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Can Machine Learning Potentials Be More Accurate Than the Methods They Are Trained On?

Pavlo O. Dral, Yi-Fan Hou, Yuxinxin Chen, Hassan Nawaz, Mateusz Witkowski, Szymon Śmiga, Ireneusz Grabowski

arXiv:2610.18027v1physics.chem-phcs.AIcs.LGphysics.comp-ph

TL;DR

MLP accuracy is commonly considered limited by the QM methods used for training, while experimental refinement is the established route described here for surpassing them. This Perspective proposes MLPs transferable across QM methods; its focal-point MLP achieves lower relative-energy error against the TBE than its most accurate training method.

  • Problem

    MLPs are commonly believed unable to exceed the accuracy of their training QM methods, while experimental refinement requires experiments and offers no systematic theory-only improvement route.

    Source: Introduction · p. 4

  • Method

    The Perspective proposes extending MLP generalization to QM-method space so models can predict at methods absent from training.

    Source: Introduction · p. 4

  • Results

    0.017 kcal/mol MAE in relative energies against the TBE—five times smaller than CCSD(T)/aug-cc-pVQZ’s 0.085 kcal/mol—and the same scheme on true QM energies gives 0.010 kcal/mol.

    Source: isting extrapolation schemes for MLPs · pp. 9, 11

  • Takeaways & Limitations

    One MLP can return predictions for each training QM method for use in extrapolation schemes as a post-processing step.

    Source: isting extrapolation schemes for MLPs · p. 8

    The focal-point MLP’s accuracy along the ammonia inversion profile is limited by training-geometry coverage and does not establish accuracy at arbitrary geometries.

    Source: Ammonia · p. 49

Abstract

from arXiv · show

Although machine learning potentials (MLPs) generalize across configurational and chemical compound spaces, their accuracy is commonly believed to be limited by their training methods, with improvements beyond this limit achieved so far only by refitting to experimental observables. In this Perspective, we argue that MLPs can also be made more accurate than their training methods non-empirically by generalizing across a third space, that of quantum-chemical methods. We propose routes to method-transferable MLPs that predict at unseen methods, with working implementations in both density-functional and wavefunction-based theories showing that they can be more accurate than any of their training methods. Aiming beyond the accuracy of the training methods could also make universal MLPs more accurate than single-method models, even when the method-transferable MLPs remain less accurate than those QM methods. Method-transferable MLPs will shape the construction of training datasets and the evaluation of models, pushing the accuracy limits of chemical simulations.

Introduction

MLPs generalize across geometries and chemical compounds, but their accuracy is commonly considered bounded by the QM methods used for training. The Perspective proposes extending generalization to QM-method space, enabling non-empirical accuracy gains from QM data alone.

Introduction

  • Figure 1 contrasts generalization across configurations, compounds, and methods, and depicts the target as accuracy beyond every training QM method.Its schematic also indicates that method-transferable MLPs can improve on single-method MLPs even when they do not reach that target.Source: Figure 1 · p. 3
  • MLPs predict unseen geometries and compounds, but the lack of generalization to unseen QM methods is believed to cap their accuracy at the training method.Configurational-space learning maps nuclear geometry to potential energy; compound-space learning supports predictions for molecules absent from training.Source: Introduction · pp. 2, 4
  • Experimental refinement can make MLP predictions of observables more accurate than their training methods, but it requires experiments and offers no systematic theory-only improvement route.Spectroscopic refinement of potential-energy surfaces against measured rovibrational levels has done this for decades.Source: Introduction · p. 4
  • Known relationships among QM methods, including Jacob’s ladder and Pople’s diagram, could guide models transferable across methods.These orderings concern density functionals and the treatment of correlation and basis sets, respectively.Source: Introduction · p. 4
  • The Perspective proposes method-transferable MLPs that predict at unseen QM methods and can exceed their training methods’ accuracy using QM data alone.This extends MLP generalization across configurations and compounds to a third space: QM methods.Source: Introduction · p. 4

Existing MLPs learning across methods

Existing approaches learn from heterogeneous QM-method data, often to reduce expensive high-level calculations or predict at methods represented in training. The Perspective identifies prediction at unseen methods as the major remaining limitation.

