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

physics.chem-phOct 13, 2026

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

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

Organizations: State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry and Chemical Engineering, and Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, Xiamen University, Xiamen, Fujian 361005, China · Institute of Physics, Faculty of Physics, Astronomy, and Informatics, Nicolaus Copernicus University in Toruń, ul. Grudziadzka 5, 87-100 Toruń, Poland · Institute of Advanced Studies, Nicolaus Copernicus University in Toruń, ul. Wileńska 4, 87-100 Toruń, Poland · Aitomistic, Shenzhen 518000, China

Abstract

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.

Figures & tables

Figure 1: Exceeding the accuracy of the training methods with MLPs. (a) Generalization across configurations, compounds and QM methods. FCI includes all determinants (all excitation ranks) in the chosen basis. Throughout this Perspective, filled/open markers denote training/unseen cases; noncircular method symbols distinguish them from atoms and neurons. (b) Schematic MLP error against the same more accurate reference as the number of training geometries grows. The dashed levels are limits of exact reproduction of the training methods. The orange arrow marks the accuracy gain; green shading indicates the target: accuracy beyond every training QM method. Even when this target is not reached, method-transferable MLPs can improve on single-method MLPs. (c) Empirical refinement compares simulated observables, such as IR spectra, with experiment to refine a QM-pretrained MLP. Röcken and Zavadlav (2024) ; Gumber et al. (2025) ; Han and Yu (2025) (d) Known relationships between QM methods, represented by Pople’s diagram and Jacob’s ladder, could guide method-transferable MLPs.

Figure 2: Existing approaches for learning in method space.

E

denotes energy;

L

M

and

H

denote low, intermediate and high levels.

\Delta E_{H-L}

is a learned high-minus-low energy correction; numbered subscripts and

m

identify individual QM methods. (a)

\Delta

-learning; (b) transfer learning; (c) low-fidelity properties as features; (d) hierarchical machine learning; (e) co-kriging, where

K_{LL}

and

K_{HH}

are low- and high-fidelity covariance blocks and the off-diagonal blocks couple them; (f) the combination technique CQML, whose third axis is training-set size; (g) multi-task outputs; (h) fixed or learned method descriptors as input to MLP; and (i) meta-learning followed by adaptation using new-method training data.

Figure 3: One MLP, extrapolation after training: the inversion of ammonia. (a) Inversion through the planar saddle point. (b) One MLP predicts energies at geometry

\mathbf{R}

for every training QM method

m

(method and basis set). Numbers count labelled training geometries used for training; purple squares and green diamonds identify energies entering the focal-point (Eq. 1 ) and direct-extrapolation (SI, Eq. S3 ) schemes, respectively;

Y

= aug-cc-pVTZ,

X

= aug-cc-pVQZ. (c) Mean absolute error (MAE) of relative energies against the theoretical best estimate (TBE) on the held-out test set, for all training methods and both schemes. Bars: true QM; symbols: corresponding MLP predictions (filled: training methods; open: unseen schemes); dashed line: best training QM method. (d) One-dimensional inversion splitting on each relaxed surface, divided by the TBE-surface splitting; bars and symbols as in (c). (e) Relaxed inversion-profile errors against the TBE for both MLP schemes and the best training QM method in true QM. The umbrella angle

\varphi

measures N–H tilt out of the hydrogen plane; N–H distances are relaxed on each surface. Dotted lines bound the training geometries.

Figure 4: The fraction of exact exchange as the method descriptor . (a) The proton transfer in malonaldehyde: the B3LYP minimum, transition state and equivalent minimum as 2D and 3D structures. (b) The MLP of Eq. 3 , trained at three values of $a$ and queried at any $a$ . (c) The space of Eq. 2 with its five named functionals, B3LYP beside it, and $a_{\mathrm{opt}}$ . (d) The barrier at the two B3LYP stationary geometries against $a$ : the real functionals (DFT with Eq. 2 ), the MLP and CCSD(T), and where each meets CCSD(T). (e) The energy along the B3LYP reaction path, with the mean absolute error (MAE) against CCSD(T) over its 12 symmetry-unique geometries. (f) The tunneling splitting of malonaldehyde (H) and of its deuterated isotopologue (D) from ring-polymer instanton calculations on the MLP against $a$ , the range over the ensemble’s five MLPs at $a_{\mathrm{opt}}$ , and experiment (H Baba et al. (1999) and D Baughcum et al. (1984) ).

