Comparison of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water interactions through many-body expansions

• Accurate molecular models that reproduce water from the gas to condensed phase remain difficult to develop.Source: §I · p. 3
• Pairwise-additive models often fail to reproduce both gas- and condensed-phase water properties because many-body effects are important.Source: §I · pp. 3–4
• 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.Source: §I · p. 4
• MB-pol has reproduced water properties from the dimer and small clusters through liquid water, ice, and interfacial systems.Source: §I · p. 5
• The study compares PIP, Behler-Parrinello neural-network, and Gaussian-approximation representations for MB-pol short-range two-body and three-body terms.Source: §4.5 · pp. 5–6
• The BPNN- and GAP-based MB-pol expressions achieve essentially the same accuracy as the original PIP-based expression.Source: §4.5 · p. 6

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.Source: §II · p. 6
• 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.Source: §II · p. 6
• 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.Source: §II · p. 6

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.Source: §II · p. 7

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.Source: §II · p. 7

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.Source: A. Training sets and reference energies · p. 8

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.Source: A. Training sets and reference energies · p. 8

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.Source: A. Training sets and reference energies · p. 8

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.Source: A. Training sets and reference energies · p. 8

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.Source: B. Water cluster test sets · p. 9

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).Source: B. Water cluster test sets · p. 9
• 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.Source: B. Water cluster test sets · p. 9

• PIPs use intermolecular and intramolecular distance variables transformed into exponential or Coulomb-type basis functions.Source: A. Permutationally invariant polynomials · p. 9
• Symmetrized monomials make the polynomial representation invariant to permutations of equivalent atoms and water molecules.Source: A. Permutationally invariant polynomials · p. 9
• The 2B PIP contains 31 basis functions and 1153 symmetrized monomials, while the 3B PIP contains 1163 symmetrized monomials.Source: A. Permutationally invariant polynomials · pp. 9–10
• Linear and nonlinear parameters are optimized by singular value decomposition and simplex minimization of a regularized squared-error objective.Source: A. Permutationally invariant polynomials · p. 10
• 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.Source: A. Permutationally invariant polynomials · p. 10

• A BPNN represents total energy as a sum of atomic-network outputs, multiplied by MB-pol switching functions for short-range interactions.Source: B. Behler-Parrinello neural networks · pp. 10–11
• Each atomic network receives atom-centered symmetry functions encoding local atomic positions and preserves rotational, translational, and permutation invariance.Source: B. Behler-Parrinello neural networks · p. 10
• Water systems use two simultaneously trained subnetworks, one shared by all hydrogen atoms and one shared by all oxygen atoms.Source: B. Behler-Parrinello neural networks · pp. 10–11
• The 2B and 3B models contain 10542 and 4798 weight-and-bias parameters, respectively.Source: B. Behler-Parrinello neural networks · p. 12
• Training minimizes mean squared error, retains the validation-best model to reduce overfitting, and evaluates generalization on an unseen test set.Source: B. Behler-Parrinello neural networks · pp. 12–13

• GAP is implemented as Gaussian-process regression, representing atomic energy as a function of neighboring-atom geometry.Source: C. Gaussian Approximation Potentials · p. 13
• SOAP describes each atomic environment through species-specific neighbor densities formed from Gaussians centered on neighboring atoms.Source: C. Gaussian Approximation Potentials · p. 13
• The neighbor densities are expanded in spherical harmonics and radial functions, whose coefficients form rotationally invariant power spectra.Source: C. Gaussian Approximation Potentials · p. 14
• The GAP kernel compares complete neighbor geometries, is normalized self-similarity, and uses exponent ζ = 2.Source: C. Gaussian Approximation Potentials · p. 14
• Separate atomic-energy fits use representative environments selected by CUR decomposition, with 9000 points for 2B and 10000 for 3B.Source: C. Gaussian Approximation Potentials · p. 14

• All three methods achieve similar accuracy for 2B and 3B interaction-energy fitting across the reported datasets.Source: A. 2B and 3B interactions, and the… · pp. 15, 17, 18

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.Source: A. 2B and 3B interactions, and the… · p. 17
• 2B errors are largest where interaction energies are high and training-point density is low.Source: A. 2B and 3B interactions, and the… · p. 17

Training-data structure

• Farthest-point sampling reduces BPNN test RMSE by up to a factor of five compared with random subset selection.Source: A. 2B and 3B interactions, and the… · p. 17

3B interactions

• For 3B interactions, PIP and GAP achieve RMSE ≈0.05 kcal/mol, followed by BPNN at RMSE ≈0.06 kcal/mol.Source: A. 2B and 3B interactions, and the… · p. 18

Training-data structure

• Improving training-set sampling density and uniformity is identified as the most effective strategy for further error reduction.Source: A. 2B and 3B interactions, and the… · p. 19

• 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.Source: B. Water clusters · p. 20
• 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.Source: B. Water clusters · p. 20
• 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.Source: B. Water clusters · pp. 20–21
• 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.Source: B. Water clusters · p. 21

• PIP, BPNN, and GAP representations were evaluated for MB-pol short-range two-body and three-body interaction energies.Source: §V · p. 21
• The three models were assessed against large CCSD(T)/CBS datasets and small water-cluster energetics.Source: §V · p. 21
• 1 kcal/mol: small water-cluster energetics were consistently within chemical accuracy for all three models.Source: §V · p. 21
• The three models were effectively equivalent, consistently showing similar performance for many-body interactions in water within MB-pol.Source: §V · p. 21
• Further accuracy improvements require increasing reference calculations and optimizing training sets to cover relevant configuration space more uniformly.Source: §V · p. 22
• 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.Source: §V · p. 22