Pushing the limits of unconstrained machine-learned interatomic potentials

Filippo Bigi; Paolo Pegolo; Arslan Mazitov; Jonathan Schmidt; Michele Ceriotti

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Pushing the limits of unconstrained machine-learned interatomic potentials

Filippo Bigi, Paolo Pegolo, Arslan Mazitov, Jonathan Schmidt, Michele Ceriotti

arXiv:2601.16195v3Jan 22, 2026physics.chem-phstat.ML

TL;DR

The paper asks whether unconstrained MLIPs remain accurate and useful as models and datasets scale. It evaluates large-scale unconstrained architectures across benchmarks and static simulation workflows, finding accuracy and speed comparable to constrained models, with practical caveats for non-conservative forces and symmetry handling.

Problem
The paper addresses whether physical symmetries and energy conservation must be enforced in MLIPs when training scales to large, diverse datasets.
Method
The study scales unconstrained PET-based architectures and evaluates them on benchmark accuracy, geometry optimization, lattice dynamics, and inference-time symmetry corrections.
Results
Unconstrained models achieve accuracy on par with state-of-the-art equivariant networks and can be more efficient at inference time.
Takeaways & Limitations
Accurate unconstrained models can support practical atomistic workflows when simple inference-time or workflow modifications address symmetry-related artifacts.
Takeaways & Limitations
Non-conservative force models often fail to match conservative models’ quantitative downstream performance despite equal or better test-set accuracy.

Abstract

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Machine-learned interatomic potentials (MLIPs) are increasingly used to replace computationally demanding electronic-structure calculations to model matter at the atomic scale. The most commonly used model architectures are constrained to fulfill a number of physical laws exactly, from geometric symmetries to energy conservation. Evidence is mounting that relaxing some of these constraints can be beneficial to the efficiency and (somewhat surprisingly) accuracy of MLIPs, even though care should be taken to avoid qualitative failures associated with the breaking of physical symmetries. Given the recent trend of scaling up models to larger numbers of parameters and training samples, a very important question is how unconstrained MLIPs behave in this limit. Here we investigate this issue, showing that -- when trained on large datasets -- unconstrained models can be superior in accuracy and speed when compared to physically constrained models. We assess these models both in terms of benchmark accuracy and in terms of usability in practical scenarios, focusing on static simulation workflows such as geometry optimization and lattice dynamics. We conclude that accurate unconstrained models can be applied with confidence, especially since simple inference-time modifications can be used to recover observables that are consistent with the relevant physical symmetries.

I. INTRODUCTION

Interatomic potentials underpin atomistic simulations, while MLIPs seek to provide near-first-principles accuracy at much lower cost. This work examines whether physical constraints can instead be learned from data in large-scale unconstrained architectures.

Interatomic potentials describe atomic interactions relevant to stability, reactivity, and thermodynamic behavior.
MLIPs train on quantum-mechanical reference data to approach first-principles accuracy at substantially reduced computational cost.
Large and diverse datasets have enabled general-purpose MLIPs spanning much of the periodic table and millions of configurations.
Common architectures explicitly enforce geometric symmetries and energy conservation, whereas unconstrained models learn these properties from training data.
Fully unconstrained models can scale to large diverse datasets, match equivariant-network accuracy, and improve inference efficiency, but may require more training epochs.Their practical use is demonstrated for structural optimization and lattice dynamics, with potential pitfalls corrected by simple modifications.

B. Machine-learned interatomic potentials

MLIPs occupy a middle ground between cheap but limited classical potentials and accurate but expensive quantum-mechanical methods. Recent universal MLIPs motivate architectures that learn physical properties rather than enforcing every constraint explicitly.

Classical potentials are cheap but have limited transferability, whereas DFT is accurate but restricted by computational cost.
Universal MLIPs trained on increasingly large and diverse datasets can generalize across systems and support direct application or inexpensive fine-tuning.
The paper focuses on learning physical properties from data instead of enforcing all of them in the functional form.
Most applications use interatomic forces, which may be obtained by differentiating energy or predicted directly by an additional model head.
Direct-force prediction avoids backward differentiation, reducing memory use and providing a 2–3× speedup, but does not guarantee energy conservation.