system	QM method	descriptor	labels	share
$\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$ \textperiodcentered $}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$	HF/cc-pVTZ	$\mathbf{e}_{1}$	energies, forces	100%
	MP2/cc-pVTZ	$\mathbf{e}_{2}$	energies, forces	100%
	CCSD/cc-pVTZ	$\mathbf{e}_{3}$	energies, forces	60%
	CCSD(T)/cc-pVTZ	$\mathbf{e}_{4}$	energies	60%
	full CI/cc-pVTZ	$\mathbf{e}_{5}$	energies	25%
	HF/cc-pVQZ	$\mathbf{e}_{6}$	energies, forces	63%

Table S1 : Training QM methods, method descriptors and labels of the MLPs of both routes.

\mathbf{e}_{k}

is the one-hot vector with 1 at position

k

of 9 (

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

) or 8 (ammonia); the share is that of the training geometries used for training with the QM method (percentages rounded to integers) (p. Training data and p. Training data and active learning ).

training QM method	3.3 Å	3.5 Å	4.0 Å
HF/cc-pVTZ	0.034	0.026	0.031
MP2/cc-pVTZ	0.022	0.022	0.023
CCSD/cc-pVTZ	0.024	0.022	0.027
CCSD(T)/cc-pVTZ	0.028	0.029	0.031
full CI/cc-pVTZ	0.033	0.031	0.040
HF/cc-pVQZ	0.027	0.029	0.026

Table S2 : Choice of the angular cutoff of the MLP for the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction by five-fold cross-validation: MAE of relative energies on the validation fold in kcal/mol, averaged over the folds, for each training QM method and their mean (standard deviation over the folds in parentheses). The five MLPs of the chosen cutoff (3.5 Å) form the ensemble.

training QM method	3.3 Å	3.5 Å	4.0 Å
HF/aug-cc-pVTZ	0.005	0.005	0.006
MP2/aug-cc-pVTZ	0.004	0.005	0.005
CCSD/aug-cc-pVTZ	0.005	0.005	0.004
CCSD(T)/aug-cc-pVTZ	0.008	0.009	0.008
HF/aug-cc-pVQZ	0.009	0.012	0.010
MP2/aug-cc-pVQZ	0.006	0.007	0.008

Table S3 : Choice of the angular cutoff of the MLP for the ammonia by five-fold cross-validation: MAE of relative energies on the validation fold in kcal/mol, averaged over the folds, for each training QM method and their mean (standard deviation over the folds in parentheses). The five MLPs of the chosen cutoff (3.3 Å) form the ensemble.

cutoff (Å)	RMSE	standard deviation over the folds
3.3	0.179	0.030
3.5	0.192	0.038
4.0	0.165	0.055

Table S4 : Choice of the angular cutoff of the atomic environment vectors of the malonaldehyde MLP: root-mean-square error (RMSE) of the energies in five-fold cross-validation and its standard deviation over the five folds, in kcal/mol, on 450 geometries of an early design study of the initial data set, labelled with the three training functionals. These geometries are not the training set of the reported MLP. The four angular shifts are spread evenly from 0.9 Å to the cutoff.

pair of QM methods	first	second	difference	correlation
full CI/cc-pVTZ $-$ CCSD(T)/cc-pVTZ	0.010	0.010	0.006	0.95
CCSD(T)/cc-pVTZ $-$ CCSD/cc-pVTZ	0.010	0.014	0.015	0.67
CCSD(T)/cc-pVQZ $-$ CCSD/cc-pVQZ	0.011	0.013	0.013	0.56
MP2/cc-pVQZ $-$ MP2/cc-pVTZ	0.012	0.010	0.005	0.95
CCSD(T)/cc-pVQZ $-$ CCSD(T)/cc-pVTZ	0.011	0.010	0.004	0.98
HF/cc-pVQZ $-$ HF/cc-pVTZ	0.014	0.010	0.008	0.88

Table S5 : Errors of the MLP (the ensemble’s averaged prediction) against true QM on the validation set of the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction for pairs of training QM methods: the MAE of the first and of the second QM method, the MAE of their difference, and the correlation coefficient of the two errors; MAEs in kcal/mol. A difference whose error is smaller than that of either QM method means an error shared by both, which the schemes do not amplify.

