Gaussian Process Regression for Machine Learning on Effective Crystal Graphs of Body-Centered Cubic Iron

Abstract

Most machine learning algorithms operate on vectorized data with Euclidean structures because of the significant mathematical advantages offered by Hilbert space, but improved representational efficiency may offset more involved learning on non-Euclidean structures. Recently, a method that integrates the marginalized graph kernel into the Gaussian process regression framework was used to learn directly on molecular graphs. Here, we describe an implementation of this method for crystalline materials based on effective crystal graph representations: the molecular graphs of 128-atom supercells of body-centered cubic (BCC) iron with periodic boundary conditions. Regressors trained on hundreds of time steps of a density functional theory molecular dynamics (DFT-MD) simulation achieved root mean square errors of less than 5 meV/atom. The mechanical stability of BCC iron was investigated at high pressure and elevated temperature using regressors trained on short DFT-MD runs, including at conditions found in the inner core of the earth. Phonon dispersions obtained from the short runs show that BCC iron is mechanically stable at 360 GPa when the temperature is above 2500 K. Atoms in the super cell were displaced in the direction of the first, second, and third nearest-neighbors from selected configurations that included thermal atomic displacements, and forces exerted on the displaced atoms were computed by numerical differentiation of the regressors.

1. Introduction

The development of machine learning (ML) and its applications is transforming the field of materials [1], although mathematical frameworks and techniques that can be recognized as ML or ML-like have been used in alloy theory and atomistic simulations of materials for many decades [2]. For example, scalar properties of lattices are routinely expanded as linear combinations of orthogonal cluster correlation functions, which allows interpolation in truncated cluster space [3,4]. Data science techniques based on rigorous statistics have been developed in the past twenty-five years and are widely used by the community to predict formation energies, electronic band-gaps, and other quantities [5].

Machine learning potentials with functional forms that can be systematically improved are increasingly favored over legacy physics-based interatomic potentials with fixed functional forms [2]. However, despite their success, widely used approaches such as the Gaussian Approximation Potential (GAP) [6], Spectral Neighbor Analysis Potential (SNAP) [7], and Moment Tensor Potential (MTP) [8] still face challenges in data efficiency and generalization across varying thermodynamic states. The Gaussian Process Regression model with a marginalized graph kernel (GPR-MGK) [9] addresses these limitations by incorporating a probabilistic kernel that directly measures similarity between graphs representing atomic configurations through random-walk correlations. Here, we present a graph-based machine learning potential that uses GPR-MGK to investigate the mechanical stability of body-centered cubic (BCC) iron at geophysically relevant pressures and temperatures.

The inner core of the earth is a solid (quasi) sphere of iron with a small fraction of lighter elements located at the center of the planet and enclosed by molten material in the outer core [10]. It grows by solidification of the molten material, a process that releases heat and helps drive the convection currents in the outer core that generate the magnetic field of the earth [11]. Unfortunately, it is challenging to study the inner core directly via seismology, and no material can be retrieved from such region, so many aspects remain controversial [12]. Pure iron adopts a BCC crystal structure with ferromagnetism at ambient pressure and temperature [13]. When pressure is kept at zero, it becomes paramagnetic at 1043 K, transforms to the face-centered cubic (FCC) phase at 1185 K before reverting back to BCC at 1667 K (sans the ferromagnetism), and melts at 1811 K [14]. At low temperature, it transforms to the hexagonal-closed packed (HCP) phase at 11 GPa [15,16,17], which remains stable into the TPa range [18,19]. The crystal structure of iron at elevated pressures and temperatures is an active area of research [20]. Although many studies suggest that HCP is thermodynamically stable at earth’s core conditions [21,22,23,24,25], the BCC structure is dynamically stable [26,27] and more recent studies show that there might be a thermodynamically stable BCC phase between HCP and melting [28,29,30]. In this study, we investigate the mechanical stability of the BCC structure.

2. Methods

2.1. Ab Initio Molecular Dynamics

Ab initio molecular dynamics (AIMD) simulations of non-spin-polarized iron were carried out with canonical (NVT) ensembles for two lattice parameters, $a_0=2.49$ Å and $a_0=2.36$ Å using supercells of 4 × 4 × 4 conventional BCC cells (totaling 128 atoms) with Born-von Kármán (BvK) periodic boundary conditions (PBC). The simulations were performed in the Born-Oppenheimer approximation with Quantum Espresso [31,32], using a PBE exchange-correlation functional [33] with the projector-augmented wave (PAW) method [34]. The kinetic energy cutoff was set to 71 Ry, the charge density cutoff to 496 Ry, and sampling was restricted to the $\Gamma$ point of the Brillouin zone. The temperature was controlled using stochastic velocity rescaling [35] with a relaxation time of 0.02 ps and thermostated at 800 K, 1250 K, 2500 K, 3700 K, and 5000 K for each lattice parameter. A total of 200 steps with time steps of 1 fs were simulated for each combination of temperature and lattice parameter. The average pressure over each trajectory is in Table 1, and throughout this article, we refer to these 10 runs as the ‘science dataset’. An additional run at $a_0=2.77$ Å and 1043 K that generated a pressure of 7 GPa was also carried out, but used a PBEsol exchange-correlation functional with a kinetic energy cutoff of 40 Ry and a charge density cutoff of 320 Ry. This run consisted of 1000 steps and was used to thoroughly investigate the performance of the GPR-MGK methodology, so we refer to it as the ‘examination dataset’.

