First-Principles ML Potentials

• First-Principles-Based ML Potentials are force fields trained on DFT data that integrate quantum mechanical accuracy with scalable atomistic simulations.
• They employ advanced descriptor schemes such as symmetry functions, SOAP, and graph neural networks to ensure physical invariance and capture complex many-body interactions.
• Applications include high-throughput property prediction, thermal transport simulation, and materials design, achieving significant speedups compared to direct ab initio methods.

First-principles-based machine learning potentials (MLIPs) are atomistic force fields trained exclusively on reference data calculated from electronic-structure theory, most commonly density functional theory (DFT). These models inherit first-principles accuracy and quantum-mechanical transferability, while extending atomistic simulation capabilities to system sizes (10³–10⁸ atoms), time scales (ns), and compositional complexity that are far beyond the reach of direct ab initio methods. MLIPs have become indispensable for high-throughput property prediction, materials design, and multiscale modeling in modern condensed matter and computational materials science (Ceriotti, 2022, Mortazavi et al., 2020, Berger et al., 9 Apr 2025).

1. Mathematical Foundations and Representation Schemes

The core mathematical ansatz of first-principles-based MLIPs is the locality decomposition: $E_{\text{tot}} = \sum_{i=1}^{N} E_{i}(q_{i})$ where $q_i$ is a vector of descriptors encoding the chemical environment within a cutoff radius, and $E_i$ is a local energy function (Ceriotti, 2022). Descriptor schemes fall into several principal classes:

• Symmetry Functions: Behler–Parrinello-type two-body ($G^2$) and three-body ($G^3$) symmetry functions sample distances and bond angles, ensuring invariance under translation, rotation, and permutation (imbalzano et al., 2018, Ceriotti, 2022).
• Smooth Overlap of Atomic Positions (SOAP): Constructs neighbor-density power spectra via spherical harmonic and radial basis expansions, yielding a continuous fingerprint (Veit et al., 2018).
• Moment-Tensor Potentials (MTP), Atomic Cluster Expansion (ACE): Systematically expand scalar contractions of tensor products of neighbor vectors, forming a complete polynomial basis up to arbitrary body-order (Mortazavi et al., 2020, Mortazavi et al., 2020, Ceriotti, 2022).
• Graph Neural Descriptors: Employ E(3)-equivariant message passing to directly encode many-body geometry and chemical embedding (e.g. nequIP, MACE, EquiformerV2) (Shuang et al., 5 Feb 2025, Bandi et al., 29 Feb 2024, Berger et al., 9 Apr 2025).

The potential $E_i(q_i)$ is mapped either via kernel regression (Gaussian Process, e.g. GAP), polynomial expansion, or neural networks, including graph or transformer architectures.

2. Model Training, Data Generation, and Regression

All first-principles-based MLIPs are trained on DFT (occasionally higher-level, e.g. CCSD(T)) reference data, comprising structures’ total energies, atomic forces, and sometimes stress tensors (Mortazavi et al., 2020, Togo et al., 31 Jan 2024, Bandi et al., 29 Feb 2024, Jinnouchi et al., 17 Sep 2024). Standard workflows include:

• Data Generation: AIMD, normal mode sampling, enhanced-sampling, or active learning to collect statistically representative, thermodynamically relevant, and out-of-equilibrium structures at targeted temperatures and pressures (Unglert et al., 13 Dec 2025).
• Loss Functions: Weighted least-squares regression combining energy and force errors, e.g.:

$L(\theta) = \sum_{k=1}^K \left[ w_E (E^{ML}_k - E^{\text{DFT}}_k)^2 + w_F \sum_{i} \|F_{k,i}^{ML}-F_{k,i}^{\text{DFT}}\|^2 + w_S \|\sigma^{ML}_k - \sigma^{\text{DFT}}_k\|^2$

with typical weight choices $w_E \sim 1$, $w_F \sim 0.1$–1, $w_S \ll w_E$ (Mortazavi et al., 2020, Mortazavi et al., 2020).

• Regression Methods:
  • Linear or ridge regression for polynomial basis models;
  • Gradient-based stochastic optimization for neural potentials;
  • Kernel ridge regression (for GAP, SOAP) (Veit et al., 2018, Bandi et al., 29 Feb 2024).
• Active Learning: Iteratively augmenting the dataset by querying high-uncertainty or extrapolative structures encountered during sampling or nested sampling (RENS) (Unglert et al., 13 Dec 2025).

3. Descriptor Construction and Physical Invariance

Physical invariance under translation, rotation, and permutation of identical atoms is embedded at the descriptor level, with additional strategies for modeling long-range electrostatics or equivariant tensorial responses where necessary (Ceriotti, 2022, Bandi et al., 29 Feb 2024). Recent initiatives incorporate E(3)-equivariant neural architectures to permit accurate learning of vector and tensor observables (forces, stress) in addition to scalar properties (Shuang et al., 5 Feb 2025, Bandi et al., 29 Feb 2024).