pair of QM methods	first	second	difference	correlation
CCSD(T)/aug-cc-pVTZ $-$ CCSD/aug-cc-pVTZ	0.008	0.003	0.007	0.61
CCSD(T)/aug-cc-pVQZ $-$ CCSD/aug-cc-pVQZ	0.009	0.007	0.002	0.97
MP2/aug-cc-pVQZ $-$ MP2/aug-cc-pVTZ	0.004	0.002	0.004	0.50
CCSD(T)/aug-cc-pVQZ $-$ CCSD(T)/aug-cc-pVTZ	0.009	0.008	0.004	0.88
HF/aug-cc-pVQZ $-$ HF/aug-cc-pVTZ	0.008	0.003	0.008	0.34

Table S6 : Errors of the MLP (the ensemble’s averaged prediction) against true QM on the validation set of the ammonia inversion for pairs of training QM methods: the MAE of the first and of the second QM method, the MAE of their difference, and the correlation coefficient of the two errors; MAEs in kcal/mol. A difference whose error is smaller than that of either QM method means an error shared by both, which the schemes do not amplify.

	cc-pVXZ				aug-cc-pVXZ
QM method	DZ	TZ	QZ	5Z	DZ	TZ	QZ	5Z
HF	16.92	17.56	17.61	17.61	16.81	17.57	17.61	17.61
MP2	13.22	13.27	13.09	13.03	12.56	13.10	13.01	12.99
CCSD	10.41	10.30	10.10	10.05	9.49	10.07	10.02	10.02
CCSD(T)	10.22	10.01	9.79	9.73	9.26	9.77	9.70	9.70
full CI	10.15	9.94	9.71	–	9.18	9.69	9.63	–

Table S7 : Barrier of the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

{}\mathrel{\mathop{\makebox[0.0pt]{\rightarrow\displaystyle}}\limits}{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

{}+{}

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

reaction in kcal/mol calculated with true QM methods at one saddle-point geometry (

r_{\mathrm{HH}}=0.930

Å) relative to

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

(

r_{\mathrm{HH}}=0.7414

Å); unrestricted HF reference for the open-shell species. The exact barrier is 9.61 kcal/mol. Full CI is exact for a given basis set, i.e., its error is only due to the incompleteness of the basis set. Dashes: not calculated.

	cc-pVXZ				aug-cc-pVXZ
QM method	DZ	TZ	QZ	5Z	DZ	TZ	QZ	5Z
HF	7.16	5.30	4.97	4.72	5.06	4.71	4.66	4.65
MP2	7.98	5.92	5.37	4.96	5.31	5.06	4.88	4.81
CCSD	8.27	6.18	5.63	5.29	5.67	5.42	5.23	5.17
CCSD(T)	8.51	6.36	5.78	5.41	5.80	5.53	5.34	5.28

Table S8 : Inversion barrier of ammonia in kcal/mol calculated with true QM methods (frozen core) at the reference geometries of the original focal-point analysis.

	$\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$ \textperiodcentered $}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ (with full CI $-$ CCSD(T))		ammonia
	cc-pVXZ	aug-cc-pVXZ	cc-pVXZ	aug-cc-pVXZ
direct extrapolation	9.53	9.56	5.52	5.22
focal point	9.60	9.52	5.56	5.25
TBE	9.59	9.61	5.23	5.23

Table S9 : Barriers from the two extrapolation schemes (Eqs. S3 and 1 ) applied to the true QM energies with the TZ and QZ basis sets, and the theoretical best estimate (TBE), Eq. S3 with the QZ and 5Z basis sets, all in kcal/mol. For the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction, all three include the difference between full CI and CCSD(T) calculated with the TZ basis set (Eq. S6 ), so that they approximate full CI/CBS; with the QZ basis set, this difference changes the barrier by 0.001 kcal/mol. The exact barrier of this reaction is 9.6089 kcal/mol: the TBE is 0.02 kcal/mol from it with cc-pVXZ, and the focal-point scheme happens to be closer at this one geometry (0.01 kcal/mol). The TBE is the reference of the SI because it can be calculated at every geometry; the exact value is known only for the barrier. Ammonia: at the reference geometries of Table S8 .

	true QM		the MLP		MLP vs
	MAE	max	MAE	max	true QM
focal-point scheme	0.010	0.035	0.017	0.103	0.017
direct-extrapolation scheme	0.013	0.071	0.018	0.070	0.015
CCSD(T)/aug-cc-pVQZ	0.085	0.275	0.083	0.271	0.010
MP2/aug-cc-pVTZ	0.092	0.400	0.092	0.399	0.003
CCSD/aug-cc-pVTZ	0.101	0.363	0.101	0.375	0.003

