Risk-Controlled DFT Selection for Machine-Learned Interatomic Potentials

Kun Ryu; Yeo Jin Jung; Shyue Ping Ong; Claire Donnat; Ying Shirley Meng

stat.MLOct 14, 2026

Risk-Controlled DFT Selection for Machine-Learned Interatomic Potentials

Authors: Kun Ryu, Yeo Jin Jung, Shyue Ping Ong, Claire Donnat, Ying Shirley Meng

Organizations: Pritzker School of Molecular Engineering, The University of Chicago · Department of Statistics, The University of Chicago · Department of Material Science and Engineering, National University of Singapore

Abstract

Automated materials discovery requires evaluating large pools of candidate configurations, with density functional theory (DFT) as a major throughput bottleneck. Pretrained machine-learned interatomic potentials (MLIPs) approximate DFT cheaply, but their errors vary by orders of magnitude across configurations, making it difficult to decide when to trust the MLIP predictions and when to fall back to DFT. We frame this decision as a risk-controlled selection problem and solve it within a conformal risk control (CRC) framework. Each configuration is mapped from atomic numbers and Cartesian coordinates to a fixed geometric descriptor, which a lightweight predictor uses to estimate an MLIP-DFT disagreement score in energy, average force, and maximum force. CRC then calibrates this score into a selection threshold for DFT labeling such that the MLIP error on the unselected configurations, which keep their MLIP labels, stays below a user-chosen fraction $α$ of the expected error. Under an identical guarantee across eight dataset-MLIP pairs, our predictor routes 7% and 14% fewer configurations to DFT at $α=0.3$ than committee baselines. Because committee disagreement overlooks errors the MLIPs share, our gains are largest when the DFT labels and MLIP training data differ in DFT functional and basis set, and narrow when the two agree. Without model access, the structure-only predictor also ranks errors as well as the target MLIP's latent representation and outperforms widely used descriptors such as Smooth Overlap of Atomic Positions (SOAP) and Atom-Centered Symmetry Functions (ACSF).

Figures & tables

Figure 1: Overview of the MLIP–DFT selection framework. A molecular configuration $R$ (caffeine in this example) is converted into a chemistry-derived structural vector and passes through a lightweight predictor to estimate energy and force residuals. These estimates are normalized and combined into a predicted error score $\hat{s}(R)$ . Conformal risk control then calibrates a threshold that retains reliable MLIP predictions and sends higher-risk configurations to DFT. The score and the capped loss are fixed before calibration.

Dataset	# configs.	# identities	# atoms (mean)	elements
COMP6-GDB07to09-FULL [ Smith et al., 2018 ]	36,000	1,500	7–27 (17.0)	H, C, N, O
MACE-OFF23-FULL [ Eastman et al., 2023 ]	50,195	15,769	3–150 (30.9)	H, C, N, O, F, P, S, Cl, Br, I
QCML-6950-9 [ Ganscha et al., 2025 ]	62,550	6,950	2–23 (11.0)	H, C, N, O, F, P, S, Cl, Br, I

Table 1: Evaluation datasets. A configuration is a geometry and a molecule identity is the exchangeable unit; configurations per identity differ by nearly an order of magnitude across datasets. MACE-OFF23 contains

228

single-configuration groups without a recorded parent identity.

Figure 2: Every score meets the same guarantee; the DFT cost differs. Three representative cells. Top row : risk ratio

r(\alpha)

against the bound

r=1

. Middle row : DFT calls

k(\alpha)

, with the random-to-oracle headroom shaded and each cell’s

G

boxed. Bottom row :

N^{3}

-weighted DFT cost (% of all-DFT), a size-scaling proxy, not wall time. Lines are means over 20 calibration/test reallocations, and bands are

\pm 1

SD for ours and the committees. Dotted verticals mark

\alpha=0.30

Figure 3: Structure beats disagreement where the MLIPs share error; the gap closes as references converge. (a) Per-cell

