ML Potentials for Hydrogen in Iron

• Machine-learning potentials for hydrogen in iron are advanced models that interpolate high-dimensional DFT energy surfaces to simulate complex H interactions with near-quantum accuracy.
• They integrate diverse frameworks, including neural network potentials, GAPs, ACE, and charge-density models, to reliably predict defect energetics, hydrogen diffusion, and embrittlement phenomena.
• These models offer orders-of-magnitude acceleration over direct quantum computations, enabling large-scale atomistic simulations of crack propagation, dislocation interactions, and other hydrogen-induced failures.

Machine-learning potentials (MLPs) for hydrogen in iron represent a central technology in multiscale simulation of hydrogen embrittlement and hydrogen-driven phase phenomena in iron and steels. These potentials interpolate high-dimensional potential energy surfaces (PES) from density functional theory (DFT) data, capturing complex many-body interactions and enabling large-scale atomistic simulations with near-DFT accuracy and orders-of-magnitude acceleration over direct quantum computations. A rapidly developing suite of models—neural network potentials, Gaussian Approximation Potentials, Atomic Cluster Expansion, and charge-density-based regressors—achieves quantitative agreement with DFT benchmarks for defect energetics, H diffusion, dislocation interaction, and crack propagation in iron, while attaining computational costs near those of empirical interatomic potentials.

1. Methodological Foundations of Machine-Learning Potentials for Fe–H

Modern machine-learning interatomic potentials for Fe–H employ architectures that map multielement atomic environments into high-dimensional descriptor spaces, followed by regression of atomic energies or forces. Leading frameworks include:

• Behler–Parrinello neural-network (NN) potentials, which decompose the system energy as $E_\text{tot}(\{r\}) = \sum_i E_i(G_i; \theta)$, with $G_i$ a vector of symmetry functions (16–70, including radial and angular channels) encoding the atomic neighborhood and $\theta$ the NN weights. Typically, multi-layer feedforward NNs (70–20–20–1, with tanh activation) are employed (Tahmasbi et al., 2023).
• Gaussian Approximation Potentials (GAPs) and tabulated GAP (tabGAP) models, where $E_\mathrm{tot}$ is constructed as a sum of a short-range Ziegler–Biersack–Littmark (ZBL) repulsion plus kernel regressions over two- and three-body descriptors and EAM-like terms. The regression kernel is $$K_\text{se}(\mathbf{q},\mathbf{q}') = \exp\bigl(-\|\mathbf{q}-\mathbf{q}'\|^2/2\theta^2\bigr)$$; the sparsification and tabulation in tabGAP enable near-EAM computational cost (Makkonen et al., 26 Nov 2025).
• Atomic Cluster Expansion (ACE), which expresses $E_\text{tot}$ as a truncated, symmetry-adapted, linear expansion over body-ordered basis functions (up to order 5), using real spherical harmonics and orthogonal radial functions for descriptor construction. Cutoffs are 6.5 Å (Fe–Fe) and 3.5 Å (Fe–H, H–H) (Egorov et al., 14 Dec 2025).
• Deep neural-network potentials with DeepMD descriptors, using dual-channel local embeddings (up to 120 neurons, 6.5 Å cutoff) and three-layer fitting nets (320 neurons per layer, tanh activations), trained with a weighted sum of energy and force losses (Zhang et al., 2023).
• Charge-density-based ML models, in which DFT-computed differential electron charge densities $\Delta\rho(\mathbf{r})$ for Fe–H guide the construction of radial distribution function (RDF) descriptors and Bader charge/volume/radius features. Regression is performed with kernel ridge regression or shallow NNs to predict properties such as H migration barriers (Massa et al., 2023).

Loss functions are typically quadratic, measuring both energies and forces, with relative weights (e.g., $\alpha:\beta \approx 1:0.1$) chosen to balance global PES fidelity and local force accuracy. Training is performed on tens of thousands of DFT-labeled configurations representing bulk, defect, surface, dislocation, and crack geometries, as well as various H concentrations.