Table S10 : Held-out test set of the ammonia inversion: 297 geometries never used during development. Relative energies against the TBE, on every surface measured from that surface’s own energy at the reference geometry: MAE and maximum absolute error in kcal/mol of the true QM calculations and of the MLP (the ensemble’s averaged prediction), and the MAE of the MLP against the same QM method or scheme in true QM.

	true QM		the MLP		MLP vs
	MAE	max	MAE	max	true QM
focal-point scheme	0.010	0.040	0.015	0.123	0.015
direct-extrapolation scheme	0.011	0.069	0.017	0.089	0.013
CCSD(T)/aug-cc-pVQZ	0.077	0.305	0.076	0.294	0.009
MP2/aug-cc-pVTZ	0.092	0.323	0.092	0.319	0.002
CCSD/aug-cc-pVTZ	0.097	0.381	0.096	0.375	0.003

Table S11 : Relative energies of the 296 geometries of the validation set (used during development, not a test set) of the ammonia inversion against the TBE, on every surface measured from that surface’s own energy at the reference geometry, for the MLP as for the QM methods: MAE and maximum absolute error in kcal/mol of the true QM calculations and of the MLP (the ensemble’s averaged prediction), and the MAE of the MLP against the same QM method or scheme in true QM.

Figure S1 : The MLP over the validation geometries. Errors against the TBE of the relative energies of the validation geometries (the reaction path, the saddle region, the reactants and thermal distortions around them; lines for the series along the two paths, symbols for every geometry) of the $\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$\textbf{\textperiodcentered}$}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ reaction (a; against the antisymmetric stretch, zero at the saddle point) and of the inversion of ammonia (b; against the H–N–H angle), for the MLP with both schemes and for the most accurate training QM method, CCSD(T) with the larger basis set, in true QM (filled symbols: training QM method; open symbols: unseen schemes); below each panel, the MAE over all validation geometries. The dip of the direct-extrapolation scheme at the saddle point in (a) is that of the scheme itself, which the MLP reproduces ( $-0.06$ kcal/mol in true QM). — Figure S1 : The MLP over the validation geometries. Errors against the TBE of the relative energies of the validation geometries (the reaction path, the saddle region, the reactants and thermal distortions around them; lines for the series along the two paths, symbols for every geometry) of the $\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$ **\textperiodcentered** $}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ reaction (a; against the antisymmetric stretch, zero at the saddle point) and of the inversion of ammonia (b; against the H–N–H angle), for the MLP with both schemes and for the most accurate training QM method, CCSD(T) with the larger basis set, in true QM (filled symbols: training QM method; open symbols: unseen schemes); below each panel, the MAE over all validation geometries. The dip of the direct-extrapolation scheme at the saddle point in (a) is that of the scheme itself, which the MLP reproduces ( $-0.06$ kcal/mol in true QM).

Figure S2 : Two extrapolation schemes applied to the same training QM methods, in true QM and on the MLP. (a) The barrier of the $\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$\textbf{\textperiodcentered}$}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ reaction in kcal/mol with every QM method and the cc-pVXZ basis sets at the benchmark saddle geometry ( $r_{\mathrm{HH}}=0.930$ Å), Anderson (2016) in parentheses and in color the error against the exact barrier of 9.6089 kcal/mol; the training QM methods are framed (full CI with cc-pV5Z was not calculated). Below the table: the TBE and both schemes applied to the true QM energies and to the MLP’s, all including $\Delta E_{\mathrm{FCI}}$ . (b) The same for the inversion barrier of ammonia with the augmented basis sets at the geometries of the original focal-point analysis, Császár et al. (1998) with the errors against the TBE. — Figure S2 : Two extrapolation schemes applied to the same training QM methods, in true QM and on the MLP. (a) The barrier of the $\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{$ **\textperiodcentered** $}}$ ${}+{}$ $\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}$ reaction in kcal/mol with every QM method and the cc-pVXZ basis sets at the benchmark saddle geometry ( $r_{\mathrm{HH}}=0.930$ Å), Anderson (2016) in parentheses and in color the error against the exact barrier of 9.6089 kcal/mol; the training QM methods are framed (full CI with cc-pV5Z was not calculated). Below the table: the TBE and both schemes applied to the true QM energies and to the MLP’s, all including $\Delta E_{\mathrm{FCI}}$ . (b) The same for the inversion barrier of ammonia with the augmented basis sets at the geometries of the original focal-point analysis, Császár et al. (1998) with the errors against the TBE.

