3. Surrogate modeling landscape

Occupant response prediction in crashworthiness and impact biomechanics is expensive: a single high-fidelity finite element (FE) run of a full vehicle model with a detailed Hybrid III or THOR dummy can take hours to days, and design-space exploration may require hundreds or thousands of such runs. A surrogate model — also called a metamodel or response surface — replaces the expensive simulator with a cheap analytical approximation that maps design or loading parameters to outputs of interest. This chapter maps the surrogate landscape along two axes that matter most in practice: the output type (scalar injury metrics versus full time histories) and the modeling philosophy (purely data-driven interpolation/regression versus physics-informed formulations).

Outputs first: scalar metrics versus time series

Most regulatory and consumer-rating criteria are scalars derived from a time history: HIC36 or HIC15, 3 ms chest acceleration clip, chest deflection peak, Nij for the neck, and AIS-correlated injury risk probabilities. Predicting these scalars directly is attractive because the target is low-dimensional and the surrogate can be trained on a modest design of experiments (DoE). The trade-off is that a scalar surrogate discards the temporal information that explains why a design is good or bad, and it cannot be reused if the injury criterion changes.

Time-series surrogates predict the full signal, e.g. the head resultant acceleration a(t) or chest deflection d(t) sampled at 1 kHz over 150 ms. These are far more informative — they support re-derivation of any scalar criterion, allow visual validation against corridors, and expose phase and peak-timing errors that scalar metrics hide. The cost is a much higher-dimensional output: a 150 ms signal at 1 kHz is 150 samples per channel, and naive regression on that vector is ill-posed. Practical time-series surrogates therefore combine a dimensionality reduction step (PCA, functional PCA, or autoencoders) with a regressor on the latent coefficients, or use sequence models directly.

Interpolation: exact reproduction of known designs

Interpolants pass exactly through every training sample. Radial basis functions (RBF), kriging/Gaussian process regression, and spline-based interpolants belong here. Kriging is the workhorse for deterministic computer experiments because it provides both a prediction and a variance estimate, which enables efficient adaptive sampling (e.g. expected improvement). For a scalar output y at input x, the ordinary kriging predictor is

y_hat(x) = mu + r(x)^T * R^-1 * (y - mu*1)

where:
  R[i,j] = exp(-sum_k theta_k * |x_i,k - x_j,k|^p)   # correlation matrix
  r(x)[i] = correlation between x and training point x_i
  mu      = generalized least-squares mean
  theta, p = fitted hyperparameters (MLE or cross-validation)

Interpolation is the right choice when the simulator is deterministic, the DoE is space-filling (Latin hypercube or maximin), and the sample count is small relative to what a neural network would need. Its weakness is sensitivity to noise and to clustered samples: a poorly spaced DoE produces correlation matrices that are near-singular, and the resulting surface can oscillate between points. Always inspect the hyperparameter fit and the condition number of R; a condition number above roughly 1e10 signals redundant or duplicated designs.

Regression: smoothing, generalization, and many samples

Regression surrogates do not reproduce training points exactly; they fit a trend and tolerate residual error. Polynomial response surfaces, support vector regression, gradient-boosted trees, and feed-forward neural networks fall in this category. They shine when the response is noisy (e.g. stochastic material parameters, mesh-induced scatter), when the sample count is large, or when the input dimension is high enough that kriging hyperparameter fitting becomes unstable.

For time-series outputs, a common and robust pipeline is PCA plus regression. Concretely, given a matrix Y of shape (n_runs, n_time) holding, say, chest acceleration for each run:

1. Center Y by subtracting the column mean, then compute the SVD and retain the first k principal components that capture at least 99% of variance (often k = 3–6 for a smooth acceleration pulse).
2. Project each run onto the retained basis to obtain coefficient vectors C of shape (n_runs, k).
3. Train one regressor per coefficient, mapping design parameters to c_1 … c_k. Kriging for small n, gradient boosting or a small MLP for large n.
4. At prediction time, reconstruct y_hat(t) = mean(t) + sum_i c_i_hat * phi_i(t).

This decomposition is interpretable: the first component usually captures overall pulse amplitude, the second captures peak timing or pulse width. A validation plot of predicted versus measured peak acceleration, plus a corridor plot of reconstructed signals, tells you quickly whether the surrogate is usable. Typical failure symptoms are a good scalar fit but a phase-shifted reconstructed pulse — a sign that too few components were retained or that the coefficient regressors were trained independently and their errors do not cancel.

Physics-informed approaches

Physics-informed surrogates inject known mechanics into the model so that fewer data are needed and extrapolation is better behaved. Three variants are common in occupant safety:

• Constrained regression: enforce known limits or monotonicity, e.g. chest deflection must increase with impact velocity over the calibrated range, or HIC must be non-negative. This is implemented as a penalty term or by construction (monotonic splines, monotone neural networks).
• Physics-informed neural networks (PINNs): add a residual of the governing equation (e.g. a lumped-mass occupant model, m·a = F_restraint − F_inertia) to the loss. The network learns the response while satisfying the dynamics, which improves behavior in sparse-data regions.
• Hybrid or gray-box models: combine a reduced-order physical model (mass-spring-damper, multi-body) with a data-driven correction term that absorbs the discrepancy between the reduced model and the FE reference. This is often the most practical option when a credible low-fidelity model already exists.

Choosing and validating a surrogate

Model selection should be driven by the intended use. For scalar injury metrics used in optimization loops, accuracy near the constraint boundary matters more than global average error, so report errors on the subset of runs close to the threshold. For time-series surrogates, report both a signal-level metric (RMSE or normalized RMS error over the pulse) and derived scalar errors (peak value, time-to-peak, HIC). Always hold out runs the model never saw; k-fold cross-validation on a space-filling DoE is standard, but leave-one-out is cheap for kriging and often sufficient.

The figure below summarizes how the three families relate to output type and data regime, and what validation signal to watch for each.

Two practical cautions close this overview. First, a surrogate is only valid inside the convex hull of its training DoE; extrapolation to a new impact angle or a new restraint configuration is a modeling assumption, not a prediction, unless a physics-informed structure supports it. Second, always keep the FE model as the arbiter: when a surrogate flags a promising design, re-run the high-fidelity simulation before committing to hardware testing. The surrogate accelerates the search; it does not replace the reference.