Learning Weak Interactions: Multifidelity Gaussian Process Regression for Potential Energy Surfaces

Sutirtha Paul and Adrian Del Maestro

Abstract

Weak non-covalent potential energy surfaces require smooth interpolation at energy scales at the single-wavenumber level. Using helium-benzene as a benchmark, we compare linear and radial-basis interpolation with standard and multifidelity Gaussian process (GP) regression on CCSD(T)/complete-basis-set data. In separate legacy random-holdout archives, the multifidelity GP has a pooled mean absolute error of 0.780 cm^-1, approximately 42% lower than the standard-GP value of 1.353 cm^-1; because the archives do not share a split manifest, this difference is descriptive. In a separate retrospective one-dimensional benchmark, posterior-variance sampling reaches the selected error target with a lower median high-fidelity budget for the multifidelity model. These results illustrate how dense lower-fidelity information can condition high-fidelity predictions in the studied settings, but they do not establish spatially blocked generalization of the three-dimensional model.

Description

This repository includes code, processed data, and generated figures for a paper on interpolation and active learning of the helium-benzene potential energy surface. It contains:

• linear, radial-basis-function, standard-GP, and multifidelity-GP interpolation comparisons;
• an error-risk analysis that resolves the tail of the prediction-error distribution;
• one-dimensional standard, autoregressive, and multifidelity GP demonstrations;
• a replicated active-learning comparison at a fixed validation-MAE target; and
• manuscript figures and PNG previews.

The underlying CCSD(T)/CBS and PBE0-D4 electronic-structure calculations are described in the companion work Accurate Helium-Benzene Potential: from CCSD(T) to Gaussian Process Regression and its code and data repository.

data/       Processed reference data and saved prediction archives
fig/        Generated PDF figures and README PNG previews
include/    Shared Matplotlib style
src/        Jupyter notebooks and plotting utilities

Data

File	Contents
data/all_pts_2545.npy	float64 array with shape (2545, 4) and columns [x, y, z, V_H]. Coordinates are in angstrom and the CCSD(T)/CBS interaction energy is in cm-1. The benchmark retains the 2,521 rows with V_H < 1000 cm^-1.
data/STGPCC.npy, data/STGPGP.npy	Saved standard-GP reference and prediction energies, each with shape (4545,), in cm-1.
data/GPCC.npy, data/GPGP.npy	Saved multifidelity-GP reference and prediction energies, each with shape (4543,), in cm-1.
data/active_learning_cut_reference.npz	Versioned one-dimensional reference table containing 501 acquisition candidates, 2,000 disjoint validation points, 100 fixed lower-fidelity points, budgets from 3 through 35, and static equal-spacing designs. Coordinates are in angstrom and energies are in cm-1.

Requirements

The figure-generation notebooks use Python, NumPy, SciPy, pandas, scikit-learn, Matplotlib, Pillow, IPython/Jupyter, PyTorch, GPyTorch, and BoTorch. The plotting style in include/aps.mplstyle currently uses a system LaTeX installation with amsmath, siunitx, and svrsymbols.

The current environment can be created with:

conda env create -f environment.yml
conda activate jmr-env

Some current figure notebooks use the external HeBz package:

python -m pip install --no-deps \
  'git+https://github.com/paulsphys/HeBz.git'

Support

This research was primarily supported by the National Science Foundation Materials Research Science and Engineering Center program through the UT Knoxville Center for Advanced Materials and Manufacturing under Award No. DMR-2309083.

Figures

Figure 01: High-fidelity He-benzene configurations

Generated by src/HighFidelityDataBallAndStick.ipynb. View the vector PDF.

Figure 02: Interpolation parity comparison

Generated by src/SMAEComparisons.ipynb. View the vector PDF.

Figure 03: Error-risk comparison

Generated by src/ErrorRiskCompanion.ipynb. View the vector PDF.

Figure 04: Interpolated potential-energy surface

Generated by src/SurfacePlot.ipynb. View the vector PDF.

Figure 05: Standard, autoregressive, and multifidelity GP comparison

Generated by src/MultifidelityGP.ipynb. View the vector PDF.

Figure 06: Active selection of high-fidelity data

All three panels are generated by src/ActiveLearningComparison.ipynb. View the vector PDFs for the learning curves and representative fits.

The same notebook also generates the companion diagnostics fig/ActiveLearningComparison.pdf and fig/ActiveLearningRepresentativeFits.pdf.

Citation and licensing

The repository software is released under the MIT License.