	true QM				the MLP
	barrier	NH 3	ND 3	15 NH 3	barrier	NH 3	ND 3	15 NH 3
HF/aug-cc-pVTZ	4.828	3.530 (2.45)	0.322 (3.29)	3.377 (2.47)	4.816	3.551 (2.47)	0.324 (3.31)	3.397 (2.48)
MP2/aug-cc-pVTZ	5.053	1.701 (1.18)	0.123 (1.26)	1.620 (1.18)	5.048	1.701 (1.18)	0.123 (1.26)	1.621 (1.18)
CCSD/aug-cc-pVTZ	5.419	1.287 (0.89)	0.084 (0.86)	1.223 (0.89)	5.410	1.289 (0.89)	0.084 (0.86)	1.225 (0.89)
CCSD(T)/aug-cc-pVTZ	5.532	1.068 (0.74)	0.066 (0.67)	1.014 (0.74)	5.533	1.062 (0.74)	0.065 (0.67)	1.009 (0.74)
HF/aug-cc-pVQZ	4.785	3.687 (2.56)	0.341 (3.49)	3.528 (2.58)	4.778	3.688 (2.56)	0.341 (3.48)	3.529 (2.58)

Table S12 : Inversion barrier (kcal/mol) and splittings (cm -1 ) of ammonia and its isotopologues on every surface, in true QM and from the MLP (the ensemble), with the same one-dimensional treatment of the relaxed inversion path; in parentheses the ratio to the splitting of the true QM TBE surface, so that every surface is compared under the same treatment. The measured splitting of NH 3 is 0.79 cm -1 .

surface		$N=1024$	$N=512$ (%)	experiment	ratio
direct extrapolation	NH 3	0.7645	+1.7	0.7934	0.96
	ND 3	0.04867	+0.9	0.053	0.92
	15 NH 3	0.7295	+1.7	0.7577	0.96
focal point	NH 3	0.7539	+1.7	0.7934	0.95
	ND 3	0.04730	+0.9	0.053	0.89
	15 NH 3	0.7192	+1.7	0.7577	0.95

Table S13 : Full-dimensional inversion splittings (cm -1 ) of ammonia and its isotopologues from ring-polymer instanton theory (half ring polymer of

N

beads at 25 K, networks in double precision) on the final

\text{NH}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{3}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{3}}\text{}}

MLP (five-model ensemble trained on the archived and the active-learning geometries, without the errors of the schemes in the training loss) with the direct and focal-point schemes and at its highest training method, CCSD(T)/aug-cc-pVQZ; beside them, experiment. Change with

N

: splitting at

N=512

relative to

N=1024

. On one-dimensional ammonia double wells, the same instanton calculation is 7–10 % above the exact splitting.

	true QM		the MLP		MLP vs
	MAE	max	MAE	max	true QM
direct-extrapolation scheme	0.019	0.070	0.022	0.410	0.014
focal-point scheme	0.047	0.198	0.059	0.292	0.018
CCSD(T)/cc-pVQZ	0.118	0.483	0.116	0.458	0.013
CCSD/cc-pVQZ	0.184	0.790	0.185	0.857	0.014
full CI/cc-pVTZ	0.261	1.062	0.261	1.060	0.013

Table S14 : Held-out test set of the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction: 189 geometries never used during development. Relative energies against the TBE, on every surface measured from that surface’s own energy at the reference geometry: MAE and maximum absolute error in kcal/mol of the true QM calculations and of the MLP (the ensemble’s averaged prediction), and the MAE of the MLP against the same QM method or scheme in true QM.

	true QM		the MLP		MLP vs
	MAE	max	MAE	max	true QM
direct-extrapolation scheme	0.023	0.077	0.025	0.325	0.011
focal-point scheme	0.037	0.288	0.042	0.219	0.014
CCSD(T)/cc-pVQZ	0.120	0.472	0.117	0.462	0.011
CCSD/cc-pVQZ	0.212	0.823	0.212	0.861	0.013
full CI/cc-pVTZ	0.261	1.028	0.261	1.067	0.010

Table S15 : Relative energies of the 274 geometries of the validation set (used during development, not a test set) of the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction against the TBE, on every surface measured from that surface’s own energy at the reference geometry, for the MLP as for the QM methods: MAE and maximum absolute error in kcal/mol of the true QM calculations and of the MLP (the ensemble’s averaged prediction), and the MAE of the MLP against the same QM method or scheme in true QM.

