioned on the true peak load magnitude. Does accuracy degrade for the highest loads — the ones that matter most? If so, implement importance sampling: regenerate training data with more samples in the high-load region and retrain. Compare the tail accuracy before and after this targeted data augmentation.

Validate the model against known physical relationships. Peak load should increase with gust intensity (monotonically, all else equal). Discrete gust loads should peak when the gust gradient length H matches a structural natural frequency (resonance tuning). Loads should increase with dynamic pressure (higher speed = more aerodynamic force). Plot partial dependence curves and verify these trends hold.

Create gust envelope plots: for a fixed flight condition, sweep gust gradient H from 30 to 350 feet and plot the predicted vs. actual peak bending moment. The envelope should show a clear peak at the resonant gradient length. This is the "tuned gust" that drives structural sizing — if the ML model captures this peak accurately, it's learning the right physics.

Implement MC Dropout (dropout at inference time with 50 forward passes) to estimate prediction uncertainty for each gust case. Map the uncertainty: is it highest for gust profiles near resonance (where small changes in input cause large changes in output)? For near-resonance cases, the model should report high uncertainty — this is a physically meaningful result.

Write a research-quality report addressing: model architecture comparison, accuracy on extreme loads, physics consistency, uncertainty quantification, and regulatory implications. Discuss how the FAA's advisory circulars on loads analysis (AC 25.341-1) would need to evolve to permit ML-augmented gust loads analysis. The report should acknowledge that ML surrogates are currently suitable for preliminary design exploration and loads trend studies, not final certification loads — and explain what additional validation would be needed to change that.