Table 1. Pressure predicted by AIMD simulations for each combination of temperature and lattice parameter in the science dataset. See Section 2.1 for details.

Temperature (K)	Pressure (GPa) at ${\mathit{a}}_0=2.36$ Å	Pressure (GPa) at ${\mathit{a}}_0=2.49$ Å
800 K	344 GPa	159 GPa
1250 K	346 GPa	162 GPa
2500 K	355 GPa	171 GPa
3700 K	362 GPa	179 GPa
5000 K	376 GPa	194 GPa

2.2. Harmonic Ensemble Lattice Dynamics

The phonon dispersion relations of BCC iron at each of the conditions (temperature and pressure) in the science dataset were calculated from the MD simulations using harmonic ensemble lattice dynamics (HELD) [27,36], and the results are shown in Figure 1. HELD fits the atomic displacements from equilibrium in an MD step and the forces exerted on the atoms on that same step to the second-order Born–von Kármán (BvK) model [37] up to fifth nearest neighbors. The BvK model is a generalization of Hooke’s law for crystals, given in global matrix form by $\mathcal{F}=\mathcal{C}\times \mathcal{P}$, where each row of vector $\mathcal{F}$ contains a Cartesian component of the force exerted on an atom in the supercell, each row of vector $\mathcal{P}$ contains the force constant between crystallographically equivalent pairs of atoms, and the displacements are in the matrix $\mathcal{C}$. HELD builds $\mathcal{C}$ from an MD step by translating all displacements and forces that are crystallographically equivalent to the first octant. The effective force constants account for thermal disorder since they come from MD simulations and are found by solving the overdetermined system of equations $\mathcal{P}={\mathcal{C}}^{-1}\times \mathcal{F}$. The specific force constants used and symmetries are on page 22 of Ref. [27].

Figure 1. Phonon dispersion relations for BCC iron at five temperatures (top to bottom) $T=\{800\mathrm{K},1250\mathrm{K},2500\mathrm{K},3700\mathrm{K},5000\mathrm{K}\}$ and two lattice parameters (left to right) $a_0=\{2.36\AA ,2.49\AA \}$ obtained by fitting ab-initio molecular dynamics data to a harmonic Born-von Kármán model. See Section 2.2 for details.

For each temperature and pressure condition, a subset of the relevant AIMD data consisting of fifty time steps randomly selected from the second half of each trajectory in the science dataset were aggregated, so $\mathcal{F}$ consisted of $50\times 128\times 3=\mathrm{19,200}$ rows that were used to obtain the 14 force constants in $\mathcal{P}$. Phonon dispersions were obtained with a customized version of the Phonopy code [38,39] from dynamical matrices assembled from the Fourier transforms of the cumulant force constant matrices $\mathcal{B}=\mathcal{D}\times \mathcal{P}$. The symmetry engine Spglib [40] is used to find $\mathcal{D}$, an operator that enforces crystallographic point-group relations, ensuring that the resulting dynamical matrices are Hermitian and physically consistent with the space-group symmetry of the crystal [41]. The results are in good agreement with similar studies [26,42], demonstrating that the AIMD data is of similarly good quality.

2.3. Effective Crystal Graphs

The molecular graph $G=\{V=\left\{v_i\right\},E=\left\{e_{ij}\right\}\},i,j\in \{1,\dots ,n\}$ of a molecule of n atoms is an undirected graph that uses its set of vertices V to describe atoms and its set of edges E to describe interactions between atoms. This and related representations offer major advantages on interpretability and expressive power, although at relatively high computational cost [43,44]. Compared to molecules, crystals have lattice periodicity and additional space group symmetries that are commonly represented with multigraphs, which allow multiple edges between pairs of vertices [45]. The molecular graph representation is a mathematical requirement of GPR-MGK, so effective crystal graphs were constructed as detailed below.

Crystals were modeled in the AIMD simulations using supercells of $n=128$ atoms with PBC. Initially, every atom i in a supercell was assigned a number between 1 and n, and the vertex $v_i\in \left\{\mathrm{Fe}\right\}$ was labeled with the chemical species of the associated atom. Any chemical species is admitted as a label by GPR-MGK, but our supercells consisted of iron atoms exclusively. Then, the edge $e_{ij}\in \mathbb{R}$ between vertices $v_i$ and $v_j$ was labeled with the distance $d_{ij}=\left|\right|\overrightarrow{r_i}-\overrightarrow{r_j}\left|\right|$ between atoms i and j and was assigned a weight calculated using the adjacency rule [46]

$$\mathcal{A}\left(s\right)={\displaystyle \frac{-b{s}^a+a{s}^b}{a-b}}; a>b\ge 2,$$

(1)

which depends on the atomic positions through the convenience function $s(\overrightarrow{r_i}-\overrightarrow{r_j})=max\{0,1-d_{ij}/d_c\}$, where $d_c=3a_0$ is a custom cutoff distance. Finally, edge labels and weights were computed according to $d_{ij}^{\mathrm{MIC}}$, the distance between atoms i and j in the minimum image convention (MIC), and replaced those from the previous step if $d_{ij}^{\mathrm{MIC}}<d_{ij}$. The adjacency matrix of the graph G is thus given by

$$A_{ij}=\left\{\begin{array}{cc}\mathcal{A}\left(s\left(d_{ij}\right)\right)\hfill & \mathrm{if}{d}_{ij}<d_{ij}^{\mathrm{MIC}}\hfill \\ \mathcal{A}\left(s\left(d_{ij}^{\mathrm{MIC}}\right)\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