	true QM				the MLP
	barrier	$\nu^{\ddagger}$	bend	$k/k_{\mathrm{TBE}}$ (300, 500, 1000 K)	barrier	$\nu^{\ddagger}$	bend	$k/k_{\mathrm{TBE}}$ (300, 1000 K)
HF/cc-pVTZ	17.58	2298i	914	0.00, 0.00, 0.03	17.56	2305i	940	0.00, 0.02
MP2/cc-pVTZ	13.23	1968i	930	0.02, 0.04, 0.16	13.22	1988i	929	0.02, 0.16
CCSD/cc-pVTZ	10.30	1539i	917	0.31, 0.46, 0.64	10.29	1540i	955	0.26, 0.58
CCSD(T)/cc-pVTZ	10.01	1510i	917	0.45, 0.59, 0.74	10.01	1521i	919	0.45, 0.73
HF/cc-pVQZ	17.63	2302i	903	0.00, 0.00, 0.03	17.60	2311i	925	0.00, 0.02

Table S16 : The collinear saddle point of every surface of the

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

reaction, in true QM and of the MLP (the ensemble, from its energies at the same geometries): barrier in kcal/mol, the imaginary and the bending frequency of

\text{H}{-}\text{H}{-}\text{H}

in cm -1 , and the rate constant of

\text{D}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

{}+{}

\text{H}{\vphantom{\text{X}}}_{\vphantom{\text{2}}\smash[t]{\text{2}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{2}}\text{}}

{}\mathrel{\mathop{\makebox[0.0pt]{\rightarrow\displaystyle}}\limits}{}

{}+{}

\text{H}{\vphantom{\text{X}}}^{\vphantom{\smash[t]{\text{2}}}\hphantom{\text{}}\text{

\textperiodcentered

}}

from the same transition-state theory with Eckart tunneling as the ratio to that of the TBE surface at 300 and 1000 K; the same treatment is used for every surface, with its residual effect on the ratios assessed in Methods.

method	own	CCSD(T) there	difference
	cc-pVDZ
BLYP	1.20	3.57	$-2.36$
B1LYP	2.65	3.88	$-1.23$
BLYP35	3.28	3.91	$-0.62$
BHandHLYP	4.23	3.87	$+0.36$
HFLYP	6.96	3.64	$+3.33$

Table S17 : Relaxed barriers in kcal/mol: each method’s barrier between its own minimum and transition state, the CCSD(T) barrier in the same basis at the same two geometries, and the difference.

method	role	MAE
BHandHLYP	not trained	0.130
MLP at $a_{\mathrm{opt}}$		0.138
real functional at $a_{\mathrm{opt}}$		0.147
parabola in $a$ at $a_{\mathrm{opt}}$	null model	0.168
line B1LYP–HFLYP at $a_{\mathrm{opt}}$	null model	0.200
BLYP35	not trained	0.303

Table S18 : Errors of every method against CCSD(T)/cc-pVDZ along the B3LYP/cc-pVDZ IRC, the validation set (used during development): mean absolute error (MAE) in kcal/mol of the energies of the interior points of the IRC (the transition state and the two ends left out), relative to the same method at the B3LYP minimum. The geometries are the same for every method and none is chosen by the MLP. The two halves of the IRC are mirror images, so its 24 interior points are 12 symmetry-unique geometries. The two null models need the DFT energies of the trained functionals at every geometry.

method	role	MAE
BLYP35	not trained	0.409
line B1LYP–HFLYP at $a_{\mathrm{opt}}$	null model	0.424
MLP at $a_{\mathrm{opt}}$		0.425
real functional at $a_{\mathrm{opt}}$		0.428
parabola in $a$ at $a_{\mathrm{opt}}$	null model	0.433
BHandHLYP	not trained	0.468

Table S19 : Held-out test set of malonaldehyde: 132 geometries never used during development, displaced from the 12 symmetry-unique interior points of the B3LYP/cc-pVDZ IRC by classical normal-mode motion at 50 K. Mean absolute error (MAE) in kcal/mol against CCSD(T)/cc-pVDZ of the energies relative to the same method at the B3LYP minimum.