G

. Brackets give the relation between dataset and MLIP training references. Right of the dashed line, ANI-2x

\times

COMP6 is a different MLIP and population, compared with its native and supervised ANI committees. The supervised committee on QCML

\times

MACE-OFF23(M) (

G=-0.28

) is drawn clipped. (b) DFT saved by ours at matched realized loss

\rho

(committee minus ours, pp of pool), averaged over the common attained

\rho

range; the mean excludes ANI-2x. (c) Equal-cell mean

G

(small markers: cells) for every score under the same CRC rule; the right column gives MLIP evaluations per configuration and required model access.

Appendix figures & tables34 assets

Supplementary material from the paper’s appendix.

Appendix

Feature family	Dim.	Type	Description
$\mathrm{ElementCount}$	10	Discrete	Global counts of the allowed elements.
$\mathrm{Coordination}$	52	Discrete	Element-resolved proximity-graph coordination states.
$\mathrm{PairState}$	156	Discrete	Element-pair contacts classified as compressed, near nominal separation, or elongated.
$\mathrm{ShortRange}$	21	Continuous	Atomic-number-weighted severity of short interatomic contacts.
$\mathrm{Extreme}$	12	Continuous	Most severe local proximity, strain, and angular-distortion environments.
$\mathrm{RBF}$	18	Continuous	Smooth multiscale pair-distance channels with global and local pooling.

Appendix

Table 2: Feature families of the structural vector. The canonical Coordination and PairState blocks contain

60

and

165

coordinates; a fixed compaction removes eight and nine of them, leaving

52

and

156

. The descriptor contains the nine families and has

350

dimensions.

dataset	evaluation reference	target MLIP	MLIP training reference
COMP6	$\omega$ B97X/6-31G(d), Gaussian 09	MACE-OFF23(M)	$\omega$ B97M-D3(BJ)/def2-TZVPPD
		AIMNet2	$\omega$ B97M-D3/def2-TZVPP
		AceFF-2	$\omega$ B97M-V/def2-TZVPPD, VV10
QCML	PBE0, FHI-aims	MACE-OFF23(M)	$\omega$ B97M-D3(BJ)/def2-TZVPPD
		AIMNet2	$\omega$ B97M-D3/def2-TZVPP
		AceFF-2	$\omega$ B97M-V/def2-TZVPPD, VV10

Appendix

Table 3: Electronic-structure references. Each dataset is paired with the reference used to train each target MLIP; at least one component differs in every combination [ Smith et al., 2018 , Kovács et al., 2025 , Anstine et al., 2025 , Farr et al., 2026 , Ganscha et al., 2025 ] .

Figure 4: Every score meets the same risk guarantee in every cell. Risk ratio

r(\alpha)

(realized capped test loss over the budget

\epsilon_{0}=\alpha\bar{\ell}_{\rm fit}

) for all eight cells and five scores. Faint lines are the

20

calibration/test reallocations; bold lines and bands are means

\pm 1

SD; the dashed line marks

r=1

. Single draws above

1

are expected because the bound is on the expectation. On QCML

\times

MACE-OFF23(M) the supervised committee routes the whole pool for

\alpha\leq 0.25

, so its

r

is exactly

0

there.

Figure 5: The realized retained loss tracks the target for every score. Retained fraction

\rho(\alpha)

, the capped loss left uncorrected over the pool’s own total, for all eight cells; the dashed line is

\rho=\alpha

, a visual reference rather than a guarantee. Faint lines: the

20

reallocations; bold: mean

\pm 1

SD. Points below the diagonal (MACE-OFF23 cells; the tied supervised committee on QCML) mean that more DFT was spent than the target required.