The adjacency matrix in Equation (2) differs from the one in Ref. [9] in two ways: the polynomial smoothing function in Equation (1) is used instead of the square exponential, and a conditional is used to implement PBC. For polynomial smoothing, the free parameters a and b control the smoothness and width of the kernel. We investigated the effect of several pairs of values on the predictions using the examination dataset, and we describe the results in the next section. For the science dataset, we used $a=3$ and $b=2$, which minimized the RMSE (1.9 meV/atom) while avoiding over sensitivity to local fluctuations at high temperature. Higher a values caused some instability, whereas smaller b values reduced correlation between first- and second-neighbor interactions. These parameters provide a smooth yet discriminative decay profile, consistent with optimal GPR-MGK behavior reported in [47].

Periodic boundary conditions were implemented using the naive MIC algorithm in ASE [48]. Succinctly, atoms i and j that are in the supercell are nth nearest-neighbors; copies of the supercell (images) are created around it using translation vectors, so i and $j^{\prime }$ (the copy of j) are $n^{\prime }$th nearest-neighbors. If $n^{\prime }<n$, the distance between i and $j^{\prime }$ is used instead of the distance between i and j. Regarding the choice of $d_c$, lattice dynamics of metals are typically accurately represented by a BvK model of 5 coordination shells, and this is the case for iron as demonstrated by the HELD results (Section 2.2 and Figure 1 ) and other studies [49,50]. For the BCC structure, the 9th coordination shell is at a distance of $3a_0$, so the cutoff distance $d_c$ safely includes all expected physically relevant interatomic interactions and the totality of the BvK model. The length of the main diagonal of a cube of length $a_0$ is $\sqrt{3}{a}_0$, so with PBC, most pairs of atoms in the supercell will be closer than 9th nearest-neighbors, so increasing $d_c$ beyond $3a_0$ results in higher computational cost without materially affecting the prediction quality.

2.4. Graph Kernel and Gaussian Process Regression

Most ML codes operate on fixed-length feature vectors because they live in Hilbert space, so linear algebra can be used at the cost of imposing a vector representation that might be suboptimal. Graphs are comparatively rich and flexible, but their inner product is not universally defined and search and compute operations are computationally expensive [51]. Graph kernels are functions that compute the inner product between two graphs in Hilbert space and can be used to interface graph data to ML codes by generating vector space representations [47]. In our case, the marginalized graph kernel [52] was integrated into the Gaussian process regression process, so the graph data were read directly.

The kernel between labeled graphs G and $G^{\prime }$ is obtained by solving the generalized Laplacian of their Kronecker product [9]

$$K(G,G^{\prime })=G\otimes G^{\prime }={\mathbf{p}}_{\times }^{⊺}{({\mathbf{D}}_{\times }{\mathbf{V}}_{\times }^{-1}-{\mathbf{A}}_{\times }\odot {\mathbf{E}}_{\times }^{-1})}^{-1}{\mathbf{D}}_{\times }{\mathbf{q}}_{\times },$$

(3)

where ${\mathbf{D}}_{\times }=\mathbf{diag}(\mathbf{d}\otimes {\mathbf{d}}^{\prime })$ is the degree matrix of the product graph (so ${\mathbf{d}}_i={\sum }_j{\mathbf{A}}_{ij}$); ${\mathbf{V}}_{\times }=\mathbf{diag}(\mathbf{v}\otimes {\mathbf{v}}^{\prime })$ is a diagonal matrix with elements given by the vertex base kernel, which is a Kronecker delta ${\kappa }_v(v,v^{\prime })=1$ if $v=v^{\prime }$ and 0 otherwise (all the vertices represented iron atoms, so ${\mathbf{V}}_{\times }^{-1}=\mathbf{I}$, the identity matrix); ${\mathbf{A}}_{\times }=(\mathbf{A}\otimes {\mathbf{A}}^{\prime })$ is the adjacency matrix of the product graph; and ${\mathbf{E}}_{\times }=(\mathbf{E}\otimes {\mathbf{E}}^{\prime })$ is a product graph with elements given by the edge base kernel, which is a square exponential on edge lengths ${\kappa }_e(e,e^{\prime })=exp\left[-{(e-e^{\prime })}^2/2{\lambda }^2\right]$ (evaluates to 1 if two edges are of the same length and tends to 0 as the length difference increases, controlled by the hyperparameter $\lambda$).

The marginalized graph kernel can also be computed by comparing the sequences of vertices and edges traversed by an infinite number of random walkers [52]. New walks start at vertices selected by sampling $\mathbf{p}$ and ${\mathbf{p}}^{\prime }$, the vectors containing the individual vertex starting probabilities for each graph, and enter Equation (3) through ${\mathbf{p}}_{\times }=\mathbf{p}\otimes {\mathbf{p}}^{\prime }$, the starting probability of a Markovian random walk process from each node of the product graph. Walks have a probability of ending at each vertex they visit, decided by evaluating individual stopping probabilities stored in vectors $\mathbf{q}$ and ${\mathbf{q}}^{\prime }$, which enter the equation above through ${\mathbf{q}}_{\times }=\mathbf{q}\otimes {\mathbf{q}}^{\prime }$, the stopping probability of the random walk process on each node of the product graph. We chose the starting and stopping probability distributions to be uniform. Their values are assigned via the hyperparameters s and q shown in Table 2.

