Message Passing Neural Networks (MPNN)

• MPNNs are graph-based neural networks that process structured data by iteratively exchanging messages between nodes.
• They employ distinct message, update, and readout functions to effectively capture and aggregate local and global graph features.
• MPNN models are widely applied in molecular chemistry, social network analysis, and recommendation systems for scalable and accurate predictions.

The Atomic Cluster Expansion (ACE) is a systematically improvable, mathematically complete framework used to represent many-body interactions in atomistic systems with explicit enforcement of physical symmetries, notably permutation, rotation, and translation invariance. Since its formalization by Drautz (2019), ACE has been widely adopted for developing high-accuracy machine-learned interatomic potentials, providing a basis that recovers standard linear expansions in electronic structure theory and underpins active learning, uncertainty quantification, and algorithmic innovations across computational chemistry and materials science (Ortner, 2023).

1. Mathematical Structure and Symmetry Principles

ACE expresses atomic-scale properties, particularly the total energy, as a sum of local site energies, each expanded over a complete set of many-body invariant basis functions. Formally, for an $N$-atom structure,

$$E_{\rm tot} = \sum_{i=1}^N E_i,\qquad E_i = \sum_{\nu=0}^{\nu_{\max}}\sum_{u} c_u^{(\nu)}\,B_u^{(\nu)}(i),$$

where $B_u^{(\nu)}(i)$ is a $\nu$-body, permutation- and rotation-invariant basis function built from the relative positions and species of neighbors of atom $i$ within a cutoff. These invariants are constructed hierarchically:

• First, one-particle features $$A_{n\ell m}^{(i)} = \sum_{j\ne i} R_{n\ell}(r_{ij}) Y_\ell^m(\hat r_{ij})$$ capture the projection of the local environment onto an orthonormal radial basis and the spherical harmonics.
• Tensor products of these features up to order $\nu$ represent explicit $\nu$-body correlations.
• Rotational invariance is obtained by contracting over the $m_t$ indices with generalized Clebsch–Gordan coefficients, enforcing total angular momentum zero.

Key symmetries are built-in:

• Translation invariance: Basis functions depend only on relative atomic positions.
• Permutation invariance: Achieved via symmetrization over neighbor indices in basis construction (Goff et al., 2022).
• Rotational invariance: Imposed through angular momentum coupling.

2. Basis Construction, Truncation, and Independence

The ACE basis is formed as follows:

• One chooses a set of orthonormal radial functions $R_n(r)$, e.g., Chebyshev or spherical Bessel, defined on $[0, r_{\rm cut}]$.
• Angular channels are parsed via $Y_\ell^m$.
• Descriptor functions are symmetrized tensor products of these, forming body-ordered invariants $B_{u}^{(\nu)}$.
• High-rank invariants are subject to rotation and permutation symmetry enforcement; recent developments (e.g., PA-RPI, permutation-adapted rotation and permutation-invariant bases) give analytic blockwise independent bases, sidestepping the need for expensive singular value decompositions. Analytical ladder recursions based on generalized Wigner symbols are used to identify and select linearly independent invariants within each block of quantum-number labels (Goff et al., 2022).

Truncation is applied by specifying maximum body order $\nu_{\max}$, maximum radial $n_{\max}$ and angular $\ell_{\max}$ indices, and polynomial degree, allowing systematic improvement of accuracy by extending the basis. Completeness and independence are rigorously established within these truncation settings.

3. Regression, Regularization, and Self-Interaction

ACE coefficients $c_{u}^{(\nu)}$ are fitted by minimizing regularized least squares over reference quantum mechanical data: $$L(\{c_u\}) = \sum_{k} \left| E^{\rm ACE}(X_k) - E^{\rm ref}(X_k) \right|^2 + w_F \sum_{i} \|\mathbf F_i^{\rm ACE}(X_k) - \mathbf F_i^{\rm ref}(X_k)\|^2 + \lambda \sum_u |c_u|^2,$$ where $w_F$ weights force fitting and $\lambda$ enforces smoothness and controls overfitting.

Canonical ("non-self-interacting") ACE, which omits terms where the same atom enters multiple times in a body-ordered invariant, has demonstrably improved numerical conditioning and more robust regression properties than the original "self-interacting" ACE, especially for high-rank expansions. Orthogonalization and purification of the basis can be achieved via sparse transformation operators exploiting the linear algebra of the constituent one-particle basis (Ho et al., 3 Jan 2024).

