Unified theory of atom-centered representations and message-passing machine-learning schemes.

Data-driven schemes that associate molecular and crystal structures with their microscopic properties share the need for a concise, effective description of the arrangement of their atomic constituents. Many types of models rely on descriptions of atom-centered environments, which are associated with an atomic property or with an atomic contribution to an extensive macroscopic quantity. Frameworks in this class can be understood in terms of atom-centered density correlations (ACDC), which are used as a basis for a body-ordered, symmetry-adapted expansion of the targets. Several other schemes that gather information on the relationship between neighboring atoms using "message-passing" ideas cannot be directly mapped to correlations centered around a single atom. We generalize the ACDC framework to include multi-centered information, generating representations that provide a complete linear basis to regress symmetric functions of atomic coordinates, and provide a coherent foundation to systematize our understanding of both atom-centered and message-passing and invariant and equivariant machine-learning schemes.

Comment on "Manifolds of quasi-constant SOAP and ACSF fingerprints and the resulting failure to machine learn four-body interactions" [J. Chem. Phys. 156, 034302 (2022)].

The "quasi-constant" smooth overlap of atomic position and atom-centered symmetry function fingerprint manifolds recently discovered by Parsaeifard and Goedecker [J. Chem. Phys. 156, 034302 (2022)] are closely related to the degenerate pairs of configurations, which are known shortcomings of all low-body-order atom-density correlation representations of molecular structures. Configurations that are rigorously singular-which we demonstrate can only occur in finite, discrete sets and not as a continuous manifold-determine the complete failure of machine-learning models built on this class of descriptors. The "quasi-constant" manifolds, on the other hand, exhibit low but non-zero sensitivity to atomic displacements. As a consequence, for any such manifold, it is possible to optimize model parameters and the training set to mitigate their impact on learning even though this is often impractical and it is preferable to use descriptors that avoid both exact singularities and the associated numerical instability.

Optimal radial basis for density-based atomic representations.

The input of almost every machine learning algorithm targeting the properties of matter at the atomic scale involves a transformation of the list of Cartesian atomic coordinates into a more symmetric representation. Many of the most popular representations can be seen as an expansion of the symmetrized correlations of the atom density and differ mainly by the choice of basis. Considerable effort has been dedicated to the optimization of the basis set, typically driven by heuristic considerations on the behavior of the regression target. Here, we take a different, unsupervised viewpoint, aiming to determine the basis that encodes in the most compact way possible the structural information that is relevant for the dataset at hand. For each training dataset and number of basis functions, one can build a unique basis that is optimal in this sense and can be computed at no additional cost with respect to the primitive basis by approximating it with splines. We demonstrate that this construction yields representations that are accurate and computationally efficient, particularly when working with representations that correspond to high-body order correlations. We present examples that involve both molecular and condensed-phase machine-learning models.

Incompleteness of Atomic Structure Representations.

Many-body descriptors are widely used to represent atomic environments in the construction of machine-learned interatomic potentials and more broadly for fitting, classification, and embedding tasks on atomic structures. There is a widespread belief in the community that three-body correlations are likely to provide an overcomplete description of the environment of an atom. We produce several counterexamples to this belief, with the consequence that any classifier, regression, or embedding model for atom-centered properties that uses three- (or four)-body features will incorrectly give identical results for different configurations. Writing global properties (such as total energies) as a sum of many atom-centered contributions mitigates the impact of this fundamental deficiency-explaining the success of current "machine-learning" force fields. We anticipate the issues that will arise as the desired accuracy increases, and suggest potential solutions.

Recursive evaluation and iterative contraction of N-body equivariant features.

Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible N-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different order (generalizations of N-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically improvable, symmetry adapted representations for atomistic machine learning.

Local invertibility and sensitivity of atomic structure-feature mappings.

Background: The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the variety of representations that can be used to convert them into a finite set of symmetric descriptors or features. Methods: Here, we analyze the sensitivity of the mapping to atomic displacements, using a singular value decomposition of the Jacobian of the transformation to quantify the sensitivity for different configurations, choice of representations and implementation details.  Results: We show that the combination of symmetry and smoothness leads to mappings that have singular points at which the Jacobian has one or more null singular values (besides those corresponding to infinitesimal translations and rotations). This is in fact desirable, because it enforces physical symmetry constraints on the values predicted by regression models constructed using such representations. However, besides these symmetry-induced singularities, there are also spurious singular points, that we find to be linked to the incompleteness of the mapping, i.e. the fact that, for certain classes of representations, structurally distinct configurations are not guaranteed to be mapped onto different feature vectors. Additional singularities can be introduced by a too aggressive truncation of the infinite basis set that is used to discretize the representations. Conclusions: These results exemplify the subtle issues that arise when constructing symmetric representations of atomic structures, and provide conceptual and numerical tools to identify and investigate them in both benchmark and realistic applications.

Retina remodeling following diode laser.

BACKGROUND: The purpose of this study was to characterize the sequential development of focal and surround retinal injury and repair following transscleral diode laser to rat retina. METHODS: Transscleral laser photocoagulation of the retina was induced with a diode laser (DioPexy Probe, 810 nm, 200 mW, 2 seconds) in adult Long-Evans rats. The right eye of rats with survival times of 0 days (n = 4), 5 days (n = 6), 2 weeks (n = 4), 6 weeks (n = 6), and 12 weeks (n = 4) was studied. Using serial sections, detailed pathological changes in laser-treated and surrounding retinal and choroidal areas were compared with the control fellow eye. RESULTS: Photocoagulation damage was limited to the retina, sparing Bruch's membrane, with minimal choroidal involvement in almost all cases (23/24 eyes). Following damage to the neural retina, the sequence of major remodeling processes was consistent and included inflammatory response, reparative changes, and formation of glial-vascular scar with neovascularization. INTERPRETATION: This new laser model caused reproducible injury, inflammation, and scarring confined to the retina, and may be a tool to help test the effects of candidate neuroprotective/regenerative agents on retinal degeneration to prevent vision loss.

Capillary hemangioma of the testis: A rare benign tumour.

Testicular capillary hemangioma is a rare benign vascular tumour. We report a case of a 66-year-old man who underwent an uncomplicated radical orchiectomy for a painless left testicular mass. Pathology showed capillary hemangioma of the testis. There are only 22 cases reported in the English literature, including the presented case. Appropriate intra-operative recognition of this entity is vital to assess for potential testicular-sparing surgery.