Table 2. Hyperparameters of the GPR-MGK models.

Hyperparameter	Symbol	Description	Value
Element similarity Factor	$\nu$	Baseline similarity assigned between atoms of different elements. Must be within (0, 1).	0.3
Spatial adjacency scale	$\zeta$	Controls how quickly edge weights decay with interatomic distance, affecting neighborhood range.	1.0
Edge length sensitivity	$\lambda$	Determines sensitivity of kernel to bond length differences, differentiating bond types.	0.05
Random walk start Probability	s	Determines the probability of starting a random walk from each vertex. Affects kernel strength.	250
Random walk stop Probability	q	Controls the length of random walk paths. Along with $\zeta$, it determines neighborhood range.	0.05

Finally, given a training set of molecular graphs $m=\{M_1,\dots ,M_m\}$ and their associated energies $\{E_1,\dots ,E_m\}$, the GPR-MGK prediction for the energies $\{E_1^{*},\dots ,E_n^{*}\}$ of a set of n unknown atomic configurations $\{M_1^{*},\dots ,M_n^{*}\}$ is given by the regression-like equation [9]

$$E^{*}={\left[\begin{array}{c}K[M^{*},M_1]\\ K[M^{*},M_2]\\ ⋮\end{array}\right]}^T{\left[\begin{array}{ccc}K[M_1,M_1]& K[M_1,M_2]& \dots \\ K[M_2,M_1]& K[M_2,M_2]& \dots \\ ⋮& ⋮& \ddots \end{array}\right]}^{-1}\left[\begin{array}{c}{E}_1\\ E_2\\ ⋮\end{array}\right].$$

(4)

2.5. Potential Energy Surfaces

The potential energy surface (PES) of a system can be predicted by training the model laid out in Equation (4), which requires potential energies and similarities between pairs of molecular graphs. In our implementation, the training set originates with the atomic positions of each step in an AIMD simulation and the potential energy of such atomic configuration. These quantities are recorded at every time step of a simulation, so the number of data points in the training set is the total number of time steps in all AIMD simulations, which is 3000. ASE [48] was used to instantiate relevant quantities from the raw data as python objects compatible with our software ecosystem. The methodology described in Section 2.3 was implemented in GraphDot [47] and used to construct the molecular graph of every data point in the training set.

The kernel in Equation (3) computes the average similarity the expectation of the path similarity between all pairs of paths that can be obtained by performing simultaneous random walks on two graphs G and G’. We used the open-source code GraphDot [47] to train the models and to perform inference. The code implements the methodology in Ref. [9], which we followed with the adjustments described in Section 2.3. GraphDot runs on graphics processing units (GPU) and implements state-of-the-art GPU acceleration algorithms [47].

GraphDot was also used to train GPR-MGK models on subsets of the training data to learn the PES of BCC iron at specific temperature T and lattice parameter $a_0$. An AIMD run of 1000 time steps that simulated BCC iron at low pressure was used to optimize the GPR-MGK modeling by assessing the impact of parameters and methods on the performance of the model. The remaining 2000 time steps are divided among ten AIMD runs of 200 time steps each that probe ten distinct phase diagram coordinates ($a_0$, T). A GPR-MGK model of the PES of BCC iron at each of those coordinates was trained on the data and used to assess the mechanical stability of the BCC structure and the forces felt by individual atoms when all atoms are thermally displaced from their ideal lattice positions. The values of the hyperparameters are shown in Table 2 and the performance of the models is described in the results section.

3. Results

3.1. Evaluation of the Machine-Learning Model

The predictive performance of the proposed GPR-MGK framework was evaluated using configurations obtained from ab initio molecular dynamics (AIMD) simulations of BCC iron. A total of 1000 AIMD time steps were generated, from which 800 equilibrated configurations were selected for model training. The root-mean-square error (RMSE) of the predicted energy systematically decreased with increasing training-set size, falling from approximately 35 meV/atom at $N=200$ to below 5 meV/atom at $N=800$. This monotonic decrease demonstrates that model accuracy improves as additional atomic environments are incorporated into training. A training set of roughly 400 configurations was sufficient to achieve low RMSE values, providing an effective compromise between computational cost and predictive accuracy (Figure 2). Table 3 compares the overall accuracy of the GPR-MGK potential with popular state-of-the-art machine learning interatomic potential (MLIP) frameworks applied to BCC iron, including the Smooth Overlap of Atomic Positions (SOAP)-based GAP, the Moment Tensor Potentials (MTP and mMTP), the Spectral Neighbor Analysis Potential (SNAP), and the Deep Potential model implemented in the DeepMD-kit. The GPR-MGK model produces an energy RMSE of 5 meV/atom with a training set of 800 configurations, corresponding to (comparatively large) pressures of 159–194 GPa and temperatures of 800–5000 K. This performance is comparable to that of GAP (1–5 meV/atom) and DeepMD-kit (3.3–3.7 meV/atom), while surpassing the accuracies reported for MTP (30 meV/atom) and SNAP (28–58 meV/atom). Notice that the RMSE increases with the number of distinct atomic environments due to, e.g., pressure and temperature and goes down with the size of the training set.

