Core Under Pressure: Machine Learning Illuminates Earth’s Inner Furnace

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Author: Denis Avetisyan

A new physics-informed machine learning approach is unlocking the secrets of iron’s behavior at the extreme temperatures and pressures of Earth’s core.

The predicted melting curve of iron, derived from two-phase simulations, aligns with and refines existing data from diverse experimental studies [emcAnzellini2013,emcSinmyo2019,emcLi2020,emcKraus2022,emcBalugani2024] and computational investigations [aimcAlfe2009,aimcGonzalez2023,aimcSun2022,aimcSun2023,aimcBelonoshko2021,aimcStixrude2014,aimcAlfe2002,aimcBelonosko2000,aimcWu2024,aimcSola2009], demonstrating the model’s capacity to synthesize and potentially transcend established knowledge of iron’s behavior under extreme conditions.
The predicted melting curve of iron, derived from two-phase simulations, aligns with and refines existing data from diverse experimental studies [emcAnzellini2013,emcSinmyo2019,emcLi2020,emcKraus2022,emcBalugani2024] and computational investigations [aimcAlfe2009,aimcGonzalez2023,aimcSun2022,aimcSun2023,aimcBelonoshko2021,aimcStixrude2014,aimcAlfe2002,aimcBelonosko2000,aimcWu2024,aimcSola2009], demonstrating the model’s capacity to synthesize and potentially transcend established knowledge of iron’s behavior under extreme conditions.

Researchers have combined density functional theory with dynamical mean-field theory and machine learning to accurately predict the iron melting curve at Earth’s core conditions, resolving long-standing discrepancies.

Constraining the behavior of iron at Earth’s core conditions remains a significant challenge due to the computational cost of accurately modeling strongly correlated electronic effects. This study, detailed in ‘Melting curve of correlated iron at Earth’s core conditions from machine-learned DFT+DMFT’, introduces a machine-learning accelerated density functional theory plus dynamical mean-field theory (DFT+DMFT) approach to efficiently calculate iron’s melting curve. By training equivariant graph neural networks to predict key electronic structure parameters, we achieve a substantial reduction in computational effort and determine a melting temperature of 6225 K at 330 GPa. Will this physics-informed machine learning framework unlock new insights into the composition and dynamics of Earth’s deep interior?

The Illusion of Order: Exploring Correlated Electrons

Certain materials, dubbed correlated electron systems, defy conventional understanding of how electrons behave, giving rise to remarkable and often unexpected phenomena. Unlike traditional metals where electron interactions are minimal, these systems exhibit strong correlations – electrons powerfully influence each other’s movements – leading to properties such as high-temperature superconductivity, where electricity flows with zero resistance at relatively warm temperatures, and unconventional magnetism, where magnetic order doesn’t align with established patterns. These behaviors fundamentally challenge the established framework of condensed matter physics, which often treats electrons as independent particles, and necessitate new theoretical approaches to accurately describe and predict the materials’ characteristics. The exploration of these correlated systems promises not only a deeper understanding of quantum mechanics but also the potential for revolutionary technologies based on these exotic properties.

The pursuit of materials with novel functionalities hinges on a deep understanding of correlated electron systems, yet these very systems present a formidable challenge to materials scientists and computational physicists. Their complex interplay of electron interactions defies simple modeling, demanding computational resources that often strain even the most powerful supercomputers. Accurately predicting a material’s behavior requires simulating the collective quantum effects of countless electrons, a task exponentially more difficult with increasing system size. This computational bottleneck hinders the rational design of materials with tailored properties – such as zero electrical resistance or enhanced magnetic responsiveness – necessitating innovative algorithms and approximation techniques to bridge the gap between theoretical prediction and practical material discovery.

The precise theoretical description of correlated electron systems hinges on accurately determining the ‘self-energy’, a quantity encapsulating the complex, frequency-dependent interactions between electrons and the material’s lattice. This isn’t a simple task; the self-energy isn’t a fixed value, but rather a function that changes with the energy of the electron, demanding intensive computation to map its behavior across all relevant frequencies. Consequently, many simulations rely on approximations to the self-energy, simplifying the interactions to make calculations tractable. However, these simplifications can obscure crucial details, potentially leading to inaccurate predictions of material properties. Researchers are actively developing more sophisticated computational techniques – including dynamical mean-field theory and quantum Monte Carlo methods – to directly calculate the frequency-dependent self-energy, pushing the boundaries of what’s computationally feasible in the quest to understand these enigmatic materials and unlock their potential for technological innovation.

Self-energy predictions from <span class="katex-eq" data-katex-display="false"> ext{DFT+DMFT}</span> iterations accurately converge for Fe, FeO, and NiO, as demonstrated by the agreement between predicted (red and blue) and calculated (black) real and imaginary components for representative structures.
Self-energy predictions from ext{DFT+DMFT} iterations accurately converge for Fe, FeO, and NiO, as demonstrated by the agreement between predicted (red and blue) and calculated (black) real and imaginary components for representative structures.