C. Applications and benchmarks

The paper frames unconstrained MLIPs around the practical requirements of geometry optimization, lattice dynamics, and benchmarking. It emphasizes that learned symmetries vary in difficulty and that energy conservation is especially costly to enforce or test directly.

Applications and benchmarks: Geometry optimization locates nearby local minima, while lattice-dynamics calculations assess vibrational behavior around optimized structures.
Applications and benchmarks: Standardized benchmarks complement practical simulation tests by evaluating accuracy and efficiency across atomistic workflows.
Learnability of symmetries: The difficulty of learning a symmetry depends on the number of transformed structures needed to resolve it at the relevant spatial scale.
Learnability of symmetries: Rotational learning requires roughly 20,000 augmentations under the stated estimate, whereas permutation learning with 30 neighbors would require 30! augmentations.
Learnability of symmetries: Energy conservation corresponds to a symmetric force Jacobian, but checking this constraint is too expensive in practice for non-conservative models.

B. Architecture

The proposed PET-based architecture adapts transformer components, feature sizes, training schedules, force prediction, and cutoff handling for large-scale datasets. The paper combines these design choices with inference-time corrections to manage symmetry-related artifacts in applications.

Architecture: The architecture uses transformer processing of atomic-neighborhood edges and modern normalization, activation, and pre-normalization components.
Architecture: Direct forces are predicted through an additional three-component output head, while some models use a smooth adaptive cutoff with linear neighbor scaling.
Physical observables: Inference-time augmentation or ensembling can reduce symmetry errors for compact-group symmetries.
Physical observables: Random rotations and inversions at each molecular-dynamics step can avoid systematic long-time errors with negligible overhead.
Physical observables: Phonon calculations require symmetry-aware optimization, relaxed detection thresholds, or fixed high-symmetry Brillouin-zone paths.

A. Large-scale materials datasets

PET-OAM scales unconstrained modeling to large materials datasets while supporting strong benchmark performance and practical geometry and lattice-dynamics workflows. Conservative models are safer for phonons, whereas unconstrained relaxations require attention to detected symmetry.

Large-scale training: PET-OAM is trained across MPtrj, Alexandria, and OMat24, reaching 730M parameters without sacrificing computational efficiency.The architecture uses large node representations and an unconstrained design.
Training protocol: The model is pretrained on non-conservative forces and stresses, then fine-tuned on conservative counterparts.This two-step protocol can reduce computational effort and improve accuracy relative to training protocols using conservative targets throughout.
Geometry optimization: 70% of conservative-model and 16% of non-conservative-model relaxed WBM structures retain initial symmetry at a 0.01 Å tolerance.At 0.1 Å, the corresponding fractions are 88% and 81%; O(3)-averaging raises the non-conservative 0.01 Å result to 77%.
Lattice dynamics: Phonon outcomes depend mainly on relaxed geometry, while non-conservative models can produce significantly different dispersion curves even after Hessian symmetrization.The authors therefore consider conservative models safer for phonon evaluation, despite accurate conservative rotationally unconstrained models performing well on vibrational benchmarks.
Benchmark performance: State-of-the-art results are reported for materials discovery, optimized geometries, and phonon-related static properties on matbench-discovery.The evaluation uses DAF, classification accuracy, F1, geometry RMSD, and κSRME metrics, with rotational symmetrization applied.

B. The SPICE molecular benchmark

The SPICE benchmark evaluates whether unconstrained MLIPs retain accuracy and efficiency for molecular systems. The proposed models exceed state-of-the-art test accuracy and offer a strong accuracy–inference-speed trade-off, while conservative models remain preferable for quantitative downstream use.