Figure 2. Root mean square error (RMSE) as a function of training set size for the GPR-MGK models.

Table 3. Comparison of GPR-MGK with other machine learning interatomic potentials for body-centered cubic (BCC) iron.

Framework	Training Set Size	Supercell Size	Pressure (GPa)	Temperature (K)	Energy RMSE (meV/atom)
This work	800	128	159–194	800–5000	5
SOAP (GAP) a	6001	1	0	300	1
SOAP (GAP) a	12,474	54	0	400–1400	1
SOAP (GAP) a	11,520	128	0	800	1
MTP b	9351	54	0	1000	30
mMTP b	22,000	16	0	800	2
SNAP (B1) c	500	128	0–12	0–200	28
SNAP (B2) c	500	128	0–15	200–750	58
DeepMD-kit d	799	128, 250	75–400	4000	3
DeepMD-kit d	640	128, 250	175–400	5000	4
DeepMD-kit d	1222	128, 250	250–400	6000	3
DeepMD-kit d	300	128, 250	350–450	7000	4
DeepMD-kit d	165	128, 250	450–650	7600	3

^a [53]. ^b [54]. ^c [55]. ^d [56].

3.2. RMSE vs. Size of the Training Set

The predictive accuracy of the GPR-MGK model was assessed as a function of training set size by calculating the RMSE across subsets of the ab initio molecular dynamics data. As shown in Figure 2, the RMSE systematically decreases with larger training sets, confirming that model performance improves as more atomic configurations are included in the training process. Using the full 1000 time steps results in RMSE values as low as 2 meV/atom, while smaller training sets yield higher errors up to 35 meV/atom. When the first 100 non-equilibrated time steps are omitted, the error range narrows considerably, suggesting that equilibrated data enhances model stability. Importantly, a training set size of 400 configurations is sufficient to achieve relatively low errors, indicating an effective compromise between computational cost and predictive accuracy.

Clever integration with the marginalized graph kernel gives GPR the ability to operate on mathematical graphs, and carefully crafted rules produce mathematical graphs that accurately represent many systems based on the type and location of its atoms. A GPR model that used empirically discovered optimal hyperparameters values was trained on 800 randomly selected data points from the low-pressure iron dataset that contains 1000 data points. The rest of the data points were used to test the model by predicting the potential energies of the remaining atomic configurations and comparing them to their known value. The upper panel of Figure 3 presents the parity plot comparing GPR-MGK predicted energies to DFT reference values for all validation configurations. The points cluster tightly along the $y=x$ line, indicating excellent calibration and correlation. The lower panel of Figure 3 shows the corresponding residual-error histogram (Pred − DFT), which is symmetric and centered near zero (mean $\mu \approx -1.1\times {10}^{-2}$ eV; standard deviation $\sigma \approx 3.19\times {10}^{-1}$ eV), confirming that prediction errors are random and unbiased.

Figure 3. (Upper): Parity plot of GPR-MGK predicted energies versus DFT reference energies for the validation set. The model achieves $R^2=0.9980$, $\mathrm{RMSE}=3.19\times {10}^{-1}$ eV, and $\mathrm{MAE}=2.60\times {10}^{-1}$ eV. (Lower): Error-distribution histogram (Pred − DFT), showing residuals centered near zero. The mean error is $\mu =-1.06\times {10}^{-2}$ eV with a standard deviation $\sigma =3.19\times {10}^{-1}$ eV.

3.3. Validation Set

A small validation set was created by displacing a single atom from the ideal lattice configuration at examination set conditions in 9 discrete steps of 0.1 Å in the direction of a first nearest neighbor. The energies of these artificially constructed configurations were obtained after the fact from Quantum Espresso with appropriate settings and then compared to the predictions made by the trained model. The model yields errors that are orders of magnitude larger for displacements larger than about 0.8 Å from equilibrium in the ideal lattice, but this is to be expected since thermal displacements of that magnitude are extremely rare and are essentially out-of-training-set. The results are shown in Table 4.

Table 4. Error in the energy prediction of single atomic displacements from equilibrium in the ideal lattice configuration at examination set conditions.

Displacement from Equilibrium (Å)	Absolute Energy Prediction Error (meV/atom)
0.0	0.0
0.1	0.0
0.2	0.1
0.3	0.3
0.4	0.3
0.5	0.1
0.6	0.7
0.7	0.2
0.8	4.0
0.9	10.0

3.4. Phonon Dispersions and Dynamical Stability

The phonon dispersions obtained with HELD for AIMD trajectories in the science data set are shown in Figure 1. The averages of the phonon dispersions derived from the simulations were obtained by taking 50 random steps within the final 0.1 picosecond of the simulation period. There are no mechanical instabilities for $a_0=2.49\AA$ in the temperature range investigated, but for $a_0=2.36\AA$ there are imaginary phonon modes at 800 K and 1250 K that are stabilized at 2500 K and above, in agreement with previous studies that showed that temperature dynamically stabilizes the BCC structure of iron at high pressure [26,42]. Those studies predict higher stabilization temperatures by a factor of two, but this variation is reasonable considering how different methodologies were used to compute the dispersions [57]. The longitudinal phonon branch has a dip at 2/3 of the way along the $\Gamma$-H direction. The dynamical transformation from BCC to the omega phase is achieved via the $L{\displaystyle \frac{2}{3}}⟨111⟩$ phonon [58]. Our phonon dispersion results demonstrate that BCC Fe becomes dynamically stabilized at high temperatures, with imaginary modes disappearing above 2500 K at 360 GPa. This trend agrees qualitatively with the ab initio lattice dynamics study of Ref. [26], which predicted stabilization of BCC Fe above 4500 K. A comparison is shown in Figure 4, along with the experimentally determined melting curve from Ref. [59].

