Accurate Surrogate Amplitudes with Calibrated Uncertainties

TL;DR

This work develops calibrated neural surrogates for LHC loop amplitudes by predicting both the amplitude $A(x)$ and its uncertainties. It systematically compares heteroscedastic losses, Bayesian neural networks, and repulsive ensembles, and evaluates activation-function strategies using Kolmogorov-Arnold Networks (KANs). The results demonstrate that symmetry-aware architectures (DSI, L-GATr) and learned uncertainties yield precise, well-calibrated predictions with systematic uncertainties reaching the $10^{-2}$ to $10^{-5}$ level, while REs show excellent central values but require calibration for reliable uncertainty estimates. The methods enable fast, reliable higher-order predictions for event generation, with practical guidance on architecture choices, uncertainty modeling, and calibration procedures.

Abstract

Neural networks for LHC physics have to be accurate, reliable, and controlled. Using neural surrogates for the prediction of loop amplitudes as a use case, we first show how activation functions are systematically tested with Kolmogorov-Arnold Networks. Then, we train neural surrogates to simultaneously predict the target amplitude and an uncertainty for the prediction. We disentangle systematic uncertainties, learned by a well-defined likelihood loss, from statistical uncertainties, which require the introduction of Bayesian neural networks or repulsive ensembles. We test the coverage of the learned uncertainties using pull distributions to quantify the calibration of cutting-edge neural surrogates.

Figures (15)

• Figure 1: Accuracy on a logarithmic scale for the MLP, DSI, and KAN networks.
• Figure 2: Learned activation functions for the GroupKAN-1 MLP and DSI networks.
• Figure 3: Comparing GELU and ReLU as possible activation functions for a deterministic network (Det).
• Figure 4: Relative systematic and statistical uncertainties learned by the BNN as a function of the dataset size, for 5%, 2%, and zero artificial noise on the training data.
• Figure 5: Left: relative uncertainty versus artificial noise for different network architectures. The 'noise only' curve shows the scaling of the systematics, assuming the added noise is the only source of uncertainty. The exact numbers are given in Tab. \ref{['tab:noise']}. Right: extracted noise, defined in Eq.\ref{['eq:sig_noise']}, as a function of the input noise for BNN, REs, and a heteroscedastic deterministic network.