OpenCSP: High-Pressure Crystal Structure Prediction

• OpenCSP is a machine learning framework for crystal structure prediction that integrates a pressure-resolved dataset and uncertainty-guided sampling to enhance stress accuracy under compression.
• It employs a dual graph architecture—combining atom–bond and bond–angle graphs—to jointly predict energy, forces, and the virial tensor for reliable high-pressure modeling.
• The framework’s pressure-aware data acquisition and efficient DFT labeling yield superior CSP performance, enabling precise pressure-controlled relaxations from 0 to 200 GPa.

Searching arXiv for the OpenCSP paper and closely related benchmark models to ground the article in current literature. OpenCSP is a machine learning framework for crystal structure prediction (CSP) spanning ambient to high-pressure conditions. It was introduced to address a central limitation of many large atomistic models: training on near-ambient, equilibrium data degrades stress accuracy at tens to hundreds of gigapascals and leaves sparse coverage of pressure-stabilized stoichiometries and dense coordination motifs. The framework comprises an open-source pressure-resolved dataset and a suite of publicly available atomistic models jointly optimized for energy, force, and stress prediction. Its core design couples randomized high-pressure sampling with uncertainty-guided concurrent learning, with the stated goal of enabling data-efficient, high-fidelity CSP from 0 to 200 GPa (Wang et al., 12 Sep 2025).

1. Scope and problem setting

High-pressure CSP underpins work in condensed matter physics, planetary science, and materials discovery. Within this setting, OpenCSP is positioned specifically around the observation that most large atomistic models are trained on near-ambient, equilibrium data, which leads to degraded stress accuracy under strong compression and weak representation of pressure-stabilized chemistries and dense motifs (Wang et al., 12 Sep 2025).

The framework is defined by two coupled components: a pressure-resolved dataset and atomistic graph models trained jointly on energy, force, and stress targets. The paper presents OpenCSP as spanning ambient to high-pressure CSP tasks rather than as a generic interatomic potential benchmark. This emphasis is important because the relevant target quantity for structure ranking under pressure is enthalpy, and reliable virial prediction becomes central to pressure-controlled relaxation and stability assessment (Wang et al., 12 Sep 2025).

A plausible implication is that OpenCSP should be understood less as a single model than as an integrated CSP infrastructure: data generation, model architecture, training objectives, and evaluation are all configured around compression-aware structure search. The paper’s benchmark design reinforces this interpretation by focusing on pressure-controlled relaxations, high-pressure enthalpy ranking, and success rates in identifying on-hull compositions.

2. Dataset construction and pressure-aware sampling

The OpenCSP dataset is constructed through randomized high-pressure sampling and then iteratively refined with an uncertainty-guided concurrent learning strategy. Initial structures are proposed by the CALYPSO particle-swarm optimizer without heuristics on composition, symmetry, or density. In the first 102 DP-GEN iterations, three-element systems of 8–16 atoms per cell are generated, and in the last 11 iterations this is extended to four-element systems of 16–32 atoms. At each sampling iteration, target pressures are drawn uniformly from 0 to 100 GPa, and structures are optimized under these pressures to their local minima using one model of the ensemble with L-BFGS (Wang et al., 12 Sep 2025).

Concurrent learning is implemented through an ensemble of four DPA3 models trained on the current dataset. During exploration, each candidate, or selected points along its relaxation trajectory, is evaluated by the ensemble to obtain an uncertainty estimate $\sigma$, defined as the standard deviation of forces or of enthalpy. Candidates with $\sigma$ above a threshold are removed as too far from any training domain; the paper gives $>1.0 \ \mathrm{eV/\AA}$ in early iterations as an example. The remaining structures are ranked by descending $\sigma$, and a fixed number, 5,000–20,000 of the top 10% most uncertain but not out-of-domain structures, are selected for full DFT labeling. The stated effect is to enrich underrepresented compression regimes while suppressing redundant labeling of already covered structures (Wang et al., 12 Sep 2025).

The final dataset contains 1,522,206 DFT-labeled configurations. System sizes span 5–32 atoms per cell. The pressure distribution is described as approximately uniform sampling from $-10$ to $+100$ GPa, in contrast to the strong ambient bias of earlier corpora. Chemical coverage is reported as $1 \leq Z \leq 83$, with 26% of configurations containing oxygen (Wang et al., 12 Sep 2025).

This dataset design is central to the framework’s claims of data efficiency. Rather than maximizing corpus size, OpenCSP emphasizes targeted acquisition in poorly covered pressure regimes. The paper explicitly attributes performance gains to this pressure-aware sampling strategy.