Figure 4. Comparison of mechanical stability based on the lack of of imaginary phonon dispersions at different temperatures and pressures obtained by this work with previous studies [26,59]. Experimental melting curve measurements included for reference.

3.5. Force–Displacement Analysis

The force–displacement curves show how atoms in BCC Fe respond when they are slightly disturbed from their lattice positions. In Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, the white lines represent the forces on individual atoms, while the orange line shows the average force for the whole system. A narrow spread of white lines means that atoms respond in a similar way, while a wide spread means that atoms respond differently. We use these curves to test for stability in real space. A stable system has an average force (orange line) that is close to linear with a negative slope and intercepts the y-axis near zero for small displacements, with little variation between atoms. An unstable system shows a curved average force and a wide spread of individual atomic forces. The results are shown for two lattice parameters, $a_0=2.36$ Å and $a_0=2.49$ Å. At $a_0=2.36$ Å, the 1NN, 2NN, and 3NN curves (Figure 5, Figure 6 and Figure 7) show stronger curvature and wider spreads, especially at lower temperatures. This means the denser lattice is less stable. At $a_0=2.49$ Å, the curves (Figure 8, Figure 9, Figure 10) are closer to linear and the spread is smaller, showing better stability. This comparison shows that the larger lattice spacing ($a_0=2.49$ Å) helps the system remain more stable under thermal disorder.

Figure 5. Forces along the direction of the first nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.36 Å (pressure in Table 5).

Figure 6. Forces along the direction of the second nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.36 Å (pressure in Table 5).

Figure 7. Forces along the direction of the third nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.36 Å (pressure in Table 5).

Figure 8. Forces along the direction of the first nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.49 Å (pressure in Table 5 ).

Figure 9. Forces along the direction of the second nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.49 Å (pressure in Table 5).

Figure 10. Forces along the direction of the third nearest-neighbors felt by an atom being displaced from its ideal lattice position in the presence of thermal disorder at different temperatures for a lattice parameter of 2.49 Å (pressure in Table 5).

Table 5. Mechanical stability analysis of BCC Iron at different temperatures and configurations.

Temperature (K)	Configuration	Mechanical Stability Analysis
800	1NN	Highly unstable: Significant force fluctuations, large nonlinearity, and chaotic atomic displacements.
	2NN	Highly unstable: Pronounced deviations from linearity, loss of predictable force response.
	3NN	Highly unstable: Strong force perturbations, nonlinear atomic interactions.
1250	1NN	Unstable: Nonlinear force behavior with increasing atomic variability.
	2NN	Unstable: Increasing deviation from harmonic behavior, growing force dispersion.
	3NN	Unstable: Significant force fluctuations, curvature effects become prominent.
2500	1NN	Onset of instability: Emergence of force deviations from harmonic response.
	2NN	Onset of instability: Detectable nonlinear behavior in atomic interactions.
	3NN	Onset of instability: Small perturbations lead to measurable force anisotropy.
3700	1NN	Marginally stable: Force response exhibits minor nonlinearity but retains overall structural integrity.
	2NN	Marginally stable: Small deviations from perfect linearity, with a well-defined force response.
	3NN	Marginally stable: Predominantly linear atomic interactions with minor force anisotropy.
5000	1NN	Stable: Near-harmonic force response with minimal perturbations.
	2NN	Stable: Slight force deviations but overall linear atomic behavior maintained.
	3NN	Stable: Minimal force fluctuations, predominantly linear and predictable response.

Table 5. Mechanical stability analysis of BCC Iron at different temperatures and configurations.

Temperature (K)	Configuration	Mechanical Stability Analysis
800	1NN	Highly unstable: Significant force fluctuations, large nonlinearity, and chaotic atomic displacements.
	2NN	Highly unstable: Pronounced deviations from linearity, loss of predictable force response.
	3NN	Highly unstable: Strong force perturbations, nonlinear atomic interactions.
1250	1NN	Unstable: Nonlinear force behavior with increasing atomic variability.
	2NN	Unstable: Increasing deviation from harmonic behavior, growing force dispersion.
	3NN	Unstable: Significant force fluctuations, curvature effects become prominent.
2500	1NN	Onset of instability: Emergence of force deviations from harmonic response.
	2NN	Onset of instability: Detectable nonlinear behavior in atomic interactions.
	3NN	Onset of instability: Small perturbations lead to measurable force anisotropy.
3700	1NN	Marginally stable: Force response exhibits minor nonlinearity but retains overall structural integrity.
	2NN	Marginally stable: Small deviations from perfect linearity, with a well-defined force response.
	3NN	Marginally stable: Predominantly linear atomic interactions with minor force anisotropy.
5000	1NN	Stable: Near-harmonic force response with minimal perturbations.
	2NN	Stable: Slight force deviations but overall linear atomic behavior maintained.
	3NN	Stable: Minimal force fluctuations, predominantly linear and predictable response.

