Quantifying uncertainty in machine learning on nuclear binding energy

TL;DR

This work addresses the challenge of quantifying epistemic uncertainty in neural-network predictions of nuclear binding energy per nucleon, $E/A$. It introduces Δ-UQ, a single-model uncertainty quantification method that uses anchored inputs to yield a mean prediction and an uncertainty band with one training run, avoiding the cost of ensemble methods. Applying Δ-UQ to a two-feature model trained on DFT and AME2020 data demonstrates reliable, self-consistent uncertainty estimates that widen appropriately when extrapolating away from the training region, and it outperforms ensembles in signaling extrapolation reliability while dramatically reducing computational cost. The approach is flexible, scalable to larger networks, and suitable for guiding trust in ML-driven nuclear theory predictions.

Abstract

Techniques from artificial intelligence and machine learning are increasingly employed in nuclear theory, however, the uncertainties that arise from the complex parameter manifold encoded by the neural networks are often overlooked. Epistemic uncertainties arising from training the same network multiple times for an ensemble of initial weight sets offer a first insight into the confidence of machine learning predictions, but they often come with a high computational cost. Instead, we apply a single-model uncertainty quantification method called Δ-UQ that gives epistemic uncertainties with one-time training. We demonstrate our approach on a 2-feature model of nuclear binding energies per nucleon with proton and neutron number pairs as inputs. We show that Δ-UQ can produce reliable and self-consistent epistemic uncertainty estimates and can be used to assess the degree of confidence in predictions made with deep neural networks.

Figures (10)

• Figure 1: $\Delta$-UQ mapping: each initial input vector $\boldsymbol{X}_i$ is shifted by the anchor $\boldsymbol{C}_j$. The new input vector is formed by aggregating the shifted vector with the anchor, resulting in input dimensionality $2n_{d}$ and a dataset of size $n \times m$.
• Figure 2: The difference between DFT and AME $E/A$ data. The inset shows the average difference calculated by averaging across each isobar.
• Figure 3: Illustration of regions used for training, validation and testing. AME dataset is split into training set and validation set (blue dots) the same way as DFT dataset in region I. DFT dataset in region II forms the testing set of DFT data. The dashed gray lines indicate $A=100, 200, 300, 400$ isobars, whose results will be presented later. The magenta dots indicate the farthest integer points within 5 nuclei away from the boundary nuclei of region I, with $A\geq18$.
• Figure 4: Training on DFT or AME data: training and validation MSE loss as a function of the number of epochs. The blue (red) stars are the validation loss (training loss) values for selected epoch number with similar validation loss, which will be discussed later.
• Figure 5: Predictions of the binding energy per nucleon $E/A$ for $A=200$ isobars for the selected epochs listed in Table \ref{['table_selected_losses']}. Crosses represent the mean value and the band indicates three standard deviation estimated with $\Delta$-UQ.