3. Model architecture and training objective

OpenCSP uses a DPA3 atomistic graph architecture based on two coupled graphs. The first, $G^{(1)}$, is an atom–bond graph with vertex dimension 128, edge dimension 64, and a cutoff of 6 Å, smooth between 3–6 Å. The second, $G^{(2)}$, is a bond–angle “line” graph with angle features, edge dimension 32, and a cutoff of 4 Å, smooth between 2–4 Å. Three model depths are used: L6, L12, and L24, corresponding to 6, 12, or 24 stacked interaction blocks. Equivariance is enforced by constructing scalar, vector, and tensor features on bonds and angles, and no explicit spherical harmonics are used (Wang et al., 12 Sep 2025).

The training target is a composite prediction of per-configuration energy $E$, per-atom forces $\sigma$0, and the virial tensor $\sigma$1. The loss is a weighted sum of mean-square errors:

$\sigma$2

This training formulation reflects the framework’s CSP orientation: stress accuracy is not ancillary, but optimized jointly with energy and force fidelity (Wang et al., 12 Sep 2025).

Training follows a two-stage schedule of 500,000 steps per stage with exponential learning-rate decay. In Stage I, the learning rate decays from $\sigma$3 to $\sigma$4, while the loss weights evolve linearly from $\sigma$5 to $\sigma$6. In Stage II, the learning rate decays from $\sigma$7 to $\sigma$8 and the loss weights are fixed at $\sigma$9. Batch size is chosen so that atoms per configuration times configurations per batch is approximately 4,096 for L6 and L12, and approximately 1,536 for L24 (Wang et al., 12 Sep 2025).

A plausible implication is that the architecture and optimization protocol were tuned to balance expressivity with virial fidelity under compression. The explicit inclusion of bond–angle structure and tensorial features suggests an attempt to capture dense, pressure-induced coordination changes without resorting to a basis of spherical harmonics.

4. Quantitative performance

The paper compares OpenCSP against three large atomistic models: MACE-MPA-0, MatterSim v1 5M, and GRACE-2L-OAM. The comparison is notable because the OpenCSP models are trained on a much smaller DFT corpus.

Model	Training corpus	#DFT structures
OpenCSP-L6/L12/L24	OpenCSP	1.5 M
MACE-MPA-0	MPTrj+sAlex	12 M
MatterSim v1 5M	proprietary MS	17 M
GRACE-2L-OAM	OMat24+…+MPTrj	113 M

On the OpenCSP test set, OpenCSP-L24 achieves $>1.0 \ \mathrm{eV/\AA}$0 meV/atom, $>1.0 \ \mathrm{eV/\AA}$1, $>1.0 \ \mathrm{eV/\AA}$2 meV/Å, $>1.0 \ \mathrm{eV/\AA}$3, $>1.0 \ \mathrm{eV/\AA}$4 meV/atom, and $>1.0 \ \mathrm{eV/\AA}$5 (Wang et al., 12 Sep 2025).

For zero-shot transfer to MPTrj-S and MPTrj-L, the reported MPTrj-S metrics are especially informative. OpenCSP-L24 records $>1.0 \ \mathrm{eV/\AA}$6 meV/atom, $>1.0 \ \mathrm{eV/\AA}$7 meV/Å, and $>1.0 \ \mathrm{eV/\AA}$8 meV/atom, compared with 12.0, 33.0, and 167.6 for MACE-MPA-0; 25.1, 48.4, and 158.6 for MatterSim 5M; and 8.6, 23.8, and 160.1 for GRACE-2L-OAM. The paper summarizes this result by stating that OpenCSP models match or exceed force and virial accuracy of baselines in zero-shot transfer, and that virial MAE is approximately 20% of that of other large atomistic models (Wang et al., 12 Sep 2025).

These results indicate a sharp asymmetry between ambient-like transfer metrics and virial prediction quality. Although some baselines can outperform OpenCSP on selected energy or force metrics, OpenCSP’s virial accuracy is markedly stronger, which is directly relevant to pressure-controlled CSP tasks.

5. Pressure-controlled relaxation and CSP success rates

A central benchmark in the paper is pressure-controlled relaxation. The protocol relaxes 200 random ternary structures at each target pressure $>1.0 \ \mathrm{eV/\AA}$9 GPa and reports the final DFT pressures.