4. Discussion

4.1. Geophysics Considerations

At extreme pressures, iron transforms from BCC to HCP [60,61,62]. The stable crystalline phases of iron at pressures ranging from 150 to 1500 GPa have been a topic of longstanding debate, with some studies suggesting that the BCC structure could remain stable at temperatures approaching the melting point [26,63,64,65,66]. The theoretical rationale proposes that vibrational entropy may stabilize the BCC structure over close-packed arrangements at high pressures and temperatures [28,29]. However, most experimental evidence indicates that the stable phase is HCP [59,67,68]. The stability of iron’s crystalline phases under extreme pressures and temperatures is a crucial area of study in planetary science, with significant implications for our understanding of Earth’s core and the interiors of exoplanets. However, acquiring direct experimental observations poses challenges due to the inaccessibility of these extreme conditions, and traditional computational methods can be prohibitively costly.

High pressure and high temperature phonon studies provide valuable information on phase stability, elasticity, and thermal conductivity in planetary interiors. Earlier measurements of phonons have frequently yielded only incomplete information, like the velocities of ultrasonic phonons, shock-wave velocities, stress-strain relations and zone-center Raman phonon spectra [69]. Experimental techniques have significantly enhanced our understanding of the vibrational properties of iron under high-pressure conditions. One such method, nuclear resonant inelastic X-ray scattering (NRIXS) [70], has been utilized to determine the phonon density of states (DOS) for BCC iron at ambient pressure [71], as well as for HCP iron at pressures up to 42 GPa [72]. Recent studies have been conducted to investigate the coupling mechanisms between magnons and phonons in bcc iron. These mechanisms have significant implications for the material’s magnetic properties and thermodynamic stability [50,73].

4.2. Effect of Magnetism on the Stability of BCC Iron

The reason we did not use spin-polarization in this study or considered disordered magnetic moments to generate our datasets is the computational cost. A typically sized academic allocation on a supercomputer is not enough to generate a large enough AIMD dataset with non-collinear magnetism, that additionally has large enough supercells and covers a large range of temperatures and pressures. For this reason, paramagnetic iron is often studied via non-magnetic iron simulations. Table 3 shows how every dataset used to train ML potentials [53,54,55,56] has constrains on its size, the supercell size, or the pressure or temperature range. For example, we used about a third of our group’s yearly allocation on Perlmutter, an Energy Research Scientific Computing Center (NERSC) supercomputer, to generate the dataset used for this work. Nevertheless, the main scientific conclusions of this work are expected to be valid due to the reasons discussed next. The non-collinear magnetic moments (a three-dimensional vector at each node) can be learned by the current implementation of the GPR-MGK by using them to populate the edge label matrix $\mathbf{E}$ (see Equation (3)) if such rich dataset were to be generated (perhaps in the future).

At 0 K, ferromagnetism stabilizes the BCC structure; in fact, iron would be HCP at standard temperature and pressure if not for ferromagnetism [74]. At low temperature, pressure eventually wins and Fe transforms from BCC to HCP [75]. Studies show that the individual atoms have a measurable magnetic moment in the HCP structure up to about 40 GPa, although without magnetic order [76]. Non-collinear magnetism might still exist at even higher pressures [77,78]. At zero pressure, the magnetic long-range order of the BCC phase collapses at the Curie temperature, but the short-range order remains [73] and the magnon–phonon interaction has small but measurable thermodynamic consequences in both the alpha [50] and delta phases [79]. The computational and experimental study of the magnetism of Fe at inner core conditions is challenging, but there is evidence that atoms in the BCC configuration maintain a magnetic moment [80]. If this is the case, all evidence points towards the magnetism mechanically and thermodynamically stabilizing the BCC phase, strengthening the main scientific conclusions of this work. This occurs at lower temperature and pressure in BCC iron [27] and in the analog system FeV [81].

4.3. Contributions to Scientific Machine Learning

ML models have significantly transformed material informatics, and scientific machine learning is being established as a field [82]; an overview of methodologies in solid-state materials research is in Ref. [83]. These models are capable of predicting interatomic interactions accurately and efficiently, thus closing the gap between computation speed and accuracy [6,84]. Gaussian approximation potential, a tool for generating interatomic potentials, has been used to gain a better understanding of magnon–phonon interactions in materials such as BCC iron [85]. The objective of the research presented here is to enhance the current understanding of the mechanical stability of BCC iron by using a machine learning-based approach. Our study integrates computational thermodynamics and machine learning algorithms to model the behavior of BCC iron under various thermodynamic conditions. The aim is to provide a comprehensive analysis that improves the prediction of its stability and properties, contributing to the broader field of materials science and engineering with potential implications for technological applications. This study utilizes machine learning techniques, specifically Gaussian Process Regression combined with a Molecular Graph Kernel, to model the mechanical stability of iron in inner and outer core-like environments. The objective is to improve the efficiency and accuracy of phase behavior predictions through data-driven approaches, providing insights into the thermodynamic properties and lattice dynamics. This methodology contributes to the understanding of planetary core structures and demonstrates how machine learning can aid in accelerating materials discovery in extreme environments.

