Advances in high-pressure materials discovery enabled by machine learning Open Access

Pressure, as a fundamental thermodynamic variable, exerts a profound influence on interatomic distances, electronic configurations, and bonding patterns in materials. These effects can induce phase transitions and unlock a wide range of intriguing properties that are often unattainable under ambient conditions.¹ The investigation of materials under compression not only advances our understanding of fundamental condensed matter physics, but also provides essential insights into the behavior of planetary interiors and supports the development of functional materials, including superhard, catalytic, optoelectronic, thermoelectric, and superconducting materials.^1–3 A notable example is diamond, which, synthesized under high-pressure and high-temperature conditions, remains stable at ambient pressure and exhibits exceptional hardness.⁴ Another significant case is the theoretical prediction and subsequent experimental discovery of the high-temperature superconducting hydride LaH₁₀, which exhibits a record-breaking critical temperature of 260 K under high-pressure conditions.^5–8 

High pressure in experiments can be generated using static methods, such as diamond anvil cells (DACs) and large-volume presses, or dynamic approaches, such as shock-wave facilities. However, these techniques are subject to limitations, including the small sample volumes of DACs and the transient nature of dynamic compression, both of which complicate data collection and interpretation. In this context, crystal structure prediction (CSP) methods play a pivotal role in complementing experimental studies. They not only assist in interpreting experimental measurements and resolving complex crystal structures, but also enable efficient predictions of high-pressure phases with intriguing properties, thereby providing valuable guidance for experimental exploration.

CSP seeks to determine the most stable atomic arrangement within a crystal for a given chemical composition and set of external conditions, such as pressure. From a thermodynamic perspective, this problem can be formulated as a global optimization task on a high-dimensional energy surface, where the goal is to identify the configuration corresponding to the lowest thermodynamic potential. In recent decades, significant progress has been achieved in addressing the computational challenges associated with CSP through the development of advanced methodologies and software packages. Prominent examples include AIRSS,⁹ basin hopping,¹⁰ CALYPSO,^11,12 minima hopping,¹³ USPEX,¹⁴ MAGUS,^15,16 and XtalOpt¹⁷ (see Ref. 18 for a more comprehensive list). These methods combine first-principles structural relaxation—most commonly performed using density functional theory (DFT)—with advanced sampling techniques for exploring the potential energy surface (PES) efficiently. They employ diverse strategies, such as random structure generation, molecular dynamics, evolutionary algorithms, and swarm intelligence. These advances have enabled significant theory-driven breakthroughs,¹⁸ including the solution of the complex crystal structures of alkali metals^19–21 and the discovery of superhydride superconductors with remarkable critical temperatures.^5–8,22,23

Despite significant advances, CSP remains fundamentally constrained by its NP-hard nature, since the computational complexity increases exponentially with system size. This inherent limitation highlights the need to address two critical challenges: efficient energy evaluation and effective PES sampling. Continued innovation in these areas is crucial for advancing CSP methodologies and enabling the prediction of increasingly complex crystal structures.

In recent years, machine learning (ML), as a rapidly evolving data-driven approach, has demonstrated exceptional capabilities in regression, pattern recognition, and generative tasks across diverse data types, including natural language²⁴ and images.²⁵ These techniques have increasingly become integral to computational physics, chemistry, and materials science, offering a promising route to address the aforementioned challenges in CSP.^26–30 On the one hand, advances in supervised learning have led to the development of machine learning potentials (MLPs), which enable efficient and accurate evaluation of energies and forces, significantly accelerating atomic-scale simulations. On the other hand, the emergence of generative models (GMs) has introduced powerful tools for efficiently sampling the vast crystal configuration space, providing novel approaches for PES exploration and optimization.

In this review, we provide a brief overview of recent advances in CSP methodologies facilitated by ML techniques, with particular emphasis on MLPs and GMs. A schematic representation of the typical methods and their interrelations is provided in Fig. 1. We examine their applications to the discovery of high-pressure materials and critically analyze the associated challenges and opportunities within this rapidly evolving domain.

MLPs are among the most promising approaches for accelerating quantum-mechanical-based calculations, enabling high-throughput, large-scale, and long-timescale atomic simulations with significantly reduced computational cost.³¹ In most MLP frameworks, the total energy of a system is expressed as the sum of the energies of its constituent atoms, where each atomic energy is determined by its local chemical environment.³² This localized representation allows MLPs to efficiently model the complex interactions within a system while maintaining computational efficiency. By leveraging the superior learning capacity of neural networks to approximate high-dimensional complex functions, MLPs are trained on data generated from high-fidelity quantum mechanical calculations, such as DFT. As a result, MLPs are designed to reproduce the accuracy of these quantum mechanical methods while achieving computational speeds that are orders of magnitude faster.

