**Direct Prediction of Phonon Density of States With Euclidean Neural Networks:** According to the DOS equation, the concept of phonon density of states—also known as “vibrational density of states”—is identical to that of the electronic density of states. The total over the Brillouin zone covers all 3N phonon bands, where N indicates the number of atoms in the cell. Two well-accepted conventions exist for normalizing the phonon DOS. It can be scaled to either unity or the amount of 3N of vibrational modes.

The partial “phonon density of states” is determined by the contribution of the supplied atom to the total phonon density of states. Understanding the nature of the various fields in the phonon spectrum using this approach is helpful. Each phonon band’s participation in the partial state density of atom is assessed using the:

where is the (normalized to unit length) eigenvector connected to the energy mode? By adding up these contributions across all phonon bands, the expected density of levels is then determined. All of the anticipated phonon DOS add up to the actual phonon DOS by construction.

## Introduction to Euclidean

Euclidean distance is considered the separation between two points in mathematics. In other words, the length of the line segment connecting two locations is what is meant by the term “Euclidean distance” to refer to the separation of two points in Euclidean space. The Euclidean distance is sometimes referred to as the Pythagorean distance since it can be calculated using coordinate points and the Pythagoras theorem.

The Euclidean distance formula, as previously mentioned, aids in determining the separation between two line segments. Assume two locations in the two-dimensional complex plane, like (x1, y1) and (x2, y2).

As a result, the following is the Euclidean distance formula: Where,

- The Euclidean distance is “d.”
- The initial point’s coordinate is (x1, y1).
- The second point’s coordinates are (x2, y2).

Consider two points, P(x1, y2) and Q(x2, y2), and let d be the distance between them to obtain the formula for the Euclidean distance. Now draw a line connecting P and Q. Let’s build a right triangle with PQ as the hypotenuse so that we may calculate the Euclidean distance formula. Draw the horizontal and vertical lines from P and Q that intersect at point R.

The Pythagorean theorem must now be applied to the triangular PQR in order to determine the separation between two points. Making use of Pythagoras’ Theorem.

“Hypotenuse^{2} = Base^{2} + Perpendicular^{2}

PQ^{2} = PR^{2} + QR^{2}

Therefore, d^{2} = (x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2” }

Taking the square root of the equation’s two sides now yields

As a result, the Euclidean distance formula is derived.

## Phonon Density of States With Euclidean Neural Networks

The architecture, research, and property prediction of materials have all benefited greatly from machine learning. However, despite machine learning’s effectiveness in predicting discrete qualities, difficulties still exist in predicting continuous properties. Because of the lack of data and crystallographic symmetry issues, the problem is even more difficult in crystalline solids. Here, simply atomic species and locations are used as input to show how to forecast phonon density-of-states (DOS) directly.

**Zhantao Chen** et al., proposed a methodology for *“**Direct Prediction of Phonon Density of States With Euclidean Neural Networks”. **They utilized* limited training set of 103 samples with more than 64 atom types, Euclidean neural networks—which by design are increasingly adaptable to 3D transformations, flips, and inversion and hence capture entire crystal symmetry—are used to produce high-quality predictions. The predictive model is naturally adapted to accurately anticipate alloy systems without incurring additional processing costs because it reproduces essential characteristics of experimental results and even generalizes to substances with unseen constituents.

The network’s promise is illustrated by its ability to anticipate a wide range of materials with high phononic heat capacity capacities. The research suggests a productive method for examining the phonon structure of materials, and it may also facilitate quick searches for high-temperature storage materials and superconductors that are phonon-mediated.

In the work, researchers developed a machine learning (ML)-based predictive model that uses atomic structures as input and directly generates the “phonon density of states (DoS)”. Phonon DoS is a fundamental factor in interface thermal resistance and a major determinant of the heat capacity and vibrational entropy of materials. Additionally, it is closely related to superconductivity, thermal, and electrical transport. Due to the restricted scattered light equipment resources and high computing complexity of ab initio computations for complicated materials, the procurement of empirical and computed phonon DoS is, however, not straightforward.

Additionally, the phonon computations in alloy systems are very difficult. Without carefully controlled approximations, several established methods, such as virtual crystal approximation (VCA), can fail both qualitatively and numerically. This necessitates a method that more effectively acquires phonon DoS, particularly for alloy systems. We use a Euclidean neural network (E(3)NN) to create such a model since it naturally acts on 3D geometry and is equivalent to 3-dimensional translations, rotations, and inversions.

The entire input’s geometric information is preserved by E(3)NNs, which also do away with the necessity for pricey (by about 500 times) data augmentation. The network also maintains all of the input data’s crystallographic symmetries. They used open-source E3N repository implementations of E(3)NNs in this work. The functionals “perturbation theory (DFPT)-based phonon database [2]”, which contains phonon DoS data for over 1,500 crystalline solids, allows for the prediction of phonon DoS with high accuracy.

Even for crystal structures with hidden components, their predictive model is capable of capturing the key characteristics of phonon DoS. We generate a list of materials with large heat capacities by estimating the phonon DoS in 4,346 novel crystal structures. These predictions are supported by additional DFPT simulations. Our work provides a reliable method for directly acquiring phonon DoS from atomic structure, best suited for the design of high throughput materials with desirable phonon-related characteristics.

## FAQs

### Describe phonon density.

The number of modes for each unit of frequency and real space volume is given by the phonon density of states. The group velocity, which is determined from the dispersion relation, makes up the final denominator.

### What is the reason for phonons’ negative mass?

Researchers discovered that, in opposition to gravity, the phonons travelled on upward trajectories. This meant that the phonons were related to gravity, in contrast to classical theories of sound waves, allowing them to carry very small quantities of “negative effective gravitational mass” as they moved.