Training and accuracy: A 190M-parameter unconstrained model exceeds state-of-the-art accuracy on the SPICE molecular benchmark.It uses conservative forces throughout and is trained for three times the epochs reported for eSEN.
Inference timings: Unconstrained models form an excellent accuracy–inference-speed compromise across molecular structures of varying sizes.Figure 3 compares accuracy-speed Pareto fronts using average energy error across subsets.
Training efficiency: Non-equivariant models generally require more epochs to converge, but their lower per-step cost largely compensates for the longer training.Direct-force models instead accelerate training and can be fine-tuned into conservative models with reduced computational effort.
Practical trade-offs: Rotationally unconstrained models can match equivariant predictive accuracy on downstream tasks, whereas non-conservative models often fail to match conservative models’ quantitative performance.The authors recommend non-conservative models primarily for conservative-model pretraining or combination with conservative models in finite-temperature applications.
Conclusion: PET retains excellent performance at scale and establishes rotationally unconstrained architectures as promising alternatives for accuracy–parameter–speed trade-offs.The conclusion is qualified by the possibility that further architectural tuning and software optimization could shift efficiency comparisons.

Appendix A: Model and training details

The appendix details architectural modifications, adaptive neighbor cutoffs, and training choices for the proposed unconstrained PET architecture. These changes improve accuracy on diverse datasets while retaining comparable computational efficiency.

Architecture: The architecture replaces layer-wise prediction summation with skip connections, processed representations, and an MLP for combining ij and ji representations.These modifications are illustrated in Fig. 1.
Architecture: A 10-15% improvement in force MAE on the MAD dataset is reported compared with literature results, with comparable computational efficiency.The authors note that further architectural tuning may improve accuracy or inference timing.
Architecture: The experiments do not use the Equivariant Coordinate System Ensemble to enforce exact energy invariance.
Adaptive cutoff: The adaptive cutoff selects a smooth cutoff radius aimed at encompassing a target number of neighbors across structures with varying densities.Neighbor counts are computed over probe cutoff distances, and the cutoff is selected using a smooth interpolation procedure.
Adaptive cutoff: The baseline B(r) = n̄(r/rmax)^3 stabilizes adaptive-cutoff evaluation and limits excessive active-edge counts near the outer cutoff.The method uses adaptive neighbor-count differences to set the Gaussian width for smooth interpolation between probe radii.

3. Training

Training uses standardized schedules and dataset splits, with explicit batch-size controls for large heterogeneous datasets. The reported setup includes a 10% learning-rate warm-up and model-specific training details.

Training protocol: All training and fine-tuning runs use a cosine learning-rate scheduler with a linear warm-up covering 10% of total training steps.Runs use single precision without TensorFloat-32 operations, with target preprocessing described in the training protocol.
Training protocol: Batches exceeding 350 atoms are rejected for OAM fine-tuning, imposing a memory-related constraint on batch construction.The adaptive cutoff strategy also contributes to controlling memory usage per batch.
Datasets and splits: Official training, validation, and test splits are used for OMat24, sAlex, SPICE, MAD, and OMol25, while MPtrj is entirely added to OAM fine-tuning training data.
Hyperparameters: The evaluated models use hyperparameters listed in Table III, with node-feature dimensionality set to four times the edge-feature dimensionality.Table III also specifies a 10% linear learning-rate warm-up.

5. Pareto frontier for PET-OMat models

PET-OMat models are evaluated across cost-accuracy trade-offs, scaling and fine-tuning settings, and downstream matbench-discovery tasks. Fine-tuning improves validation errors and reduces training epochs, while conservative predictions generally perform better on downstream evaluations.

Model scaling and Pareto trade-offs: The PET-OMat model series spans a broad range of computational cost and accuracy trade-offs on OMat24.The Pareto plot uses force MAE on a hold-out test set and ASE-measured computational cost, with point color and size indicating maximum system size before GPU memory exhaustion.
Fine-tuning: Fine-tuning OMat models halves validation energy and force errors compared with training from scratch.Near-converged validation metrics require 100 epochs for a non-conservative model and 30 epochs for a conservative model, versus 700 or more from scratch.
Fine-tuning: The OMAD model shows superior accuracy and transferability compared with OAM models when evaluated consistently.
Downstream evaluation: Non-conservative models consistently perform worse on matbench-discovery downstream tasks despite often achieving better training and validation accuracies from scratch.The comparison includes conservative and non-conservative models with and without rotational averaging.
Downstream evaluation: Central finite differences can more than halve phonon-task error for non-equivariant models by averaging inconsistencies between symmetry-equivalent displacements.The conservative model achieves state-of-the-art performance, while the non-conservative model remains competitive with conservative approaches and outperforms other direct-force models.