Model	0 GPa	50 GPa	100 GPa
MACE-MPA-0	13 ± 131	108 ± 357	208 ± 397
MatterSim 5M	−1 ± 24	50 ± 8	123 ± 205
GRACE	3 ± 36	90 ± 287	218 ± 556
OpenCSP-L24	0 ± 2	50 ± 1	100 ± 1

Model	150 GPa	200 GPa
MACE-MPA-0	346 ± 958	582 ± 1824
MatterSim 5M	146 ± 13	195 ± 19
GRACE	478 ± 1200	910 ± 1972
OpenCSP-L24	150 ± 1	200 ± 3

The paper states that OpenCSP reproduces target pressure to within $\sigma$0–3 GPa even at 200 GPa, corresponding to $\sigma$1 error (Wang et al., 12 Sep 2025). This is one of the clearest empirical differentiators of the framework, because relaxation under a prescribed external pressure requires consistent enthalpy and virial behavior, not merely good local force estimates.

The CSP benchmark uses the success rate $\sigma$2, defined as the fraction of on-hull compositions found among 2,000 random trials. At 0 GPa, all models are near 60%. At 50 GPa, OpenCSP-L24 is approximately 80%, while MACE-MPA-0, MatterSim 5M, and GRACE-2L-OAM are each approximately 50%. At 100 GPa, OpenCSP is approximately 55%, MACE-MPA-0 approximately 30%, MatterSim 5M approximately 58%, and GRACE-2L-OAM approximately 25%. At 150 GPa, OpenCSP and MatterSim are each approximately 50%, compared with approximately 20% for MACE-MPA-0 and approximately 15% for GRACE-2L-OAM. At 200 GPa, OpenCSP and MatterSim are each approximately 40%, versus approximately 10% and approximately 5% for MACE-MPA-0 and GRACE-2L-OAM, respectively. The paper states that at 50–150 GPa, OpenCSP outperforms all baselines by 20–30 percentage points, with the largest gains observed in the 50 GPa regime (Wang et al., 12 Sep 2025).

A plausible interpretation is that OpenCSP’s strongest advantage appears not in generic interatomic regression, but in the coupled task of high-pressure search and ranking where virial fidelity, enthalpy ordering, and dense-coordination coverage interact.

6. Data efficiency, significance, and limitations

OpenCSP is presented as achieving comparable or superior virial and CSP performance using only 1.5 million DFT structures, described as one to two orders of magnitude fewer than competing large atomistic models (Wang et al., 12 Sep 2025). The paper attributes this efficiency to three factors: pressure-aware sampling, DP-GEN concurrent learning, and the scalable DPA3 graph architecture.

The pressure-aware sampling is described as filling voids left by ambient-biased datasets through explicit high-pressure relaxation trajectories. DP-GEN concurrent learning is described as focusing DFT effort aggressively on poorly covered regions. The DPA3 architecture is described as yielding high expressive power per parameter through message passing on atom–bond and bond–angle graphs (Wang et al., 12 Sep 2025).

These claims also delimit the framework’s scope. The dataset pressure distribution is approximately uniform from $\sigma$3 to $\sigma$4 GPa, whereas some benchmarks extend to 200 GPa. This suggests that the model’s extrapolative behavior above the labeled pressure range is part of its demonstrated performance envelope, but not necessarily equivalent to dense direct supervision throughout that regime. Similarly, the paper emphasizes three- and four-element systems with 5–32 atoms per cell, so broader applicability to larger cells or qualitatively different sampling distributions would require further evidence.

A common misconception in evaluating such systems is to treat total training-set size as the dominant predictor of CSP utility. The OpenCSP results argue against that simplification. Within the reported experiments, targeted pressure-resolved data acquisition appears more consequential for high-pressure CSP than sheer corpus scale.

7. Applications and projected extensions

The paper states that OpenCSP provides a reproducible, open-source foundation for autonomous CSP workflows spanning 0–200 GPa. Its high virial fidelity is identified as enabling reliable enthalpy rankings under compression, which is described as crucial for discovering superhard phases, hydrogen-rich superconductors, and planetary-interior materials (Wang et al., 12 Sep 2025).

Several future directions are explicitly proposed. These include extending sampling beyond 100 GPa into the TPa regime; incorporating finite-temperature NPT molecular dynamics data to capture anharmonic vibrational and thermal expansion effects; multi-fidelity learning, for example embedding DFT+$\sigma$5 or hybrid-DFT corrections for strongly correlated systems; and integration into generative CSP engines such as diffusion- or reinforcement-learning-guided searches for on-the-fly exploration of novel chemistries (Wang et al., 12 Sep 2025).

These extensions indicate how the framework could evolve from a pressure-resolved static CSP system toward a broader autonomous discovery stack. A plausible implication is that OpenCSP’s present contribution is methodological as much as empirical: it demonstrates that pressure-aware dataset design and uncertainty-driven DFT allocation can materially change the operating regime of atomistic models used for CSP under extreme conditions.