4. Extensions: Multipoles, Charges, Magnetism, and Applications

ACE’s formalism extends naturally to scalar, vector, and tensor properties and additional degrees of freedom:

• Flexible Charge and Electronegativity Models: ACE can serve as a foundation for site-dependent charge models by expanding electronegativity or other scalar parameters in the same ACE basis as the short-range energy (Goff et al., 2023, Rinaldi et al., 6 Nov 2024). Variational or self-consistent charge equilibration is supported by embedding a charge-dependent quadratic energy (QEq) coupled to local descriptors, and including long-range electrostatic interactions when required. This yields transferability and avoids redundant information in charge-constrained DFT.
• Spin-Lattice and Magnetic Systems: Generalizing the ACE framework to non-collinear magnetism involves treating atomic moments as vector degrees of freedom in the neighborhood density and introducing additional basis functions for the moment magnitude and orientation. Coupling via Clebsch–Gordan coefficients yields invariants corresponding to standard interactions (e.g., Heisenberg exchange, biquadratic couplings) (Rinaldi et al., 2023).
• Electronic Structure Representations: ACE generalizes to both symmetric (bosonic) and anti-symmetric (fermionic) wavefunction ansätze, recovering Slater determinants, configuration interaction expansions, and backflow forms through appropriate contraction and symmetrization of multi-particle basis tensors (Drautz et al., 2022, Zhou et al., 2023).

5. Practical Workflow: Fitting, Active Learning, and Uncertainty

Typical ACE model development involves:

• DFT or quantum chemical reference data collection over a diversity of configurations, species, and environments.
• Construction of the invariant basis set with chosen truncation, symmetry adaptation, and possible pruning of numerically redundant or insignificant basis functions.
• Regularized regression to fit the ACE coefficients to reference energies and forces.
• Active learning using the D-optimality criterion (MaxVol algorithm) to identify gaps in the training set. The extrapolation grade $\gamma$ provides a scalable, efficient means for uncertainty quantification and robust selection of configurations for new DFT labeling (Lysogorskiy et al., 2022, Liang et al., 2023).
• Automated iteration between MD, active structure sampling, and ACE model refitting, ensuring coverage of the relevant configuration space, crucial for reliable transferability and robust prediction, especially for catalysis, phase transitions, defects, or out-of-equilibrium phenomena.

6. Model Performance, Transferability, and Applications

ACE’s completeness and invariance properties yield broad applicability and DFT-level accuracy across a spectrum of materials and molecular systems:

• For elemental and alloy systems, such as Mg, Pt–Rh, Fe–Co, and multielement structured solids, ACE models have achieved energy RMSEs on test sets <10-20 meV/atom and force RMSEs ~50 meV/Å over diverse training/validation data (Ibrahim et al., 2023, Li et al., 2023, Liang et al., 2023).
• Phase transitions, nucleation, and defect phenomena can be quantitatively captured, with melting/freezing points, elastic constants, and activation barriers matching DFT values to within a few percent (Ibrahim et al., 2023, Qamar et al., 2022, Rinaldi et al., 2023).
• Heterogeneous and reducible systems, including flexible-charge insulators and catalytically active oxide surfaces, can utilize charge-constrained or variationally self-consistent ACE+Q models for accurate electrostatics and charge transfer (Rinaldi et al., 6 Nov 2024, Goff et al., 2023).
• Large-scale (sub-million atom) simulations, high-T MD, and non-equilibrium phenomena are enabled by linear ACE’s computational efficiency (Araki et al., 2023, Yang et al., 2023).
• Quantum chemistry applications and variational wavefunction optimization have demonstrated superior scaling of ACE-based polynomial-basis representations, both for bosonic and fermionic systems (Zhou et al., 2023, Drautz et al., 2022).

Model transferability critically depends on the diversity and coverage of the training database, judicious selection of body order, and proper uncertainty estimation. The basis can, in principle, be systematically extended to arbitrary accuracy, though in practice, balancing computational tractability and transferability is required (Goff et al., 2023, Lysogorskiy et al., 2022).

7. Algorithmic and Conceptual Innovations

Recent advances anchored in ACE include:

• Analytical identification and selection of linearly independent permutation- and rotation-invariant descriptor sets via PA-RPI techniques, avoiding numerical SVD (Goff et al., 2022).
• Orthogonal, non-self-interacting basis projections for improved regression properties and physical fidelity (Ho et al., 3 Jan 2024).
• Variational inclusion of charge (or moment) degrees of freedom, retaining adiabatic separation and avoiding redundant DFT charge-constrained calculations (Rinaldi et al., 6 Nov 2024).
• Bayesian compressive sensing for sparse selection of only the most numerically and physically relevant descriptors, often yielding highly compact, interpretable models without loss of accuracy (Goff et al., 2022).
• Extension to active learning and uncertainty quantification within ab initio MD and materials discovery workflows (Lysogorskiy et al., 2022, Liang et al., 2023).

ACE thus defines a universal, modular framework for high-precision atomistic potential construction, capturing the full hierarchy of many-body, multi-property interactions with rigorous enforcement of physical symmetries and mathematical completeness (Ortner, 2023).