Appendix D: LAMbench evaluation

LAMBench evaluates universal MLIPs across force-field generalizability, catalytic reaction barriers, elastic constants, and molecular conformer energies. PET models perform strongly across domains, but elastic properties are especially sensitive to energy conservation and underlying DFT settings.

Force-field generalizability: PET-OAM-C achieves the best overall force-field generalizability score across Molecules, Inorganic Materials, and Catalysis.It matches DPA-3.1-3M on Inorganic Materials and ranks second in Catalysis, while PET-OMAD performs worse under inconsistent theory baselines.
Catalysis: PET-OMAD ranks third for catalytic reaction energies and barriers and achieves the highest desorption success rate.The authors attribute this strength to explicit molecular and surface data in MAD, despite inconsistent DFT targets.
Elastic constants: Conservative PET-OAM achieves the best shear-modulus accuracy and competes closely with universal models on bulk-modulus prediction.Elastic moduli are sensitive to the lack of energy conservation because their calculation includes geometry optimization.
Molecular conformers: PET-OAM-NC achieves the best conformer-energy results among trained PET models and ranks second overall behind DPA-3.1-3M.The authors connect the gap to explicit molecular data in the DPA-3.1-3M training set.

Appendix E: MADBench evaluation

The study evaluates PET-OAM, PET-MPtrj, and PET-OMAD on MADBench, which compares universal MLIPs across domains using internally consistent reference theory. PET-OMAD is generally most accurate, while performance depends on dataset coverage and evaluation consistency.

Benchmark scope: MADBench spans inorganic materials and molecular subsets while maintaining internal consistency in the reference ab initio theory.The benchmark combines subsets from MAD, MPtrj, matbench-discovery, Alexandria, SPICE, and MD22.
Results: PET-OMAD demonstrates the best overall accuracy in almost all MADBench cases for both energy and force predictions.It falls behind PET-MPtrj and PET-OAM only on the MPtrj and Alexandria subsets.
Metrics: The reported accuracy tables use energy MAEs in meV per atom and force MAEs in meV/Å, with matbench-discovery force errors omitted because reference forces are unavailable.MACE and eSEN accuracies were taken from prior references.
Evaluation setup: Inference timings for PET and MACE were measured on a single Nvidia A100 80 GB GPU, while eSEN timings were reproduced from prior work.MACE-L timings were also included to assess setup-related differences.

Appendix G: The Open Molecules Benchmark

The Open Molecules Benchmark extends PET to charge- and spin-varying molecular systems and evaluates conservative and non-conservative models. The workflow combines specialized conditioning, geometry optimization, and finite-difference phonon calculations, with unconstrained relaxations sometimes changing lattice symmetry.

Model adaptation: PET models trained on OMol-1 use separate charge and spin embeddings projected into the node-feature dimensions.The projection layer is initialized to zero to stabilize training and combine conditioning with pretrained models.
Training: Conservative OMol models were fine-tuned for 7 epochs, and the authors state that additional training would likely improve them.Training duration was limited by available GPU resources.
Geometry optimization: Geometry optimizations use ASE with metatomic calculators, while constrained optimizations apply symmetry fixing and freeze off-diagonal strain degrees of freedom.Convergence is set to 10^-5 eV/Å for forces and appropriately scaled stress components.
Lattice dynamics: Phonon force constants use central finite differences with a 0.03 Å displacement to avoid instabilities from residual forces at symmetry-constrained minima.The phonon bands are evaluated using ASE’s phonons module.
Relaxed structures: Unconstrained relaxations return FCC and HCP structures from the C model but FCO and BCO structures from corresponding NC starting configurations.The reported lattice assignments depend on whether the initial structure is BCC or HCP.