The first MLP applied to crystalline systems was the Behler–Parrinello neural network (BPNN).³² In this approach, the local atomic environment is represented using hand-crafted descriptors known as atom-centered symmetry functions (ACSFs).³² ACSFs capture two- and three-body interactions by projecting the atomic environment into a set of basis functions, effectively encoding the local geometry and chemical surroundings of each atom. Using ACSFs as input, a feedforward fully connected neural network is employed to regress the atomic energies, which are then summed to obtain the total energy of the system. Subsequently, many MLPs based on carefully devised descriptors—referred to as descriptor-based models—have been developed, including Gaussian approximation potentials (GAPs),³³ moment tensor potentials (MTPs),³⁴ and spectral neighbor analysis potentials (SNAPs),³⁵ among others.^36,37 These models offer robust performances for a wide range of applications within the scope of their designed capabilities. Among descriptor-based approaches, the Deep Potential (DP) model³⁸ introduced a significant innovation by employing deep neural networks to learn more flexible and expressive representations of atomic environments during the training process.

Meanwhile, graph neural networks (GNNs)³⁹ have emerged as powerful deep learning techniques for modeling interatomic potentials, in what are referred to as graph-based models. In this framework, a crystal structure is represented as a graph, with atoms treated as nodes and the interactions between them as edges. These graph-based models typically incorporate two-body (distances),^40–44 three-body (angles),^45–48 or even higher-order^49,50 information. In the manner of message passing neural networks (MPNNs),⁵¹ messages between atoms are updated through multiple layers of the networks, and finally the regression target, typically the energy, is read out. A key factor contributing to the success of MLPs is their ability to exploit the inherent symmetries of crystals, including permutation, rotation, and translation invariance. While invariance under translation and rotation can be easily achieved by feeding invariant features (e.g., relative distance), some graph-based MLPs utilizing equivariant GNNs have been proposed. These equivariant models typically use specially designed architectures^42,52 or Wigner-D matrices, spherical harmonics, and tensor product methods^{44,49,50,53–55} to preserve equivariance under rotation and invariance under translation. The inclusion of high-order tensors and equivariance in models can indeed improve data efficiency and generalization and result in a better learning curve.^44,50 Furthermore, integrating transformer models with graph representations of structure also provides a promising route to the modeling of interatomic potentials. Equiformer/EquiformerV2^56,57 uses graph neural networks leveraging the strength of transformer architectures, while incorporating an equivariant graph attention mechanism to model interatomic potentials. Similar approaches that combine attention mechanisms with GNNs are also common in the literature.^58–61 

Initially, MLPs were typically developed and trained from scratch for each specific system under investigation. This approach inherently restricted their generalizability and significantly limited their applicability to broader materials discovery and exploration. To overcome these limitations, Takamoto et al.⁶² introduced a universal MLP, PreFerred Potential (PFP), capable of handling combinations of 45 elements. PFP has demonstrated promising performance in diverse applications, including modeling lithium diffusion in LiFeSO₄F and phase transitions in Cu–Au alloys. Similarly, M3GNet,⁴⁷ a model based on GNNs incorporating three-body interactions, was developed and trained on the Materials Project⁶³ database, which spans 89 elements in the Periodic Table. M3GNet successfully identified 1578 stable structures after screening over 31 × 10⁶ candidates, highlighting its effectiveness in large-scale materials discovery. Furthermore, CHGNet,⁴⁸ another universal MLP based on graph neural networks and trained on the Materials Project Trajectory (MPtrj) dataset,⁶⁴ extended the predictive capabilities of MLPs by accurately predicting energies, forces, stresses, and magnetic moments. In parallel, DPA-2⁶⁵ introduced a pretraining–fine-tuning approach to develop a universal MLP, in which a universal descriptor was first trained on a diverse dataset calculated at different levels of theory and then fine-tuned for specific chemical systems of interest. MatterSim⁶⁶ further advanced the development of universal MLPs by training on a highly diverse dataset that incorporated not only a wide range of chemical species, but also varying pressure and temperature conditions, thereby broadening its applicability. Other notable efforts include MACE,^50,67 GNoME-NequIP,⁶⁸ EquiformerV2-OMat24,⁶⁹ Orb,⁷⁰ SevenNet,⁷¹ and grACE.⁷² Collectively, these advances represent significant strides toward the realization of universal MLPs, offering transformative potential for comprehensive materials discovery, design, and exploration.