The Limits of Calculation: Dynamical Mean-Field Theory

Dynamical Mean-Field Theory (DMFT) provides a method for addressing the many-body problem in condensed matter physics, particularly for strongly correlated electron systems where traditional band structure approaches fail. However, the computational expense of DMFT scales significantly with both the number of lattice sites N and the required frequency resolution ω. This arises from the need to solve an effective impurity problem self-consistently at each lattice site and for each frequency. Specifically, the computational cost typically scales as O(N^2 \omega) or higher, depending on the specific implementation and approximations employed. This rapid scaling limits the applicability of DMFT to relatively small systems or requires substantial computational resources for larger simulations, necessitating the development of efficient algorithms and approximation schemes.

Accurate calculation of the frequency-dependent self-energy \Sigma(\omega) is central to the Dynamical Mean-Field Theory (DMFT) approach, but represents a significant computational bottleneck. DMFT maps the lattice problem onto an effective impurity problem that must be solved self-consistently for each frequency. Directly representing \Sigma(\omega) across the entire frequency range requires substantial computational resources. Consequently, approximations are often employed, including the simplification of the high-frequency behavior. The static high-frequency limit assumes that \Sigma(\omega) approaches a constant value for sufficiently large ω, effectively reducing the number of frequency points requiring explicit calculation and thereby decreasing computational cost. While this approximation can introduce inaccuracies, it provides a computationally tractable pathway for applying DMFT to larger systems and more complex materials.

Legendre coefficients offer a compact representation of the self-energy \Sigma(i\omega_n) by expanding it as a series of Legendre polynomials. This approach reduces the computational cost associated with storing and manipulating the frequency-dependent self-energy compared to direct discretization, as only a finite number of coefficients are required to achieve a specified accuracy. However, despite this compression, accurately representing the self-energy to the level necessary for quantitative predictions still demands significant computational resources. The number of Legendre coefficients needed scales with both the frequency range of interest and the desired accuracy, and the iterative solution of the DMFT equations involving these coefficients remains computationally intensive, particularly for larger systems or lower temperatures where correlation effects are stronger.

The Illusion of Efficiency: Machine Learning as a Tool

The computational cost of Dynamical Mean-Field Theory (DMFT) is often dominated by the repeated solution of the impurity problem, known as the impurity solver. Machine learning techniques are being explored to address this bottleneck by providing approximations to the impurity solver. These machine learning-based surrogate models learn the mapping between the input parameters defining the effective impurity problem and the resulting self-energy, circumventing the need for explicit iterative solution of the impurity model. This approach offers the potential for significant computational savings, as the machine learning model can provide predictions much faster than traditional iterative methods, without substantial loss of accuracy in the DMFT calculation.

The computational expense of Dynamical Mean-Field Theory (DMFT) is often dominated by solving the impurity problem at each DMFT self-consistency loop. Machine learning-based surrogate impurity solvers offer a means of significantly reducing this cost. Benchmarking across iron (Fe), iron oxide (FeO), and nickel oxide (NiO) systems demonstrates a consistent performance gain, with these surrogate models achieving a 2-4x speedup compared to traditional iterative impurity solvers. This acceleration is achieved by training the machine learning model to accurately predict the local self-energy, effectively bypassing the need for computationally intensive iterative solutions at each DMFT cycle.

E(3)-Equivariant Graph Neural Networks (EGNNs) represent a class of machine learning models specifically constructed to adhere to the rotational and translational symmetries inherent in three-dimensional space. This architectural constraint is crucial for accurately predicting the self-energy in Dynamical Mean-Field Theory (DMFT) calculations, as the self-energy is a complex quantity dependent on momentum space which exhibits these symmetries. By incorporating these symmetries directly into the network architecture, EGNNs require fewer parameters and less training data to achieve accurate predictions compared to standard neural networks. This leads to improved generalization performance and reduced computational cost when used as surrogate impurity solvers within DMFT, effectively capturing the relevant physics of the system while maintaining symmetry consistency.

Beyond the Horizon: Simulating Planetary Interiors

The Earth’s core, a realm of intense pressure and heat, profoundly influences planetary processes like the generation of the magnetic field and the dynamics of plate tectonics. Comprehending these forces necessitates accurately simulating the behavior of iron, the core’s primary constituent, under extreme conditions-reaching temperatures exceeding 5000 Kelvin and pressures exceeding 300 Gigapascals. Traditional computational methods struggle with the scale and complexity of modeling atomic interactions at such extremes, leading to uncertainties in predicted core properties. Therefore, developing robust and precise models of iron’s behavior is not merely a materials science challenge, but a fundamental step towards unraveling the mysteries of Earth’s deep interior and, by extension, understanding the evolution and characteristics of other terrestrial planets.

