Source-linked AI summary
Comparison of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water interactions through many-body expansions
Thuong T. Nguyen, Eszter Székely, Giulio Imbalzano, Jörg Behler, Gábor Csányi, Michele Ceriotti, Andreas W. Götz, Francesco Paesani
TL;DR
Accurate water simulations require potential-energy surfaces that represent many-body interactions reliably. The paper compares PIP, neural-network, and Gaussian-approximation representations within MB-pol for short-range two-body and three-body terms. All three representations show similar accuracy on reference interactions and small water-cluster energies, supporting their combined use with a physically grounded many-body framework.
Problem
Accurate representation of multidimensional water potential-energy surfaces remains necessary for realistic simulations, while pairwise models do not reproduce both gas- and condensed-phase properties.
Method
The study compares PIP, BPNN, and GAP representations of MB-pol short-range two-body and three-body interactions.
Results
The three analytical representations achieve similar accuracy for two-body and three-body reference data and small water-cluster energetics within chemical accuracy.
Takeaways & Limitations
Combining a physically sound many-body framework with flexible machine-learning representations provides a promising route to accurate, efficient, and transferable potential-energy surfaces.
Abstract
from arXiv · showhide
The accurate representation of multidimensional potential energy surfaces is a necessary requirement for realistic computer simulations of molecular systems. The continued increase in computer power accompanied by advances in correlated electronic structure methods nowadays enable routine calculations of accurate interaction energies for small systems, which can then be used as references for the development of analytical potential energy functions (PEFs) rigorously derived from many-body expansions. Building on the accuracy of the MB-pol many-body PEF, we investigate here the performance of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water two-body and three-body interaction energies, denoting the resulting potentials PIP-MB-pol, BPNN-MB-pol, and GAP-MB-pol, respectively. Our analysis shows that all three analytical representations exhibit similar levels of accuracy in reproducing both two-body and three-body reference data as well as interaction energies of small water clusters obtained from calculations carried out at the coupled cluster level of theory, the current gold standard for chemical accuracy. These results demonstrate the synergy between interatomic potentials formulated in terms of a many-body expansion, such as MB-pol, that are physically sound and transferable, and machine-learning techniques that provide a flexible framework to approximate the short-range interaction energy terms.
I. INTRODUCTION
Accurate water simulations require models that capture many-body interactions across phases. This study combines the MB-pol many-body framework with alternative machine-learning representations of short-range interactions.
- Accurate molecular models that reproduce water from the gas to condensed phase remain difficult to develop.
- Pairwise-additive models often fail to reproduce both gas- and condensed-phase water properties because many-body effects are important.
- The many-body expansion provides a rigorous framework in which low-order interaction terms can be fitted to CCSD(T)/CBS data and higher-order effects represented by classical induction.
- MB-pol has reproduced water properties from the dimer and small clusters through liquid water, ice, and interfacial systems.
- The study compares PIP, Behler-Parrinello neural-network, and Gaussian-approximation representations for MB-pol short-range two-body and three-body terms.
- The BPNN- and GAP-based MB-pol expressions achieve essentially the same accuracy as the original PIP-based expression.
II. MB-POL FUNCTIONAL FORM AND COMPUTATIONAL DETAILS
MB-pol combines a many-body expansion with classical long-range terms and flexible short-range representations for water 2B and 3B interactions. This work compares PIPs, BPNNs, and GAPs as alternative machine-learning representations of those short-range terms.
- MB-pol functional form: MB-pol explicitly combines 1B, 2B, and 3B terms with classical N-body polarization.The framework uses a many-body expansion and accounts for higher-body interaction contributions through polarization.
- MB-pol functional form: The 2B term separates classical long-range electrostatics, induction, and dispersion from short-range quantum-mechanical overlap effects.The long-range components use modified TTM4-F electrostatics and induction plus a damped C6 dispersion term.
- MB-pol functional form: The 3B term combines long-range classical induction with a short-range expression for complex interactions.Classical 3B induction captures essentially all of the 3B interaction energy at long range.