Given the optimal trade-off between accuracy and computational efficiency provided by MLPs, leveraging a well-trained MLP specifically tailored to a target system provides a practical and effective method for accelerating structure prediction by replacing DFT in energy evaluations. For instance, Deringer et al.⁷³ combined a GAP, trained on liquid and amorphous structures, with the AIRSS method, successfully discovering several new carbon allotropes. Similarly, Wang et al.⁷⁴ integrated the DP model with the CALYPSO method to conduct structure searches on binary Al–Mg alloy systems, uncovering six previously unreported metastable structures. By combining an MLP with the stochastic surface walking global optimization method,^75,76 Huang et al.⁷⁷ proposed several competitive candidates after exploring the PES of β-B, and revealed the key principles governing the filling of interstitial sites. These examples⁷⁸ highlight the growing potential of MLPs in accelerating and enhancing CSP across a wide range of material systems.

MLPs have demonstrated notable accuracy and generalization, but often suffer from significant prediction uncertainty in regions of the PES beyond the training data domain. This limitation is particularly critical for CSP tasks, which require extensive PES exploration to identify low-energy structures.⁷⁸ Integrating MLPs with CSP in an active learning (AL) framework offers a promising solution. Specifically, an AL strategy iteratively performs configuration extrapolation detection, DFT labeling, training dataset refinement, and MLP updates, thereby improving the reliability of MLP predictions and accelerating the CSP process.

In this context, Deringer et al.^79,80 combined a GAP with AIRSS⁹ in an AL framework. Starting with DFT calculations of randomly generated structures to train an initial GAP, they iteratively refined the model by using the GAP to relax new structures and incorporating selected relaxed structures into the training dataset. Enhancements, such as selecting structures on the basis of physical constraints, have enabled successful CSP applications to systems such as phosphorus, carbon, silicon, and titanium.^80,81 Tong et al.⁸² identified a novel core–shell-type boron B₈₄ cluster using an active learning scheme combining an ACSF-based GAP with CALYPSO.^11,12 Similarly, Yang et al.⁸³ iteratively constructed an ACSF-based GAP with CALYPSO, discovering a new boron phase, c-B₂₄, composed of B₆ octahedra, B₂ pairs, and B₁₂ icosahedra, which exhibits a superconducting critical temperature of 13.8 K at ambient pressure. Podryabinkin and Shapeev⁸⁴ developed an AL framework for MTPs using the D-optimality criterion for efficient configuration selection, and Podryabinkin et al.⁸⁵ subsequently integrated this with USPEX¹⁴ to accelerate CSP. This method successfully predicted crystal structures of carbon, high-pressure sodium phases, and boron allotropes, including a previously unknown 54-atom boron structure,⁸⁵ with minimal computational cost.⁸⁶ Cheng et al.⁸⁷ used a GNN to model the relationship between crystal structures and formation enthalpies, integrated three optimization algorithms to enhance the efficiency of searching for the structure with the lowest formation enthalpy, and updated the GNN prediction model using AL. By identifying crystal structures of 29 compounds, they showcased the potential of this approach to accelerate materials discovery.

The effectiveness of AL approaches is particularly evident in the discovery of novel high-pressure phases. Pickard⁸⁸ introduced the ephemeral data-driven potentials (EDDP) framework, a methodology designed to dynamically construct and refine interatomic potentials using minimal, on-the-fly generated data. A previously unreported high-pressure silane (SiH₄) phase with 12 formula units in the primitive cell was discovered through the application of EDDP and was determined to be thermodynamically stable at ∼300 GPa. Moreover, the EDDP framework facilitated the revisitation and identification of a series of stable scandium hydride phases at 300 GPa,⁸⁹ thus demonstrating its efficacy in exploring high-pressure phase stability and uncovering novel structures. Sun and co-workers used MAGUS^15,16 in combination with AL to predict a series of new compounds under planetary interior conditions, primarily high pressure and high temperature. They successfully predicted a novel ground state of silica at pressures between 645 and 890 GPa, featuring six-, eight-, and nine-coordinated silicon atoms.⁹⁰ Additionally, they identified several stable magnesium oxide–water compounds within the pressure range of 270–600 GPa, which could potentially serve as a water reservoir on the early Earth.⁹¹ By exploring the phase diagram of the lithium–aluminum system from 0 to 150 GPa, they revealed a pressure-stabilized compound exhibiting both superconducting and superionic behaviors.⁹² These findings underscore the power of MLPs in high-pressure CSP.