functional		MAD
BLYP	fit	0.009
B1LYP	fit	0.015
BLYP35	interpolation	0.013
BHandHLYP	interpolation	0.009
HFLYP	fit	0.007
real functional at $a_{\mathrm{opt}}$	at $a_{\mathrm{opt}}$	0.010

Table S20 : The MLP against DFT with the same functional along the B3LYP/cc-pVDZ IRC: mean absolute difference (MAD) in kcal/mol between the MLP at a value of

a

and DFT with the functional of the same

a

, over the 12 symmetry-unique interior geometries, for the three training functionals, for the two functionals not used in training (BLYP35 and BHandHLYP; the MLP interpolates in

a

between the training functionals), and for the functional with

a=a_{\mathrm{opt}}

. The errors of the MLP against CCSD(T) are in Table S18 .

		cc-pVDZ
method	$a$	barrier	$-$ CCSD(T)
BLYP	0.00	1.16	$-2.64$
B1LYP	0.25	2.63	$-1.16$
BLYP35	0.35	3.21	$-0.59$
BHandHLYP	0.50	4.04	+0.25
HFLYP	1.00	6.33	+2.53

Table S21 : The barrier (kcal/mol) at the two B3LYP/cc-pVDZ stationary geometries: the five named functionals of Eq. 2 , B3LYP and CCSD(T), each functional’s difference from CCSD(T), and the fraction of exact exchange at which the real functional meets CCSD(T) (linear interpolation on a grid of 0.05 in

a

Figure S3 : The MLP across the space (cc-pVDZ): the relaxed barrier between its own minimum and transition state, with and without the harmonic zero-point energy, and the imaginary frequency at the transition state, on a grid of 21 values of $a$ (lines), against the real functionals at their own stationary points (symbols; the star is the real functional at $a_{\mathrm{opt}}$ ); the numbers are the MLP minus the real functional.

method	$a$	barrier	+ZPE	$\|\omega_{\mathrm{TS}}\|$	barrier	+ZPE	$\|\omega_{\mathrm{TS}}\|$	frequencies
		real functional			MLP
BLYP	0.000	1.20	$-0.87$	902	1.23	$-0.85$	889	3.2
B1LYP	0.250	2.65	0.21	1159	2.67	0.21	1146	2.0
BLYP35	0.350	3.28	0.74	1245	3.29	0.73	1232	2.1
BHandHLYP	0.500	4.23	1.57	1356	4.22	1.53	1346	1.8
HFLYP	1.000	6.96	4.00	1638	6.94	4.00	1643	2.6

Table S22 : Relaxed barrier (kcal/mol), barrier with the harmonic zero-point energy, and imaginary frequency (cm -1 ) at each method’s own stationary points, from the real functional and from the MLP at the same

a

(cc-pVDZ), and the mean absolute difference of the harmonic frequencies of the minimum between the two (cm -1 ).

beads	splitting	$S/\hbar$
512	28.41	5.6747
1024	29.33	5.6788
2048	29.57	5.6793

Table S23 : Convergence of the tunneling splitting (cm -1 ) with the number of beads of the half ring polymer, on a test MLP at

a=0.461

(cc-pVDZ, one model), with the zero mode removed by projecting out the tangent of the path, and the action

S/\hbar

. All three runs are on one machine with the settings of the splittings of the main text: the minima optimized with the tightest criteria of geomeTRIC and polished by two Newton steps, i-PI’s convergence thresholds tightened tenfold, and the masses of the most abundant isotopes. The splitting is 28.41, 29.33 and 29.57 cm -1 with 512, 1024 and 2048 beads; the steps shrink as the inverse square of the number of beads, which extrapolates to 29.66 cm -1 , 1.1% above its 1024-bead value. In pairs of runs on one machine, the splitting changes by

+3.2

% from 25 to 12.5 K at 2048 beads, and at 512 beads by

-0.6

% when i-PI’s thresholds are tightened tenfold and by

-0.7

% with the isotope masses instead of i-PI’s standard atomic weights. At the minimum, the finite differences of the forces that give every Hessian of the calculation reproduce the analytic harmonic frequencies of the MLP to 0.16 cm -1 , which changes the zero-point energy by

-0.11

cm -1 . The determinant with the zero mode projected out, from the block factorization used for every splitting, agrees with a dense calculation on the full Hessian of 3 runs to

3\times 10^{-10}

in its logarithm.