- Switching and cutoffs: Switching functions smoothly turn off short-range corrections beyond preset oxygen–oxygen separation cutoffs, producing a smooth differentiable potential-energy surface.The switching construction is applied to the short-range 2B and 3B terms.
- Machine-learning representations: The study compares PIPs with Behler–Parrinello neural networks and Gaussian approximation potentials for representing water 2B and 3B short-range interactions.The original PIP-MB-pol representation is evaluated alongside BPNN-MB-pol and GAP-MB-pol alternatives.
A. Training sets and reference energies
The study trains and evaluates short-range interaction models on CCSD(T)-quality water dimer and trimer data sampled across relevant geometries and condensed-phase configurations. The final datasets contain 42,069 2B and 12,347 3B configurations, with isolated test sets for evaluation.
- Training-set composition: The datasets sample water dimers and trimers spanning minima, saddle points, compressed geometries, and configurations extracted from MD or PIMD simulations.The sampling includes liquid-water and small-cluster environments over relevant temperature and pressure conditions.
- Reference energies: The 2B and 3B reference energies were obtained at the complete-basis-set limit of CCSD(T).Training energies for the short-range models were formed by subtracting the MB-pol long-range baseline from the quantum-mechanical reference energies.
- Dataset filtering: 42,069 configurations remained in the 2B training set after removing binding energies at or above 60 kcal/mol and separations beyond the 6.5 Å cutoff.The trimer dataset was fully retained with 12,347 configurations.
- Dataset splitting: The final dataset contains 12,347 fully retained trimer configurations alongside the filtered dimer set.Both datasets were randomly split into training, validation, and isolated test sets at a ratio of 0.81:0.09:0.1.
B. Water cluster test sets
Water cluster test energies for n = 4−6 were assembled with a many-body expansion using CCSD(T)-quality 2B and 3B terms and explicitly correlated higher-body contributions. The resulting reference construction combines consistent low-order data with CCSD(T)-F12b estimates above 3B.
- Cluster test sets: Reference interaction energies for (H2O)n clusters with n = 4−6 used MP2 and RI-MP2 optimized geometries.The cluster geometries were taken from a prior reference study.
- Many-body reference construction: The cluster interaction energies were computed with a many-body expansion using 2B and 3B energies at the same level as the MB-pol training sets.These 2B and 3B terms were effectively evaluated at the CBS limit of CCSD(T).
- Many-body reference construction: Higher-order contributions above 3B were obtained from explicitly correlated CCSD(T)-F12b calculations with the VTZ-F12 basis set.The selected basis yields results close to the complete-basis-set limit.
III. MANY-BODY MODELS
The many-body models represent water interactions with permutationally invariant polynomial expansions built from distance-based functions and symmetrized monomials.
- PIPs use intermolecular and intramolecular distance variables transformed into exponential or Coulomb-type basis functions.
- Symmetrized monomials make the polynomial representation invariant to permutations of equivalent atoms and water molecules.
- The 2B PIP contains 31 basis functions and 1153 symmetrized monomials, while the 3B PIP contains 1163 symmetrized monomials.
- Linear and nonlinear parameters are optimized by singular value decomposition and simplex minimization of a regularized squared-error objective.
- Regularization parameters are set to 5×10^-4 for 2B and 1×10^-4 for 3B to reduce variation in linear parameters without spoiling accuracy.
B. Behler-Parrinello neural networks
The Behler–Parrinello neural-network models represent short-range 2B and 3B energies through atom-centered symmetry functions and shared atomic subnetworks.
- A BPNN represents total energy as a sum of atomic-network outputs, multiplied by MB-pol switching functions for short-range interactions.
- Each atomic network receives atom-centered symmetry functions encoding local atomic positions and preserves rotational, translational, and permutation invariance.
- Water systems use two simultaneously trained subnetworks, one shared by all hydrogen atoms and one shared by all oxygen atoms.