Advancements in computational materials science now leverage Machine-Learned Interatomic Potentials (MLIPs) to overcome the limitations of traditional methods when simulating the Earth’s core. These MLIPs, constructed using frameworks such as NequIP, effectively learn the complex relationships governing atomic interactions from high-fidelity quantum mechanical calculations. This allows researchers to perform efficient molecular dynamics simulations – capturing the movement of atoms over time – under the extreme pressure and temperature conditions found deep within planets. Unlike computationally expensive ab initio methods, MLIPs dramatically reduce the processing demands, enabling simulations of larger systems and longer timescales – crucial for understanding phenomena like iron melting and the generation of Earth’s magnetic field. The resulting potential energy surfaces accurately represent the behavior of iron at core conditions, offering a pathway to explore a wider range of planetary interiors and materials under extreme conditions.

The accuracy of materials simulations hinges on the quality of the interatomic potentials used to describe atomic interactions. Recently developed machine-learned interatomic potentials (MLIPs) demonstrate exceptional performance in modeling iron under extreme conditions. Specifically, these potentials achieve a test-set mean absolute error of only 69.2 \text{ meV/atom} for predicting the energy of atomic configurations and 76.7 \text{ meV/Å} for the forces between atoms. These remarkably low errors – representing a significant improvement over previous models – indicate the MLIPs can reliably capture the complex behavior of iron at the immense pressures and temperatures found within planetary cores, paving the way for more accurate simulations of Earth’s interior and other high-pressure environments.

Recent simulations, powered by machine-learned interatomic potentials, have successfully predicted the melting temperature of iron under conditions mirroring Earth’s core. These calculations, conducted at an immense pressure of 330 GPa, estimate the melting point to be 6225 K. This result is remarkably consistent with direct experimental measurements – notably, the 6230±500 K value obtained by Anzellini and colleagues, and corroborating data from Kraus and collaborators. The close agreement between simulation and experiment validates both the accuracy of the machine-learning approach and the underlying physics used to model iron at extreme conditions, offering new confidence in understanding the behavior of planetary interiors and solid-liquid coexistence at pressures previously inaccessible to direct observation.

At 307.8 GPa, the density profile at 20 ps shows liquid-liquid coexistence interfaces flanking a region of stable temperature after approximately 15 ps, as evidenced by the inset atomic configuration and temperature variation.
At 307.8 GPa, the density profile at 20 ps shows liquid-liquid coexistence interfaces flanking a region of stable temperature after approximately 15 ps, as evidenced by the inset atomic configuration and temperature variation.

The study meticulously constructs a pathway to navigate the complexities of density functional theory coupled with dynamical mean-field theory (DFT+DMFT), a formidable challenge in condensed matter physics. This work, leveraging machine learning techniques and E(3)-equivariant neural networks, aims to establish a more accurate understanding of material behavior under extreme conditions. As Albert Camus observed, “The struggle itself… is enough to fill a man’s heart. One must imagine Sisyphus happy.” The relentless pursuit of accurate modeling, even when faced with the computational demands of DFT+DMFT and the inherent limitations in extrapolating to Earth’s core conditions, mirrors this Sisyphean task. The researchers, by addressing discrepancies between simulation and experiment concerning the iron melting curve, demonstrate a commitment to pushing the boundaries of knowledge, finding meaning in the continuous refinement of theoretical frameworks.

What Remains to be Seen

This work, like all attempts to model the planet’s interior, arrives at a provisional truth. The acceleration of DFT+DMFT calculations via machine learning is, in itself, a familiar story – a faster path to the same eventual horizon. It resolves discrepancies, yes, but each resolved detail merely sharpens the questions that remain. The iron melting curve, predicted with greater precision, doesn’t suddenly reveal the core; it defines the boundary of what is currently unknowable, given the tools at hand.

Future iterations will undoubtedly refine the machine learning architectures, perhaps incorporating more sophisticated physics priors. Legendre polynomials and E(3)-equivariance are elegant, but elegance doesn’t guarantee correspondence with reality. A more pressing limitation lies in the inherent approximations within DFT+DMFT itself – the very foundation upon which these accelerated calculations rest. Every theory is just light that hasn’t yet vanished, and this one, however bright, will eventually encounter data that demands reconsideration.

The true challenge isn’t simply achieving greater computational speed, but acknowledging the fundamental limits of prediction. The Earth’s core isn’t yielding its secrets; it’s holding a mirror to the hubris of those who seek to decipher them. Models exist until they collide with data, and the deeper one probes, the more inevitable that collision becomes.

Original article: https://arxiv.org/pdf/2512.25061.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-05 00:23