Machine-learning-based inversion of nuclear responses

A microscopic description of the interaction of atomic nuclei with external electroweak probes is required for elucidating aspects of short-range nuclear dynamics and for the correct interpretation of neutrino oscillation experiments. Nuclear quantum Monte Carlo methods infer the nuclear electroweak response functions from their Laplace transforms. Inverting the Laplace transform is a notoriously ill-posed problem; and Bayesian techniques, such as maximum entropy, are typically used to reconstruct the original response functions in the quasielastic region. In this work, we present a physics-informed artificial neural network architecture suitable for approximating the inverse of the Laplace transform. Utilizing simulated, albeit realistic, electromagnetic response functions, we show that this physics-informed artificial neural network outperforms maximum entropy in both the low-energy transfer and the quasielastic regions, thereby allowing for robust calculations of electron scattering and neutrino scattering on nuclei and inclusive muon capture rates.

Figure 1

Schematic overview of the Phys-NN approach.

Figure 2

Training data examples of response functions exhibiting a single asymmetric QE peak.

Figure 3

Training data examples of response functions characterized by an EL narrow peak in addition to the QE peak.

Figure 4

Box plots of (left) $R^2$, (middle) ${\chi }_E^2$, and (right) $S_R$ for the Phys-NN and MaxEnt methods. The top and bottom rows refer to the one-peak and two-peaks datasets, respectively. The line in the middle of the box denotes the median, and the box represents the range between the 25% and 75% quantiles. Whiskers cover the area between the 1% and 99% quantiles; data beyond these whiskers are outliers and are indicated by circles.

Figure 5

Correlation plots of $\overline{{\chi }_E^2}$ versus $\overline{S_R}$ as obtained with the Phys-NN (top row) and MaxEnt (bottom row) methods. The left and right columns refer to the one-peak dataset and the two-peak dataset, respectively. The reference lines indicate the median $\overline{{\chi }_E^2}$ and $\overline{S_R}$ values.

Figure 6

Comparison between the Phys-NN and MaxEnt reconstructions for the one-peak dataset. The top row displays the response functions and the bottom row the corresponding Euclidean responses.

Figure 7

Same as Fig. 6 for the two-peak dataset.

Figure 8

Energy-dependent entropy for the Phys-NN and MaxEnt results for the one-peak (left panel) and two-peak (right panel) datasets.

Figure 9

Phys-NN and MaxEnt reconstruction performance with increasing level of noise in the input Euclidean responses for one-peak (top row) and two-peak (bottom row) datasets.

Figure 10

Change in the entropy from increasing the standard deviation of the Gaussian noise in the input Euclidean responses from $\sigma ={10}^{-4}$ to $\sigma ={10}^{-3}$.