Wang et al.⁹³ employed an AL scheme to construct a DP model suitable for lithium at high pressure. Using this model, they explored over 600 000 crystal structures at 50 and 60 GPa, identifying four candidate structures for previously unknown crystal phases located at the minima of the melting curve. Among these, two structures, aP160 and oP192, containing more than 150 atoms in the unit cell, with P1 and Pcc2 space groups respectively, were discovered.⁹³ These findings underscore the predictive power of global structure search methods when combined with accurate ML potentials for uncovering complex crystal structures. Furthermore, Wang et al.⁹⁴ combined CALYPSO with DP in an AL manner. They revisited the complex ternary superhydride compound Li–La–H, a material with potential for high-temperature superconductivity, at 300 GPa. After exploring 375 000 DP-based and 1475 DFT-based local minima structures, a new phase, Cmcm-Li₂La₂H₂₃,⁹⁴ was revealed. This phase is thermodynamically stable at 300 GPa and exhibits a critical temperature of ∼130 K. This approach reduced the computational cost by two orders of magnitude compared with traditional DFT-based CSP methods.

Additionally, MLPs have demonstrated considerable efficiency, making them suitable for large-scale simulations. The focus of the present paper is on CSP, and therefore a more detailed discussion of MLPs is beyond its scope. However, comprehensive reviews on this topic can be found in the literature.^27,31,95,96

Unsupervised learning GMs have demonstrated a significant capacity to approximate the underlying probability distributions of complex datasets, enabling the generation of novel data instances through sampling from these learned distributions. Traditionally, CSP methods have relied primarily on techniques such as Monte Carlo simulations, molecular dynamics, and global optimization algorithms to explore the PES and generate new structures. However, the recent emergence of comprehensive materials databases, encompassing both structural and property information, has facilitated the integration of unsupervised learning algorithms into materials design and CSP. This integration represents a paradigm shift in CSP methodologies, offering a complementary and often more efficient approach to exploring the vast configuration space. Prominent examples of GMs in ML include generative adversarial networks (GANs), variational aautoencoders (VAEs), diffusion models, flow models, and autoregressive models.

The applications of GMs to crystal structures began with descriptor-based rough/compressed representation of structures,^97–111 mostly utilizing GANs and VAEs. For instance, CrystalGAN⁹⁷ employed a 2D raw coordinate-based representation and trained a GAN on a dataset of binary hydrides to generate ternary hydrides. Hoffmann et al.⁹⁹ used a voxel representation of structures, akin to image generation techniques, employing a VAE/U-Net framework to generate voxelized structure images. Compression of crystal structure representations inevitably results in a loss of information, and therefore additional efforts are required for these methods to reconstruct structures from their representations. Voxel representation approaches^{99–102,106,109} usually employ object detection techniques to ascertain atomic positions, whereas fingerprint representation methods^{98,104,112,113} often rely on composition substitution or random generation to identify structures that match the desired fingerprint. Furthermore, neglecting the fundamental permutation, rotation, and translation invariance that are inherent in crystal structures could shorten a model’s expressive capacity. In addition, the restriction of training data to specific atomic species or lattice types constrains the ability of these models to accommodate diverse and realistic crystalline datasets.

The advent of invariant/equivalent GNNs marked a significant breakthrough in this field, particularly with the introduction of CDVAE,¹¹⁴ which utilized crystal graph representations within a VAE/diffusion framework. This has led to a series of methodologies capable of learning from comprehensive elemental datasets and efficiently generating high-quality crystalline structures.^115–125 For example, DiffCSP¹¹⁸ employs a periodic-E(3)-equivariant denoising model to generate structures under a fractional coordinate system, and FlowMM¹²¹ adopts a generalized Riemannian flow matching technique for crystal structure. The integration of symmetry-aware structure generation into graph-based models is also a crucial development.^119,126,127 CHGFlowNet¹²⁶ achieves this by using a sequential mechanism to generate symmetry-nonequivalent atoms one-by-one, and DiffCSP++¹¹⁹ enforces symmetry constraints during the diffusion process to ensure that the generated structures retain the symmetry of the input configuration. Furthermore, MatterGen,¹²⁰ Con-CDVAE,¹²⁸ Cond-CDVAE,¹²² and CrystalFlow¹²³ integrate conditional generation techniques into these graph-based models to generate structures under targeted external conditions or material properties, such as space group label, magnetic density, bandgap, bulk modulus, formation energy, and pressure, significantly advancing the fields of CSP and inverse material design.

