Dimensionality Reduction for Surrogates

Reducing inputs or outputs to their essential few directions makes high-dimensional surrogate modeling tractable and more accurate.

Shrinking the problem first

High-dimensional inputs or outputs make surrogates expensive and inaccurate. Dimensionality reduction finds a smaller set of coordinates that retains the information that matters, then builds the surrogate in that compact space. Done well, it turns an intractable problem into a routine one and often improves accuracy by removing directions that only add noise.

Reducing the output

Kronos motion — three outputs

When the output is a whole field - a temperature map, a stress distribution - proper orthogonal decomposition or an autoencoder compresses it to a few coefficients. The surrogate then predicts those coefficients from the inputs, and the full field is reconstructed on demand. This is the standard route to surrogates and reduced-order models for field-valued outputs.

Reducing the input

Sensitivity analysis discards inputs that barely affect the output
Active subspaces find the input directions along which the output changes most
Sliced inverse regression finds directions relevant to the response
Principal component analysis compresses correlated inputs

Active subspaces

The active subspace method uses the average outer product of the output gradient to identify a few linear combinations of inputs that capture nearly all of the output's variation. The surrogate is built as a function of these combinations. When an active subspace of low dimension exists, high-dimensional problems collapse to one or two effective variables.

Linear versus nonlinear

Linear methods - PCA, POD, active subspaces - are interpretable and robust and suffice when the data lies near a linear subspace. When it lies on a curved manifold, nonlinear methods - kernel PCA, autoencoders, manifold learning - capture more with fewer dimensions, at the cost of interpretability and more data.

In fusion surrogates

For Kronos models, field outputs of the machines are compressed by POD before a surrogate predicts the mode coefficients, and input screening plus active-subspace analysis reduce many physics parameters to the few effective directions that drive performance, keeping surrogate construction tractable and validated.