Existing MLPs learning across methods

  • Most existing methods for heterogeneous QM data aim to reduce expensive high-level calculations, rather than improve upon the training QM methods.The Perspective reviews these approaches as a foundation for method-transferable MLPs.Source: Existing MLPs learning across methods · p. 5
  • ∆-learning learns the difference between low- and high-level methods and needs the low-level baseline at inference, whereas transfer learning fine-tunes a low-level model on high-level data.Transfer-learned models avoid inference-time baseline calculations, but without a physical baseline may produce unphysical results; the approaches can also be combined.Source: Existing MLPs learning across methods · p. 5
  • Other approaches include low-fidelity properties as features, co-kriging across fidelity levels, and CQML grids spanning correlation treatment, basis set, and training-set size.These approaches are built to deliver the top of the hierarchy supplied to them.Source: Existing MLPs learning across methods · pp. 5, 7
  • Hierarchical machine learning stacks ∆-models across a QM-method hierarchy, reducing high-level reference-data generation by 99% and reaching spectroscopic accuracy of about 1 cm−1.These results are reported for an application of the approach and its later adoption in other applications.Source: Existing MLPs learning across methods · p. 7
  • A single MLP can learn all training methods using separate output heads or a method descriptor as input, allowing users to select a training method at inference.The authors advocate method descriptors; learned method embeddings and meta-learning are also discussed, though meta-learning requires fine-tuning before prediction at a new method.Source: Existing MLPs learning across methods · p. 7
  • The major limitation is that these approaches do not predict at a QM method absent from their training set.The Perspective identifies this as the limitation that caps MLP accuracy.Source: Existing MLPs learning across methods · p. 7

Exceeding the accuracy of the training methods

This section presents method-transferable MLPs that use post-training quantum-chemical extrapolation or continuous DFT method descriptors. Its demonstrations show predictions more accurate than the training methods and examine their use in simulations requiring many energy evaluations.

isting extrapolation schemes for MLPs

  • Reversing the order lets one MLP predict each training QM method before extrapolation schemes are applied as post-processing, with different methods labeling different geometries.When candidate schemes are known before training, including their prediction errors in the loss brings the MLP to the accuracy of those schemes on true QM energies.Source: isting extrapolation schemes for MLPs · pp. 8, 9, 12
  • 0.017 kcal/mol: the focal-point MLP’s held-out relative-energy MAE against the TBE is five times smaller than the best training method’s 0.085 kcal/mol.The same scheme applied to true QM energies gives 0.010 kcal/mol, and the MLP advantage persists along the relaxed inversion profile to about 12 kcal/mol above the minimum.Source: isting extrapolation schemes for MLPs · pp. 9, 11, Figure 3 · p. 10
  • For full-dimensional ammonia instantons, the focal-point MLP splitting is 0.754 cm−1, versus 0.669 cm−1 for the CCSD(T)/aug-cc-pVQZ MLP and 0.793 cm−1 experimentally.A full MLP calculation takes about 40 core-hours, avoiding repeated QM calculations for focal-point components.Source: isting extrapolation schemes for MLPs · p. 11
  • On held-out H· + H2 geometries, direct extrapolation gives 0.022 and focal-point extrapolation 0.059 kcal/mol; the same MLP can supply predictions for different schemes after training.For ammonia, the two schemes give 0.018 and 0.017 kcal/mol, respectively.Source: isting extrapolation schemes for MLPs · pp. 8, 11

space

  • For malonaldehyde, the fraction of exact exchange a lets one MLP query unseen hybrid functionals; aopt=0.455, chosen using two CCSD(T) energies, gives 0.05 versus 1.23 kcal/mol barrier error.The MLP is trained on energies and forces from three functionals, and the selected descriptor defines an actual DFT functional.Source: space · pp. 12–13
  • Away from the two geometries used to select aopt, the MLP’s reaction-path MAE against CCSD(T) is 0.14 kcal/mol, versus 0.46–0.84 kcal/mol for training functionals.It also predicts the withheld BLYP35 and BHandHLYP functionals with errors comparable to those at training functionals.Source: space · pp. 13, 15, Figure 4 · p. 14
  • At aopt, the DFT-trained MLP gives malonaldehyde tunneling splittings of 29.8 and 4.39 cm−1 for H and D, within 20% of the dedicated single-method MLP’s splittings.One splitting requires about 2.6 million bead energy-and-force evaluations; the MLP supplies them in about 13 core-hours, compared with an estimated 25,000–32,000 core-hours using cc-pVDZ DFT.Source: space · p. 15
  • Extending continuous method descriptors to wavefunction methods remains prospective; the proposed coordinates would represent changes in electron correlation and basis-set completeness with a meaningful joint limit.The authors say they are developing such coordinates and testing predictions at unseen methods.Source: space · p. 15

