[Paper Review] Atomic Permutationally Invariant Polynomials for Fitting Molecular Force Fields

Alice E. A. Allen, Gábor Cśanyi+2 more|arXiv (Cornell University)|Oct 23, 2020

Machine Learning in Materials ScienceMaterials Science77 references52 citations

TL;DR

The paper extends atomic permutationally invariant polynomials (aPIP) to molecular systems, combining low-body-order empirical-like terms with data-driven fits to achieve transferable, accurate force fields for small organic molecules.

ABSTRACT

We introduce and explore an approach for constructing force fields for small molecules, which combines intuitive low body order empirical force field terms with the concepts of data driven statistical fits of recent machine learned potentials. We bring these two key ideas together to bridge the gap between established empirical force fields that have a high degree of transferability on the one hand, and the machine learned potentials that are systematically improvable and can converge to very high accuracy, on the other. Our framework extends the atomic Permutationally Invariant Polynomials (aPIP) developed for elemental materials in [Mach. Learn.: Sci. Technol. 2019 1 015004] to molecular systems. The body order decomposition allows us to keep the dimensionality of each term low, while the use of an iterative fitting scheme as well as regularisation procedures improve the extrapolation outside the training set. We investigate aPIP force fields with up to generalised 4-body terms, and examine the performance on a set of small organic molecules. We achieve a high level of accuracy when fitting individual molecules, comparable to those of the many-body machine learned force fields. Fitted to a combined training set of short linear alkanes, the accuracy of the aPIP force field still significantly exceeds what can be expected from classical empirical force fields, while retaining reasonable transferability to both configurations far from the training set and to new molecules.

Motivation & Objective

• Bridge the gap between traditional transferable empirical force fields and data-driven machine-learned potentials by developing aPIP-based molecular force fields.
• Keep term dimensionality low via body-order decomposition while enabling systematic improvement up to 4-body interactions.
• Ensure smooth PES extrapolation and avoid overfitting through regularisation and iterative data fitting.

Proposed method

• Decompose the total energy into body-ordered terms up to four-body contributions with element-specific components.
• Transform Cartesian coordinates to rotation-invariant distance and angle coordinates and construct permutation-invariant polynomials via primary and secondary invariants.
• Impose distance-based cutoffs to manage computational cost and use a smooth cutoff to avoid holes in the PES.
• Fit the resulting linear combination of basis functions to energies and forces using regularised linear least-squares.
• Apply Laplace regularisation and two-sided cutoffs to promote smoothness and prevent unphysical regions in the PES.
• Use an iterative data-gathering and fitting scheme to improve PES coverage and extrapolation.

Experimental results

Research questions

• RQ1 Can aPIP-based potentials with controlled body order achieve transferability across related molecules while retaining high accuracy?
• RQ2 What is the effect of limiting body order and applying regularisation on the accuracy and extrapolation behavior of molecular force fields?
• RQ3 How do distance-angle coordinates and permutation-invariant polynomials compare to full high-dimensional ML approaches in PES fitting for small molecules?

Key findings

• aPIP-based force fields with up to generalised 4-body terms achieve high accuracy for small organic molecules, comparable to many-body ML force fields on a per-molecule basis.
• Fitted to a combined training set of short linear alkanes, the aPIP accuracy significantly exceeds that of classical empirical force fields while maintaining transferability to configurations outside the training set and to new molecules.
• Regularisation and an iterative fitting scheme improve the smoothness and extrapolation of the PES and help avoid holes.
• A distance-angle invariant representation with symmetry-adapted polynomials enables scalable treatment of multi-element systems without explicit discrete atom types.