# Machine Learning Interatomic Potentials

KEN embeds two-dimensional convolutional kernels in an E(3)-equivariant graph neural network so that an interatomic potential reads electronic spin-interaction matrices alongside geometric coordinates, sidestepping the ill-posedness of coordinate-only potentials in strongly correlated magnetic systems. On the HE26 actinide dataset, KENJI reaches a training bonding-energy mean absolute error of 6.93 meV/atom against 44.7 meV/atom for MACE-osaka26, and its training force error of 34.10 meV/Å is on par with 26.30 meV/Å. On the independent test set the energy error is 46.74 meV/atom and the force error exceeds MACE-osaka26, which the authors partly attribute to training without stress.

KEN embeds two-dimensional convolutional kernels in an E(3)-equivariant graph neural network so that an interatomic potential reads electronic spin-interaction matrices alongside geometric coordinates, sidestepping the ill-posedness of coordinate-only potentials in strongly correlated magnetic systems. On the HE26 actinide dataset, KENJI reaches a training bonding-energy mean absolute error of 6.93 meV/atom against 44.7 meV/atom for MACE-osaka26, and its training force error of 34.10 meV/Å is on par with 26.30 meV/Å. On the independent test set the energy error is 46.74 meV/atom and the force error exceeds MACE-osaka26, which the authors partly attribute to training without stress.

SO3LR Tauri extends the earlier SO3LR force field from eight elements and 4 million structures to 85 elements and roughly 148 million structures, and rewrites its short-range repulsion, electrostatics and dispersion modules. Keeping the fast SO3krates message-passing backbone, it reproduces nucleobase-pair and protein–ligand interactions and carbohydrate conformer ordering against coupled-cluster references, and yields water and organic-solvent densities, ion hydration and structures of several biomolecular classes in dynamics without system-specific fitting. The models ship in three sizes and scale to the million-atom range on a single GPU.

SO3LR Tauri extends the earlier SO3LR force field from eight elements and 4 million structures to 85 elements and roughly 148 million structures, and rewrites its short-range repulsion, electrostatics and dispersion modules. Keeping the fast SO3krates message-passing backbone, it reproduces nucleobase-pair and protein–ligand interactions and carbohydrate conformer ordering against coupled-cluster references, and yields water and organic-solvent densities, ion hydration and structures of several biomolecular classes in dynamics without system-specific fitting. The models ship in three sizes and scale to the million-atom range on a single GPU.

Latent-charge machine-learning potentials learn atomic charges as hidden variables from energies and forces, but their long-range electrostatic term had no exact periodic form, leaving net charge mixed with self-energy. Writing that term as a smeared Epstein–Hurwitz lattice sum separates the net-charge artefact from the local self-energy and yields a sphere-average theorem for cells of any shape, plus forces and virial. Projecting the latent charges onto fixed total charge gives projected latent Ewald summation (pLES); on charged NaCl clusters it recovers the total charge an unconstrained model misses by up to 1.8 and cuts a short-range model's test errors from 1.44 to 0.42 meV/atom and from 37 to 13 meV/Å.

Latent-charge machine-learning potentials learn atomic charges as hidden variables from energies and forces, but their long-range electrostatic term had no exact periodic form, leaving net charge mixed with self-energy. Writing that term as a smeared Epstein–Hurwitz lattice sum separates the net-charge artefact from the local self-energy and yields a sphere-average theorem for cells of any shape, plus forces and virial. Projecting the latent charges onto fixed total charge gives projected latent Ewald summation (pLES); on charged NaCl clusters it recovers the total charge an unconstrained model misses by up to 1.8 and cuts a short-range model's test errors from 1.44 to 0.42 meV/atom and from 37 to 13 meV/Å.

SO(3)-equivariant graph-neural-network interatomic potentials usually use one cutoff radius, so radial interactions carry a single characteristic spatial-frequency scale. MCSA fuses radial kernels with different cutoff radii before the angular tensor product, covering several scales with one angular computation. Across the Nequix, Eqnorm and EquiformerV3 backbones, energy, force and stress errors fall on both in-distribution and out-of-distribution tests; Nequix+MCSA lowers the symmetric relative mean error of lattice thermal conductivity from 0.487 to 0.420, with per-material time rising from 1.107 to 1.136 minutes.

SO(3)-equivariant graph-neural-network interatomic potentials usually use one cutoff radius, so radial interactions carry a single characteristic spatial-frequency scale. MCSA fuses radial kernels with different cutoff radii before the angular tensor product, covering several scales with one angular computation. Across the Nequix, Eqnorm and EquiformerV3 backbones, energy, force and stress errors fall on both in-distribution and out-of-distribution tests; Nequix+MCSA lowers the symmetric relative mean error of lattice thermal conductivity from 0.487 to 0.420, with per-material time rising from 1.107 to 1.136 minutes.

