An Efficient Learning Method to Connect Observables

Abstract

Constructing fast and accurate surrogate models is a key ingredient for making robust predictions in many topics. We introduce a new model, the Multiparameter Eigenvalue Problem (MEP) emulator. The new method connects emulators and can make predictions directly from observables to observables. We present that the MEP emulator can be trained with data from Eigenvector Continuation (EC) and Parametric Matrix Model (PMM) emulators. A simple simulation on a one-dimensional lattice confirms the performance of the MEP emulator. Using $^{28}$O as an example, we also demonstrate that the predictive probability distribution of the target observables can be easily obtained through the new emulator.

Figures (4)

• Figure 1: Workflow of our method compared with existing statistical procedures
• Figure 2: The eigenvalue of the three-body system $E_{3}$ as a function of that of the two-body system $E_{2}$. The energies are computed with a simple Hamiltonian in the one-dimensional lattice, see text for the details. The training points for the EC emulator are represented by orange squares; blue dots represent validation from the original problem Eq. \ref{['eq:Main']}. In panel (a), the energies are computed by solving Eq. \ref{['eq:GEM']}. The gray dots are obtained from the spectra of the original problem. Panel (b) shows the energies computed from the MEP emulator constructed with the ground-state eigenvectors. The solid (dashed) curve corresponds to ground-state $E_{2}$ and ground-state $E_{3}$ (excited-state $E_{2}$ and ground-state $E_{3}$). The triangles emphasize the corresponding $V_{0}$. The ground-state $E_{2}$ and $E_{3}$ as a function of $V_{0}$ are show in the top inset. The gray curve in the zoomed-in plot on the bottom is from inverting EC emulators according to Eq. \ref{['eq:inversion']}.
• Figure 3: Joint probability density distributions of [28]O energy differences with respect to [27]O and [24]O. The red shade and red dashed lines are our emulator prediction using $10^5$ samples from averaging 10 bootstrapping batches with input constructed directly from $A = 2-4$ NI measure Jiang:2022oba. The small orange dots are the data points used in constructing the emulators. We do not have any input from LECs. We then calibrate our joint density (blue shades and blue solid lines) with additional constraints on ground states of $A = 16, 24$ (see text). Circles are $1\sigma$ confident intervals (CIs). Red error bars are experimental values taken from Ref. Kondo:2023lty.
• Figure 4: Performance of emulators. Panels (a) and (b) show the performance for the ground-state energy for [16]O, where the emulators take the 17 LECs and observables as inputs, respectively. Panel (c) is for the MEP emulator targeting the ground-state energy difference between [24]O and [28]O. The training and validation data are given with the orange and blue circles, respectively.