Outlook: toward universal method-transferable MLPs

The Perspective argues that method-transferable MLPs can improve universal models and outlines dataset and evaluation choices for pursuing that goal. Extending the demonstrated routes to universal MLPs remains an open accuracy question.

Outlook: toward universal method-transferable MLPs

  • The demonstrations use individual molecules with dense configurational sampling, whereas universal MLPs sample many compounds’ potential-energy surfaces more sparsely.Extending the constructions requires learning method relationships across chemical compound space, and whether this can exceed training-method accuracy remains open.Source: Outlook: toward universal… · pp. 15–16
  • For universal MLPs, UMA-M approaches direct DFT accuracy on selected ligandstrain and conformer benchmarks, but reducing fitting error alone cannot systematically correct training-method errors.Predicting at more accurate, unseen methods could address this limitation.Source: Outlook: toward universal… · p. 16
  • Training on extrapolation-scheme components could let users choose the scheme after training, while continuous method descriptors support predictions between and beyond training methods.The paper demonstrates the first route for ammonia and interpolation with continuous descriptors for malonaldehyde.Source: Outlook: toward universal… · p. 16
  • Multiple-method labels can teach relationships among QM methods, and complete high-level coverage at every geometry is unnecessary.Cheaper calculations can cover many geometries, supplemented by high-level calculations at selected ones; expensive high-level labels remain a constraint.Source: Outlook: toward universal… · pp. 16–17
  • Method transfer could make universal MLPs more accurate than single-method models even when neither reaches its QM target.OMNI-P1 improves when its descriptor changes from DFT to CCSD(T)*/CBS, although both predictions remain less accurate than direct DFT calculations.Source: Outlook: toward universal… · pp. 16–17

Code and data availability

The paper’s MLatom implementation and supporting materials are openly available under specified licenses.

Code and data availability

  • The MLatom implementation is available under Apache 2.0, while scripts, source, models, data, predictions, and selected logs are deposited on Zenodo.Original data, model parameters, and results use CC BY 4.0; third-party components retain their original licenses.Source: Code and data availability · p. 17

Authors contributions

The authors describe individual research roles and state that the AI Dral Group carried out calculations, analysis, and drafting under author direction and verification.

Authors contributions

  • P.O.D. conceived and directed the study, conducted the research, wrote the manuscript, and takes responsibility for its content.Y.-F.H. proposed continuous DFT-functional learning and developed the active-learning protocol; Y.C. developed the OMNI-P architecture extension.Source: Authors contributions · p. 18
  • H.N. and M.W. conducted a preliminary study of continuous wavefunction-method descriptors, demonstrating feasibility of MLPs more accurate than their training QM methods.S.´S. and I.G. contributed discussions on extrapolation schemes and learning across DFT functionals.Source: Authors contributions · p. 18
  • The AI Dral Group carried out calculations, analysis, and drafting under the authors’ direction and verification, with results reproducible from the repository by one command.Source: Authors contributions · p. 18

Declaration of generative AI use

The paper reports substantive use of generative AI by an agent collective, with human authors directing and verifying the work.

Declaration of generative AI use

  • The AI Dral Group executed the study using Claude Code, Codex, and Protomia agents with named models from Anthropic, OpenAI, and Qwen.The human authors directed and verified every step and take full responsibility for the content.Source: Declaration of generative AI use · p. 18

Computational details

The protocols pair PySCF reference calculations with method-descriptor MLPs trained on route-specific energy targets. Active learning supplies geometries, while tunneling and rate calculations evaluate downstream properties.