MLP	isotope	$a$	barrier	$S/\hbar$	splitting
ensemble	H	0.200 †	2.36	3.875	175.96
ensemble	H	0.250 †	2.67	4.261	123.59
ensemble	H	0.300 †	2.98	4.643	86.71
ensemble	H	0.350	3.29	5.016	61.06
ensemble	H	0.400	3.60	5.379	43.21
ensemble	H	0.450	3.91	5.728	30.89

Table S24 : Tunneling splittings (cm -1 ) from ring-polymer instanton calculations on the final MLP (cc-pVDZ, 1024 beads, 25 K): the ensemble of five across the space and at

a_{\mathrm{opt}}

, and its five members alone at

a_{\mathrm{opt}}

; the barrier of the MLP (kcal/mol) and the action

S/\hbar

of the instanton. With 2048 beads the splitting at

a_{\mathrm{opt}}

is 29.98 cm -1 (H,

+0.6

% from 1024 beads) and 4.41 cm -1 (D,

+0.3

% from 1024 beads). † : below

a=0.35

the zero-point energy cancels the barrier of the MLP (at

a=0.25

it takes it from 2.7 to 0.2 kcal/mol), the proton is not localized and instanton theory, a theory of deep tunneling, does not apply; these rows are listed for completeness and are not shown in the main text.

Machine Learning Interatomic Potentials

Machine learning interatomic potentials predict atomic energies and forces for fast simulations of molecules and materials.

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Multitask learning with semiempirical orbital charges enables sample-efficient MLIPs
May 22, 2026Ihor Neporozhnii, Sjoerd Hoogland, Oleksandr VoznyyMachine Learning Interatomic Potentials Multi-Task Learning
Department of Physical and Environmental Sciences, University of Toronto, 1065 Military Trail, Toronto, M1C 1A4, Ontario, Canada. · The Alliance for AI-Accelerated Materials Discovery, 10 King’s College Road, Toronto, M5S 3G4, Ontario, Canada. · Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, M5S 3G4, Ontario, Canada.
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1Multiscale Modeling of Fluid Materials, Department of Engineering Physics and Computation, TUM School of Engineering and Design and Department of Physics, TUM School of Natural Sciences, Technical University of Munich, Munich, 80333, Germany. · 2Atomistic Modeling Center, Munich Data Science Institute, Technical University of Munich, Garching, 85748, Germany.
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May 20, 2026Ali Ramlaoui, Alexandre Duval, Hannah Bull +4Disentangled Representation Learning Machine Learning Interatomic Potentials
Entalpic, Paris, France · Universit´e Paris-Saclay, CentraleSup´elec, Inria, Gif-sur-Yvette, France

May 22, 2026J/K move · Enter open · S save

Multitask learning with semiempirical orbital charges enables sample-efficient MLIPs

Ihor Neporozhnii, Sjoerd Hoogland, Oleksandr Voznyy

Department of Physical and Environmental Sciences, University of Toronto, 1065 Military Trail, Toronto, M1C 1A4, Ontario, Canada. · The Alliance for AI-Accelerated Materials Discovery, 10 King’s College Road, Toronto, M5S 3G4, Ontario, Canada. · Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, M5S 3G4, Ontario, Canada.

Machine Learning Interatomic Potentials Multi-Task Learning Data-Efficient Learning

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Machine learning interatomic potentials (MLIPs) require generating computationally expensive, large-scale training datasets to accurately simulate materials and molecules. Incorporating electronic structure information using multitask learning improves sample efficiency, however, training on full Hamiltonian matrices, which scale quadratically with the number of atoms, is intractable for large datasets. In this work, we show that multitask learning utilizing orbitally resolved semiempirical charges significantly improves sample efficiency and accuracy in MLIPs. To efficiently predict orbital charges, we implement a specialized equivariant model, reducing charge prediction error compared to an invariant baseline. By augmenting training with computationally inexpensive GFN1-xTB orbital charges, which scale linearly with the number of atoms, our model achieves a 58% reduction in energy mean absolute error relative to energy-only baselines, and requires five times less data to match their performance. Against the stronger baseline of joint energy-and-force training, orbital charges still reduce the energy error by 12%. Furthermore, our approach matches or outperforms models trained on expensive density functional theory (DFT) atomic charges, capturing orbitally resolved electronic complexity and forcing the network to learn a physically accurate latent space that spontaneously clusters metals by shared chemical properties. Because orbital charges are only required during training, this approach preserves inference efficiency, providing a scalable recipe for developing accurate, data-efficient foundation models for complex chemical systems.

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Machine Learning Interatomic Potentials

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