2. Reference Data Generation and Validation Protocols

State-of-the-art Fe–H MLPs draw on comprehensive DFT datasets including:

• Pristine bcc Fe and Fe–H (various H concentrations up to 20 at.%).
• Defects: vacancies, vacancy clusters, self-interstitials, stacking faults, edge and screw dislocations (and H decoration thereof), and free surfaces (Zhang et al., 2023, Egorov et al., 14 Dec 2025, Makkonen et al., 26 Nov 2025).
• Grain boundaries, crack-tip models, and fracture-separation curves (Egorov et al., 14 Dec 2025, Zhang et al., 2023).
• H–H and H–Fe short-range binding in highly compressed cells.
• Charge-density profiles and NEB migration pathways for H in Fe (for charge-density-based models) (Massa et al., 2023).
• Diffuse and high-symmetry interstitial configurations (tetrahedral, octahedral) and transition states.

Key DFT parameters: VASP and GGA-PBE functionals, plane-wave cutoffs of 400–500 eV, k-point density ~0.15 Å$^{-1}$, and spin-polarization for Fe. Training/validation splits typically allocate 10–20% of data for unbiased error estimation. Reported performance includes:

Model/Study	Test RMSE (Energy, meV/atom)	Test RMSE (Force, eV/Å)	Systemic Coverage
tabGAP (Makkonen et al., 26 Nov 2025)	1.88	0.069	Bulk, defects, dislocations, H–H/H–V
ACE (Egorov et al., 14 Dec 2025)	4.78	0.069	Bulk, fracture surfaces, cracks
DeepMD (Zhang et al., 2023)	4.8 (val.)	0.072	Defects, GBs, surfaces, embrittlement
NN (Tahmasbi et al., 2023)	30	0.308	FeH phases, clusters, high pressure

These errors are within 0.02–0.05 eV (point-defect and migration energies) of DFT for benchmark properties, representing near-quantum accuracy.

3. Key Physical Predictions and Mechanistic Insights

Fe–H MLPs are validated and applied through quantitative comparison to DFT for:

• H point-defect energetics: solution energies in tetrahedral (T), octahedral (O), and vacancy (V) sites (e.g., $E_{s,T}^H=0.227$ eV; $E_{s,O}^H=0.375$ eV; $E_{s,V}^H=-0.350$ eV for tabGAP (Makkonen et al., 26 Nov 2025)).
• Migration barriers: $E_{m,T\!-\!T}^H=0.114$ eV, $E_{m,T\!-\!O\!-\!T}^H=0.150$ eV (tabGAP within $\pm0.02$ eV of DFT) (Makkonen et al., 26 Nov 2025).
• H–vacancy and H–dislocation interactions: Accurate reproduction of incremental binding energies (e.g., H–vacancy first binding: 0.577 eV DFT, 0.603 eV tabGAP), non-monotonic trapping, and dislocation core occupation (Makkonen et al., 26 Nov 2025, Zhang et al., 2023).
• Elastic moduli and their reduction with H content: tabGAP and DFT agree on the decrease of $C_{11}$ and $C_{44}$ by 1.5–1.6% per at.% H (Makkonen et al., 26 Nov 2025).
• Hydrogen embrittlement phenomena:
  • At crack tips: H segregation decreases surface energy, promoting brittle cleavage ahead of dislocation emission and facilitating a ductile-to-brittle transition at ppm H concentrations (Egorov et al., 14 Dec 2025, Zhang et al., 2023).
  • At grain boundaries: H induces nanovoid formation and intergranular fracture by lowering cohesive strength (Zhang et al., 2023).
  • Under tensile loading: Decreased notch toughness and increased vacancy concentration under H loading are observed (notched Fe–H fractures at $\varepsilon=0.069$ vs $\varepsilon=0.077$ for pure Fe (Makkonen et al., 26 Nov 2025)).
  • Mechanistic models: Modified Griffith criteria incorporating H-coverage dependent surface energy and lattice trapping term quantitatively match molecular dynamics (MD) fracture loads (Egorov et al., 14 Dec 2025).