Programs and reference QM calculations

  • PySCF reference calculations use fixed geometries for relative energies; CCSD(T) forces are excluded because its analytic gradients deviate from finite differences.The interpolation route instead references the B3LYP/cc-pVDZ minimum.Source: Programs and reference QM calculations · pp. 31–32

The method-transferable MLP

  • The ANI-type MLP uses one-hot method descriptors for extrapolation or exact-exchange fraction for interpolation, learning method-specific energies and forces where available.Extrapolation targets relative energies; interpolation targets total energies.Source: The method-transferable MLP · pp. 32–34

Active learning

  • Active learning labels only the first uncertain geometry along each simulation path, using model disagreement against a validation-set threshold, then retrains iteratively.The default threshold is the validation median plus three median absolute deviations of the disagreement.Source: Active learning · p. 36

Tunneling splittings by ring-polymer instanton theory

  • Full-dimensional instanton splittings use 1,024 beads, within 0.6% of the bead limit; the one-dimensional ammonia treatment overestimates experimental NH3 splitting 1.8-fold on TBE.The ammonia surfaces are therefore compared through their splitting ratios to TBE.Source: Tunneling splittings by ring-polymer… · p. 38

Rate constants

  • D· + H2 rates use transition-state theory with Eckart tunneling and are compared with TBE under identical treatment; at 200 K, changing the tunneling correction shifts ratios by 17%.The rate constants are for D· + H2 → HD + H· because H· + H2 is degenerate and has no measurable rate constant.Source: Rate constants · pp. 38–40

Extrapolation after training

The section applies two extrapolation schemes to MLP predictions and evaluates them on ammonia and H· + H2. Results show that these schemes can outperform the training QM methods, while accuracy remains dependent on geometry and the chosen scheme.

The schemes and the TBE

  • The direct-extrapolation scheme extrapolates HF and CCSD(T) correlation energies, while the focal-point scheme extrapolates MP2 correlation energy; the TBE uses direct extrapolation with X = 5 and Y = 4.Both schemes amplify changes between basis sets, but shared MLP errors across the two basis sets cancel; the TBE shares its rule with direct extrapolation and can favor it in comparisons between schemes.Source: The schemes and the TBE · pp. 41–42

Training data

  • Training combines grid and random geometries generated before QM calculations with active-learning additions, while validation and held-out test geometries are generated separately.Validation geometries were used during development; held-out test geometries were drawn afterward and not used for decisions, with 189 H· + H2 and 297 ammonia geometries retained for testing.Source: Training data · pp. 44–46

Ammonia

  • On the ammonia held-out test set, focal-point and direct MLP errors against TBE are 0.017 and 0.018 kcal/mol, versus 0.085 for CCSD(T)/aug-cc-pVQZ.On validation geometries farther than 0.01 Å from training geometries, the two MLP errors are 0.016 and 0.018 kcal/mol; barrier agreement alone can conceal larger surface errors.Source: Ammonia · pp. 47–48, Table S10 · p. 48
  • The focal-point MLP follows the relaxed ammonia inversion profile more closely than CCSD(T)/aug-cc-pVQZ only up to about 12 kcal/mol above the minimum.At 86°, its error is −0.134 kcal/mol versus −0.104 for CCSD(T)/aug-cc-pVQZ, and it reaches −0.52 kcal/mol by 78°; only 82 active-learning geometries extend below 93°.Source: Ammonia · p. 49
  • On the relaxed ammonia inversion path, focal-point and direct MLP splittings are 0.97 and 1.00 times the TBE splitting, versus 0.89 for the CCSD(T)/aug-cc-pVQZ MLP.Full-dimensional instanton results raise the splitting of the best training QM method by 13–19%; the surfaces omit core–valence and relativistic corrections, and full-dimensional instanton error is unbenchmarked.Source: Ammonia · pp. 49–51, Table S13 · p. 51

Hydrogen exchange: extrapolation to full CI/CBS

  • On held-out H· + H2 geometries, direct and focal-point MLP MAEs against TBE are 0.022 and 0.059 kcal/mol, versus 0.118 for CCSD(T)/cc-pVQZ.The focal-point MLP is described as twice as accurate as the most accurate training method, but it falls short of the preset 0.05 kcal/mol upper-95%-confidence-bound margin.Source: Hydrogen exchange: extrapolation to… · p. 53, Table S14 · p. 53

Rate constants.