- The 2B and 3B models contain 10542 and 4798 weight-and-bias parameters, respectively.
- Training minimizes mean squared error, retains the validation-best model to reduce overfitting, and evaluates generalization on an unseen test set.
C. Gaussian Approximation Potentials
The Gaussian Approximation Potential models use Gaussian-process regression with SOAP descriptors to interpolate atomic energies from local neighbor geometries.
- GAP is implemented as Gaussian-process regression, representing atomic energy as a function of neighboring-atom geometry.
- SOAP describes each atomic environment through species-specific neighbor densities formed from Gaussians centered on neighboring atoms.
- The neighbor densities are expanded in spherical harmonics and radial functions, whose coefficients form rotationally invariant power spectra.
- The GAP kernel compares complete neighbor geometries, is normalized self-similarity, and uses exponent ζ = 2.
- Separate atomic-energy fits use representative environments selected by CUR decomposition, with 9000 points for 2B and 10000 for 3B.
A. 2B and 3B interactions, and the structure of the training data
The study evaluates PIP, BPNN, and GAP representations of water 2B and 3B interactions using structured training data and regional error analysis. All three methods achieve similar accuracy, while sampling density strongly affects errors and farthest-point selection improves 2B fitting efficiency.
- All three methods achieve similar accuracy for 2B and 3B interaction-energy fitting across the reported datasets.
- 2B interactions: For 2B interactions, errors are below 0.01 kcal/mol for far-away molecules but reach 1 kcal/mol in the repulsive region.
- 2B interactions: 2B errors are largest where interaction energies are high and training-point density is low.
- Training-data structure: Farthest-point sampling reduces BPNN test RMSE by up to a factor of five compared with random subset selection.
- 3B interactions: For 3B interactions, PIP and GAP achieve RMSE ≈0.05 kcal/mol, followed by BPNN at RMSE ≈0.06 kcal/mol.
- Training-data structure: Improving training-set sampling density and uniformity is identified as the most effective strategy for further error reduction.
B. Water clusters
For water clusters containing 4–6 molecules, MB-pol with PIP, BPNN, or GAP short-range representations reproduces coupled-cluster interaction energies accurately. Errors remain below 0.3 kcal/mol for 2B and 3B terms, while total-energy errors increase with cluster size but never exceed 0.8 kcal/mol.
- Total interaction-energy errors increase with cluster size as individual 2B and 3B errors accumulate, especially for 2B contributions.This accumulation is associated with configurations containing repeating dimer and trimer units.
- 2B errors are similar across PIP-MB-pol, BPNN-MB-pol, and GAP-MB-pol, whereas GAP-MB-pol has smaller 3B errors.The difference is most apparent in the three-body interaction energies.
- Ring-type isomers show particularly large errors because extended hydrogen bonding and symmetry produce non-negligible higher-body contributions.These higher-body contributions can have errors comparable to the 2B and 3B terms.
- Despite these larger deviations, total interaction-energy errors never exceed 0.8 kcal/mol and the relative ordering of isomers is retained.The authors conclude that all three approaches are suitable for accurate water-cluster interaction energies within MB-pol.
V. CONCLUSIONS
The study compares PIP, BPNN, and GAP representations of MB-pol short-range two- and three-body energies. All three reproduce reference interaction energies and small water-cluster energetics with similar accuracy, supporting machine learning within a physically grounded many-body framework.
- PIP, BPNN, and GAP representations were evaluated for MB-pol short-range two-body and three-body interaction energies.
- The three models were assessed against large CCSD(T)/CBS datasets and small water-cluster energetics.
- 1 kcal/mol: small water-cluster energetics were consistently within chemical accuracy for all three models.
- The three models were effectively equivalent, consistently showing similar performance for many-body interactions in water within MB-pol.
- Further accuracy improvements require increasing reference calculations and optimizing training sets to cover relevant configuration space more uniformly.
- Combining accurate machine-learning short-range terms with physically sound long-range contributions offers a promising route to accurate, efficient, and transferable potential energy surfaces.