The efficacy of these MLPs is illustrated in large-scale (up to 10$^8$ atom) MD simulations, allowing nanosecond-scale evolution of crack growth, GB failure, and dislocation-H interactions under realistic conditions.

4. Computational Performance and Practical Implementations

Computational efficiency is a differentiating metric:

• tabGAP achieves $0.0096$–$0.0172$ ms/(atom·step) for 0.25–10% H, <1 order slower than EAM, and $\sim100\times$ faster than standard NNPs (Makkonen et al., 26 Nov 2025).
• DeepMD attains $0.64$ μs/(atom·step) on 8 GPUs, $>40\times$ faster than n2p2 NN on CPU (Zhang et al., 2023).
• ACE is $\sim15\times$ faster than NNP, enabling efficient simulation of 10$^5$–10$^6$ atom cells for crack propagation (Egorov et al., 14 Dec 2025).

All leading models retain near-DFT accuracy across bulk/defected/strained state-space. Empirical EAM potentials are marginally faster but fail in reproducing defect and embrittlement energies.

5. Descriptor Innovations and Data-Driven Approaches

The growth of charge-density-based MLPs supplements traditional geometric descriptors with ab initio-informed features:

• Charge density perturbation descriptors: RDFs of DFT differential charge densities ($\Delta\rho(\mathbf{r})$), Bader charges/volumes, and related scalar summaries (peak height, position, width) encode local H–Fe electronic structure (Massa et al., 2023).
• Regression targets and feature-importances: NEB-calculated migration energies $E_m$ are regressed on RDF/Bader features, with peak height and H Bader volume among the dominant correlates ($|r|\sim0.7$ with $E_m$).
• Such approaches enable compact, transfer-learning-capable potentials for predicting H migration and trapping in iron and iron-based alloys, with mean absolute error $<0.05$ eV (Massa et al., 2023).

A typical limitation is that long-range effects outside the immediate descriptor cutoff (elastic/magnetic/multihydrogen) are not represented unless explicitly incorporated.

6. Multiscale and Surrogate ML Models: Constitutive-Level Links

Beyond true atomistic potentials, ML surrogates have been formulated to map CPFEM simulation data—strain, H concentration, crystal orientation—to field-level quantities (stress, triaxiality):

• Support Vector Regression (SVR) Surrogates: Linear or RBF-kernel SVRs substitute for constitutive models, trained on CPFEM-generated data with physics-informed regime classification (elastic vs. plastic), yielding instantaneous predictions at any meshpoint (Siddiq, 2021).
• Applicability: These surrogates accelerate mesoscale simulation (minutes to seconds vs. hours), but lack explicit energy conservation and do not encode atomistic bond-breaking.

Transfer to higher H contents, grain-boundary phenomena, or polycrystalline response necessitates retraining or descriptor extension.

7. Current Limitations and Outlook

Despite rapid progress, present Fe–H machine-learning potentials display several limitations:

• Transferability: Models are primarily parametrized and validated for bcc Fe with H in dilute to moderate concentrations; extension to face-centered cubic phases, carbide-rich steels, and magnetic disorder calls for expanded DFT databases and retraining.
• Quantum effects: Nuclear quantum effects for H diffusion/migration below ∼200 K are neglected (Egorov et al., 14 Dec 2025).
• Elastic and surface phenomena: Explicit long-range electrostatics and charge equilibration schemes are typically absent; virial force training is not ubiquitous.
• Crack geometry and extreme strain: Validation is generally limited to {110}/{100} Mode-I cracks; caution in extrapolation is recommended.

Prospective advances include descriptor pruning and mixed-precision neural inference (for further acceleration), explicit zero-point energy correction, inclusion of multi-element alloying (C, Mn, Ni), magnetic fluctuations, and fine control of interfacial/precipitate H trapping (Makkonen et al., 26 Nov 2025, Zhang et al., 2023).

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References:

(Tahmasbi et al., 2023, Massa et al., 2023, Siddiq, 2021, Zhang et al., 2023, Egorov et al., 14 Dec 2025, Makkonen et al., 26 Nov 2025).