Relevant classes:

Descriptor	Functional Form	Invariance
G2/G3	BP radial/angle symmetry functions	Trans/rot/permutable neighbors
SOAP	Spherical harmonic power spectrum	Trans/rot/perm/chem permutation
MTP/ACE	Polynomial moment tensors	Trans/rot/perm; body-order systematic
Graph NN	Message-passing over graphs	Learned equivariance, chemical embedding

4. Applications: Multiscale Modeling, Thermal Transport, and Redox

First-principles MLIPs enable direct simulation and prediction of:

• Thermal Conductivity: MLIPs (e.g. MTP, polynomial MLPs) trained on DFT can replace thousands of DFT force evaluations in workflows for lattice thermal conductivity (LTC), both via direct force-constant extraction (ShengBTE/BTE) and equilibrium/nonequilibrium MD, offering <5% deviation from DFT at >50× speedup (Togo et al., 31 Jan 2024, Mortazavi et al., 2020, Qian et al., 2019).
• Anharmonic Phonon Dynamics: MLIPs trained on irreducible finite-difference expansions of the Born–Oppenheimer potential reproduce phonon lineshifts, linewidths, and high-order force-constants (up to fifth order) within <10–20% of DFT, crucial for accurate prediction of temperature-dependent transport (Bandi et al., 29 Feb 2024).
• Defects, Phase Diagrams, and High-Throughput Screening: Universal MLIPs (MACE, M3GNet, EquiformerV2, CHGNet) enable screening of defect formation energies, phase boundaries, and stability for >10⁵ structures at DFT-level accuracy and $10^3$–$10^5$ speedup (Berger et al., 9 Apr 2025, Shuang et al., 5 Feb 2025, Unglert et al., 13 Dec 2025).
• Electrochemical Potentials: Δ-machine learning adds corrections to DFT-based potentials using small sets of CCSD(T) or hybrid DFT points, achieving millivolt-level errors in redox and proton-insertion free energies, e.g. via thermodynamic integration and TI + TPT workflows (Jinnouchi et al., 17 Sep 2024, Nandi et al., 29 Jul 2024).
• Finite-Temperature and Disorder: MLIPs can directly model crystalline, amorphous, and interfacial systems, accurately capturing both harmonic and anharmonic vibrational properties, e.g. for silicon phases and coplanar graphene/borophene heterostructures (Mortazavi et al., 2020, Qian et al., 2019).

5. Advanced Architectures and Model Generalization

Modern developments are converging on universal, transferable graph-based MLIPs trained on massive DFT datasets spanning elements, compositions, and defect topologies (e.g. MACE, EquiformerV2, CHGNet, M3GNet, ALIGNN). These universal MLIPs achieve:

• Energy RMSE $<5$ meV/atom and force RMSE $<100$ meV/Å across broad material classes, generalizing to defects, surfaces, strain, and non-crystalline phases without retraining (Shuang et al., 5 Feb 2025, Berger et al., 9 Apr 2025).
• Pareto-optimal computational throughput—often $10^4$ times faster than DFT.
• Reliable uncertainty quantification via ensemble predictions and active learning loops (Shuang et al., 5 Feb 2025, Unglert et al., 13 Dec 2025).

Δ-machine learning strategies (editor's term) decouple the high-cost quantum correction from the base model, reducing the number of reference calculations required for chemical accuracy to a manageable O(10²–10³) points (Nandi et al., 29 Jul 2024, Jinnouchi et al., 17 Sep 2024).

Physics-informed weak supervision methods impose first- and second-order consistency constraints (e.g. via Taylor expansion or conservative force check) on the MLIP energy and force predictions, enforcing physical robustness and transferability, especially in sparse-data or fine-tuning regimes (Takamoto et al., 23 Jul 2024).

6. Benchmarks, Validation, and Best Practices

Typically reported accuracies for state-of-the-art first-principles-based MLIPs are:

• Energy MAE: 1–10 meV/atom;
• Force RMSE: 0.05–0.1 eV/Å (for targeted systems) (Ceriotti, 2022, Mortazavi et al., 2020, Bandi et al., 29 Feb 2024, Berger et al., 9 Apr 2025).
• Applications demanding higher-order properties (phonons, LTC, transition states) require validation not only on energies/forces but directly on derived observables—phonon frequencies, BTE–computed $\kappa$, lineshifts/linewidths, diffusion barriers, and phase boundaries (Bandi et al., 29 Feb 2024, Unglert et al., 13 Dec 2025, Togo et al., 31 Jan 2024).

Convergence studies routinely monitor training–test error curves as a function of model complexity, training set size, and descriptor selection (CUR, FPS, PC) (imbalzano et al., 2018, Rohskopf et al., 2023). For high-throughput workflows, domain reweighting in the loss function and committee-based model averaging are essential to control extrapolation and overfitting (Unglert et al., 13 Dec 2025).

7. Limitations and Outlook

Current challenges include:

• Transferability to chemistries/environments not present in the training data; reactivity and far-from-equilibrium events remain demanding (Ceriotti, 2022, Shuang et al., 5 Feb 2025).
• Long-range contributions; coupling to explicit electrostatics and dispersion may be required for polar/van der Waals systems (Veit et al., 2018, Jinnouchi et al., 17 Sep 2024).
• Data efficiency; new active learning and weak supervision techniques reduce the burden of extensive DFT calculations (Unglert et al., 13 Dec 2025, Takamoto et al., 23 Jul 2024).
• Quantum nuclear effects require explicit path-integral MD or effective correction schemes (Veit et al., 2018).
• Model interpretability and human-guided fine-tuning (e.g. via spline-NN architectures or sparsified descriptors) balance transparency with flexibility (Vita et al., 2023, imbalzano et al., 2018).

The field is evolving toward fully autonomous, foundation-level MLIPs with robust uncertainty quantification, active learning, and integration into end-to-end workflows for property prediction, materials screening, and functional property computation. This offers the prospect of true first-principles accuracy at length, time, and compositional scales previously unattainable in computational materials science (Berger et al., 9 Apr 2025, Unglert et al., 13 Dec 2025, Shuang et al., 5 Feb 2025).