The section evaluates reaction-rate predictions and scheme-aware training losses. The rate ratios are close to the TBE at 300 and 1000 K, and scheme-loss terms improve extrapolation accuracy in the reported settings.

Rate constants.

  • At 300/1000 K, D· + H2 rate constants are 1.01/1.00 (direct) and 1.01/1.01 (focal point) times TBE, versus 0.68/0.87 for CCSD(T)/cc-pVQZ.At 200 K, sensitivity to the tunneling treatment prevents a resolved comparison.Source: Rate constants · p. 53, Table S16 · p. 54

The errors of the schemes in the training loss

  • For H· + H2, adding scheme errors to the training loss reduces the direct-extrapolation MLP’s difference from the same scheme on true QM from 0.10 to 0.02 kcal/mol.The loss includes both schemes so either remains available after training; Hessian terms yield average errors of 1.5% in the bending frequency and 0.6% in the imaginary frequency at the saddle point.Source: The errors of the schemes in the… · pp. 54–55
  • For ammonia, scheme-loss training reduces held-out test MAEs against TBE from 0.017 and 0.018 to 0.010 and 0.013 kcal/mol for focal-point and direct schemes.The comparison holds for five-model ensembles; the authors note that the result need not hold for a single model.Source: The errors of the schemes in the… · pp. 55–56

Interpolation along exact exchange

The malonaldehyde MLP interpolates continuously over exact-exchange fraction, and the selected aopt is tested against CCSD(T) across reaction-path, relaxed-property, and tunneling calculations.

Training data and active learning

  • Active learning produced a 1,826-geometry union from two loops that had not met the convergence criterion; only the three training functionals were calculated for dataset geometries.The two withheld functionals were calculated only for test geometries.Source: Training data and active learning · pp. 57–58

The MLP and the choice of aopt

  • The MLP takes nuclear coordinates and continuous exact-exchange fraction a as inputs, training on BLYP, B1LYP, and HFLYP while withholding BLYP35 and BHandHLYP.The training data contain energies and forces for 1,826 geometries at cc-pVDZ.Source: The MLP and the choice of aopt · p. 58
  • 0.05 kcal/mol is the aopt MLP’s CCSD(T) error at its own relaxed stationary points, versus 1.23–3.33 for training functionals; aopt=0.455 was selected by matching the barrier at fixed B3LYP geometries.That matched barrier is not an independent validation of the resulting MLP.Source: The MLP and the choice of aopt · p. 59, Table S17 · p. 59

Checks along the reaction path and across the method space

  • 0.43 kcal/mol is the aopt MLP’s CCSD(T) MAE on 132 held-out geometries near the IRC, versus 0.51 for B1LYP; the confidence interval of the difference includes zero.The interval is −0.28 to +0.07 kcal/mol.Source: Checks along the reaction path and… · p. 60, Table S19 · p. 61
  • 0.138 kcal/mol is the aopt MLP’s IRC mean absolute error against CCSD(T), versus 0.455–0.838 for training functionals, 0.631 for B3LYP, and 0.200/0.168 for linear/parabolic null models.Both null models require DFT energies at every geometry.Source: Table S18 · p. 60, Checks along the reaction path and… · p. 61
  • Ensemble disagreement flagged every trajectory that broke or entered an energy hole before it did, but holes between recomputed snapshots mean hole-free counts are upper bounds.The production dynamics used 50 NVE trajectories of 2.5 ps per a value.Source: Checks along the reaction path and… · p. 63
  • At aopt, the relaxed barrier differs from true DFT by at most 0.03 kcal/mol, while minimum harmonic-frequency differences average 1.8–3.2 cm−1; zero-point energy lowers the barrier from 3.94 to 1.29 kcal/mol.Figure S3 and Table S22 compare MLP and real-functional properties across exact-exchange fractions.Source: Checks along the reaction path and… · p. 62, Figure S3 · p. 63, Table S22 · p. 64

Tunneling splittings of malonaldehyde

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