Machine-learning interatomic potentials (MLIPs) have largely kept inference in FP32 because molecular dynamics is sensitive to small force errors. For PET, an MLIP from EPFL's Lab COSMO, BF16x9 on Blackwell GPUs and TF32x3 on Ampere GPUs decompose each FP32 matrix-multiply operand into lower-precision components evaluated on Tensor Cores, with no retraining or architecture change. On a 17,280-atom water box, BF16x9 sped PET's large GEMMs by 2.2–2.5x and the full MD step by up to 1.7x; TF32x3 sped major GEMMs on A100 by about 1.7–1.9x and the MD step by up to 1.5x, while keeping energy conservation and structural relaxation close to FP32.

Machine-learning interatomic potentials (MLIPs) have largely kept inference in FP32 because molecular dynamics is sensitive to small force errors. For PET, an MLIP from EPFL's Lab COSMO, BF16x9 on Blackwell GPUs and TF32x3 on Ampere GPUs decompose each FP32 matrix-multiply operand into lower-precision components evaluated on Tensor Cores, with no retraining or architecture change. On a 17,280-atom water box, BF16x9 sped PET's large GEMMs by 2.2–2.5x and the full MD step by up to 1.7x; TF32x3 sped major GEMMs on A100 by about 1.7–1.9x and the MD step by up to 1.5x, while keeping energy conservation and structural relaxation close to FP32.

Barium titanate's electro-optic coefficient falls markedly as modulation frequency rises, limiting modulator bandwidth. Using machine-learned potentials trained on PBE and HSE data and Green–Kubo molecular dynamics for the fixed-cell ionic coefficient, the work finds that nonharmonic damping and atomic displacement correlations spanning many unit cells, rather than phonon frequency itself, set both the low-frequency amplitude and the attenuation frequency. The HSE potential gives about 590 pm/V and about 137 GHz, against experimental values of 480 pm/V and 72.6 GHz.

Barium titanate's electro-optic coefficient falls markedly as modulation frequency rises, limiting modulator bandwidth. Using machine-learned potentials trained on PBE and HSE data and Green–Kubo molecular dynamics for the fixed-cell ionic coefficient, the work finds that nonharmonic damping and atomic displacement correlations spanning many unit cells, rather than phonon frequency itself, set both the low-frequency amplitude and the attenuation frequency. The HSE potential gives about 590 pm/V and about 137 GHz, against experimental values of 480 pm/V and 72.6 GHz.

Treating density query points as virtual atoms that share one encoder with real atoms lets electron-density supervision shape interatomic-potential representations: density error on MP falls below ChargE3Net, and joint training cuts zero-shot force error on larger molecules.

Treating density query points as virtual atoms that share one encoder with real atoms lets electron-density supervision shape interatomic-potential representations: density error on MP falls below ChargE3Net, and joint training cuts zero-shot force error on larger molecules.

Chaining machine-learning potentials, accelerated sampling and mass thermodynamic integration into one workflow brings water's autoionization constant to 14.31 against the experimental 13.93 once nuclear quantum effects are included explicitly, and the same workflow computes all six dissociation constants of phosphoric acid without changing the order-parameter definitions, cutting the mean absolute error from 2.19 to 0.45.

Chaining machine-learning potentials, accelerated sampling and mass thermodynamic integration into one workflow brings water's autoionization constant to 14.31 against the experimental 13.93 once nuclear quantum effects are included explicitly, and the same workflow computes all six dissociation constants of phosphoric acid without changing the order-parameter definitions, cutting the mean absolute error from 2.19 to 0.45.

The authors introduce SPACE, an architecture that keeps locality, smoothness and symmetry priors while using efficient tensor products in a compact equivariant representation to let body order grow to the number of atoms in the receptive field, so it does not assume low-order truncation of the body-ordered series; the work is presented as an architectural demonstration that the design space of equivariant architectures still has poorly explored regions.

The authors introduce SPACE, an architecture that keeps locality, smoothness and symmetry priors while using efficient tensor products in a compact equivariant representation to let body order grow to the number of atoms in the receptive field, so it does not assume low-order truncation of the body-ordered series; the work is presented as an architectural demonstration that the design space of equivariant architectures still has poorly explored regions.

The authors present BranchIP (Branch Interatomic Potential), a single-model framework for learned adaptive tensor-product computation trained with a novel distillation loss; in experiments on two systems of physical interest, a heterogeneous catalysis system and a proton-conducting solid acid electrolyte, it accelerates equivariant machine learning interatomic potentials across model sizes by up to 2.4x while reducing memory usage by up to 2.6x, maintains physical fidelity, and uses the learned adaptive computation to reveal which interactions demand deeper computation and how computational depth relates to chemical complexity and dynamics.

The authors present BranchIP (Branch Interatomic Potential), a single-model framework for learned adaptive tensor-product computation trained with a novel distillation loss; in experiments on two systems of physical interest, a heterogeneous catalysis system and a proton-conducting solid acid electrolyte, it accelerates equivariant machine learning interatomic potentials across model sizes by up to 2.4x while reducing memory usage by up to 2.6x, maintains physical fidelity, and uses the learned adaptive computation to reveal which interactions demand deeper computation and how computational depth relates to chemical complexity and dynamics.