Modeling the PES as a function of the atomic positions is critical for atomistic simulations, and high quality calculations have traditionally relied on quantum mechanical descriptions commonly based DFT. DFT remains computationally expensive, but ML potentials that learn the PES from data generated by DFT are now used for large and long dynamics simulations, from which the materials properties and thermodynamics can be extracted [86]. Popular frameworks used to learn interatomic potentials include the Gaussian approximation potential (GAP) [87,88], the moment tensor potential (MTP) [89,90] and the Spectral Neighbor Analysis Potential (SNAP) [7,91], among others. Here, we report on a framework based on Gaussian process regression performed on the similarity between molecular graphs as determined by the marginalized graph kernel (GPR-MGK) [9], applied to a crystalline material for the first time. It is more computationally expensive than the aforementioned models, but its predictions are of similar accuracy with smaller datasets. We tested the methodology on BCC iron at high pressure and temperature.

4.4. Contributions to the Understanding of the Inner Core of the Earth

Our results reinforce Luo et al.’s [26] conclusions regarding the high-temperature stability of BCC Fe. While their study relies on direct ab initio lattice dynamics calculations, our approach utilizes HELD to obtaining phonon dispersion and the findings are similar, emphasizing the role of anharmonic effects in stabilizing the BCC structure at extreme conditions. The agreement between these independent methodologies strengthens the case for considering BCC iron as a possible phase in the Earth’s inner core under sufficiently high temperatures. Our analysis presented in Table 1 not only supports the conclusions of Luo et al. but also provides further mechanistic insights into the stability of bcc-Fe under the conditions found in Earth’s core. The strong correlation between independent methodologies namely, ab initio lattice dynamics and machine learning-based modeling bolsters the hypothesis that bcc-Fe can exist as a stable phase at sufficiently high temperatures within the Earth’s inner core.

When force acting upon individual atoms in non-spin-polarized BCC Fe at 800 K as a function of displacement from their equilibrium position in the direction of each of their second and first nearest neighbors was approximated by numerical and analytical means, stability was only observed for the first nearest neighbor interactions of the system. This finding suggests that the instability in the second nearest neighbor interactions caused by the lack of a magnetic moment has a measurable effect on the overall instability of the system as found in the literature [77]. The analysis of the stability of BCC iron at different temperatures using GPR-MGK and nearest neighbor methods offers insights that both align with and expand upon existing knowledge in the field. Previous studies, such as those by [92,93], have extensively examined the behavior of iron under extreme conditions, especially its phase transitions and stability within Earth’s core. These studies highlight that as temperature and pressure increase, iron undergoes structural changes. Our findings contribute to this body of knowledge by providing a detailed stability map for BCC iron using advanced machine learning techniques.

The energy predictions, obtained through GPR-MGK, offer a detailed insight into the forces acting on iron atoms in their specific atomic surroundings. By combining our results with existing research, we provide a comprehensive view of the stability paths of BCC iron, which improves understanding of its behavior under conditions that are relevant to Earth’s core. The force–displacement analysis across the first, second, and third nearest-neighbors highlights three key findings. Temperature dependence shows that stability is maintained at higher temperatures (3700 K and 5000 K) but deteriorates significantly at low temperatures, particularly at 800 K and 1250 K. Directionality reveals that instability emerges earlier in specific crystallographic directions, suggesting a transformation path. Finally, the lattice parameter influence indicates that at 2.36 Å, the denser structure supports higher pressures, whereas the expanded lattice at 2.49 Å shows reduced stability. This understanding can guide future studies on Earth’s geodynamics and the creation of materials tailored to endure extreme conditions. At 2.36 Å, higher pressures are recorded, indicating a denser lattice structure, while at 2.49 Å, lower pressures reflect slight lattice expansion and reduced atomic forces. These results demonstrate that pressure increases consistently with temperature, aligning with thermodynamic expectations, as summarized in Table 2.

5. Conclusions

We performed a Gaussian process regression to analyze the similarity of mathematical graphs representing cubic iron with thermal displacements at high pressures and temperatures. This analysis helped us predict the energy of arbitrary configurations. These energy predictions were then used to investigate the forces experienced by the iron atoms in their local atomic environments, helping us determine the instability path of iron under Earth’s core conditions. Stability is maintained at higher temperatures but degrades significantly at lower temperatures. The degree of instability depends on the coordination shell, emerging earlier along certain nearest neighbors, and becoming more pronounced at higher pressures. A comparison between lattice parameters 2.36 Å and 2.49 Å reveals that the denser structure supports higher pressures, whereas the expanded lattice exhibits reduced stability. These insights contribute to understanding the behavior of BCC iron under inner (and outer) core-like thermal conditions and provide a foundation for future computational and experimental studies. A comparison with Luo et al. [26] reinforces our findings. Their study shows that BCC Fe remains dynamically unstable below 3000 K but stabilizes at higher temperatures, aligning with our results. Both studies indicate that increased pressure requires higher temperatures for BCC stability, with Luo et al. predicting stability above 4500 K at 380 GPa, consistent with our transition at 3700 K and beyond. Additionally, our force–displacement analysis quantifies mechanical instability at the atomic level, enhancing our understanding of anharmonic effects. This alignment supports the consideration of BCC Fe as a viable phase in Earth’s inner core at high temperature, certainly mechanically and perhaps thermodynamically.