Abstract: High-dimensional model representation (HDMR) gives a surrogate that is interpretable by construction: with an orthonormal basis, its component functions translate directly into variance-based Sobol' sensitivity indices. The established HDMR surrogates, however, expand the response as a sum of low-order additive components, so a genuinely multiplicative (product or power law) response is reached only through a cascade of interaction terms whose count grows combinatorially with the dimension. We attach to the additive expansion a single log-additive multiplicative support S(x) = exp( Σⱼ gⱼ(xⱼ) ), in the spirit of the Enhanced Multivariance Product Representation (EMPR), and let a small neural network learn the remaining non-separable part. The support is used only when it lowers a held-out error, so the method guards itself. On clean product-structured benchmarks, the surrogate is dimension-independent in accuracy: on a Genz product peak, it holds a test-normalised mean squared error near 2 × 10⁻⁴ from five to thirty variables, while a Gaussian process and a multilayer perceptron drift from 10⁻³ up to about 0.65 over the same range. We then study two real systems. On Delft model basin yacht resistance data, the method is competitive and recovers the correct dominant variable (the Froude number). On hydrogen ignition delay over fifteen perturbed Arrhenius rate parameters, generated from a real kinetic mechanism, it again recovers the physically correct ranking (the chain branching reaction H+O₂ ⇌ O+OH) but is beaten on accuracy by the black box models. A simple linear-in-log fit quality explains and predicts this split: near one (clean products), the support brings accuracy dominance; well below one (real coupled physics), it brings only interpretability. A direct comparison with polynomial chaos expansion further locates this product structure advantage in a logarithmic transform combined with an additive truncation rather than in the choice of HDMR, and the method is positioned within the broader landscape of spectral surrogates, post hoc explainers such as SHAP, screening methods, and Shapley effects for dependent inputs. The contribution is therefore stated precisely rather than broadly.

Keywords: High-dimensional model representation, Enhanced multivariance product representation, Surrogate modelling, Sobol' indices, Global sensitivity analysis, Curse of dimensionality, Combustion

Pages: 696-704