In parallel, inspired by advances in natural language processing and large language models (LLMs), another class of LLM-based models utilizing character-type representation has rapidly emerged. These methods employ sequential tokenization of crystal structures and utilize an autoregressive mechanism—typically recurrent neural networks (RNNs) or transformer models—to achieve dialogue-oriented generation schemes.^129–140 For example, CrystalLLM¹³⁰ utilizes a LLM to generate the texts of crystallographic information files (CIFs), and CrystalFormer¹³⁵ utilizes a transformer model to generate structures by a sequence of symmetric Wyckoff sites. Such approaches are well suited for facilitating the integration of multimodal learning strategies, encompassing literature review and material synthesis path retrieval, thus demonstrating a promising direction for new material design.

As a data-driven approach, the development of GMs capable of producing crystal structures under specified high-pressure conditions necessitates a comprehensive high-pressure crystal structure dataset. In this context, a CALYPSO high-pressure dataset containing more than 657 000 structures and mainly covering pressures between 0 and 300 GPa has been curated, leading to the development of two models, Cond-CDVAE¹²² and CrystalFlow.¹²³ Cond-CDVAE employs a diffusion process to produce promising structures, while CrystalFlow utilizes a flow model architecture. Both models adopt a conditional generation approach that leverages pressure as a conditioning variable. It has been shown that with Cond-CDVAE, significantly less effort is required to identity the stable high-pressure phases of lithium, boron, and silica compared with traditional CSP methods. Meanwhile, CrystalFlow exhibits enhanced capabilities in generating local stable structures under high-pressure conditions. Despite the shared mathematical formulation of diffusion and flow models,^141,142 diffusion models benefit from the accumulated results of their application in other fields, particularly computer vision,¹⁴³ and thus at present offer a more mature approach compared with flow models. However, research on flow models is rapidly advancing.¹⁴⁴ Overall, these advances represent a substantial step forward in the data-driven exploration of crystal structures under extreme conditions, paving the way for future research in materials science.

The exploration of materials under high-pressure conditions is pivotal for understanding fundamental condensed matter physics, planetary interiors, and the development of functional materials with exceptional properties. In this review, we have highlighted the important role of CSP methodologies, particularly those enhanced by ML techniques, in addressing the challenges faced by traditional computational techniques. By leveraging advances in MLPs and GMs, researchers have made significant strides in accelerating the discovery of complex crystal structures and uncovering novel high-pressure phases.

Looking ahead, several key opportunities and challenges define the future trajectory of this rapidly evolving field. First, the development of universal and transferable MLPs that can accurately model diverse chemical systems across varying pressure and temperature conditions remains a critical goal. Recent progress in AL frameworks and pretraining–fine-tuning strategies has demonstrated the potential to enhance MLP generalizability, but further efforts are needed to expand these models to encompass more elements, complex bonding environments, and extreme conditions.

Second, generative models offer a promising avenue for efficient exploration of the vast configuration space of crystal structures. However, to fully realize their potential, future work must address challenges such as ensuring physical validity, incorporating symmetry constraints, and integrating property-driven condition generation. The incorporation of advanced techniques, such as equivariant neural networks and graph-based representations, has already shown promise in generating high-quality structures, but continued innovation is required to improve data efficiency, scalability, and the ability to target specific material properties.

Finally, as the field progresses, interdisciplinary collaboration will play a crucial role in addressing the multifaceted challenges associated with high-pressure materials discovery. Advances in computational physics and ML techniques must be synergistically combined to push the boundaries of what is achievable. Moreover, the establishment of open-access datasets, standardized benchmarks, and community-driven initiatives will be essential for fostering innovation and accelerating progress.

The work is supported by the National Key Research and Development Program of China (Grant No. 2022YFA1402304), the National Natural Science Foundation of China (Grant Nos. 12034009, 12374005, 52288102, 52090024, and T2225013), the Fundamental Research Funds for the Central Universities, and the Program for JLU Science and Technology Innovative Research Team.

The authors have no conflicts to disclose.

Zhenyu Wang: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Xiaoshan Luo: Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Qingchang Wang: Investigation (equal); Writing – review & editing (equal). Heng Ge: Investigation (equal); Writing – review & editing (equal). Pengyue Gao: Investigation (equal); Writing – review & editing (equal). Wei Zhang: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). Jian Lv: Conceptualization (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Yanchao Wang: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.