Figure 6: The finite correction keeps the budget; its price is DFT when calibration is scarce. (a) Realized risk/budget and (b) fraction of calibration draws exceeding the budget at

\alpha=0.30

, for finite CRC, a plug-in cutoff without

B/(n{+}1)

, and a cutoff fixed on

\mathcal{D}_{\mathrm{fit}}

; equal-cell means over eight cells (

200

draws for

n\leq 100

). (c) Deployment DFT at

\alpha=0.30

: faint lines are cells on their own calibration grids, the bold line the equal-cell mean over the five shared

n

. The ranking score is fixed throughout; log

x

axes.

			at $\alpha=0.30$		committee vs. ours at matched resources
Method	MLIP evals	$G$	DFT [%]	$\rho$	DFT saved [pp of pool]	rel.-energy MAE [meV]	selection success [pp]
Ours	1	0.808	57.8	0.289	—	—	—
Raw committee	3	0.497	61.9	0.282	$+3.0$	$-2.0$	$+0.7$
Supervised committee	3	0.341	67.5	0.251	$+4.0$	$-9.7$	$+1.5$
Random	1	0.000	71.6	0.284	—	—	—
Oracle (retrospective)	—	1.000	54.4	0.288	—	—	—

Appendix

Table 4: Main results, equal-weight means over the eight cells.

G

is the normalized gain Eq. ( 12 ); DFT and

\rho

are the deployment DFT fraction and realized retained loss at

\alpha=0.30

. The last three columns compare each committee with ours over the common attained support of every cell: DFT saved by ours at matched

\rho

(Appendix F.1 ), and the relative-energy MAE and selection-success change of ours at matched deployment DFT (Appendix J ).

	primary scores			label-free controls
Cell	Ours	Raw	Supervised	kNN-1	kNN-5	Max-force
COMP6 $\times$ MACE-OFF23(M)	0.879	0.223	0.274	0.261	0.267	0.134
COMP6 $\times$ AIMNet2	0.822	0.237	0.255	0.236	0.244	0.131
COMP6 $\times$ AceFF-2	0.888	0.136	0.183	0.226	0.232	0.079
QCML $\times$ MACE-OFF23(M)	0.820	0.910	-0.278	0.362	0.385	0.245
QCML $\times$ AIMNet2	0.748	0.529	0.395	0.371	0.389	0.322

Appendix

Table 5: Normalized gain

G

per cell. Each

G

uses the

20

-reallocation mean DFT-call curve on the

\alpha

grid. Among ours and the two committees, bold marks the best and underlining the second best; the last three columns are label-free controls (Appendix H ), which never exceed ours. The final row is the equal-cell mean.

Figure 7: Within the shared guarantee, ours needs the fewest DFT calls of the deployable scores in most cells. DFT calls

k(\alpha)

for all eight cells; the gray band is the random-to-oracle headroom and each panel prints

G

for ours and the two committees. Means over

20

reallocations with

\pm 1

SD bands for ours and the committees. The raw committee wins QCML

\times

MACE-OFF23(M) and the committees match or beat ours on MACE-OFF23. The oracle is retrospective, not deployable.

Figure 8: Weighting each DFT call by

N^{3}

preserves, and on QCML widens, the advantage of ours.

N^{3}

-weighted DFT cost, the summed cubed atom counts of routed configurations over the same sum for the pool, for all eight cells; the dotted line marks

\alpha=0.30

. Means over

20

reallocations with

\pm 1

SD bands for ours and the committees. The proxy captures size scaling, not measured wall time, and the comparison fixes

\alpha

, not the realized loss.

Figure 9: At equal realized loss, ours still calls DFT least in most cells. DFT calls against realized retained loss

\rho

; each point is one target

\alpha

at the

20

-reallocation mean, with

\pm 1

SD whiskers in both coordinates. The shaded interval is the common attained

\rho

range of ours and the two committees, over which Fig. 3 (b) averages the DFT difference. On QCML

\times

MACE-OFF23(M), the supervised committee’s first five

\alpha

values collapse to one all-DFT point at

\rho=0

(a) Matched-resource comparisons
Dataset	MLIP	DFT saving $\uparrow$ [pp]		MAE change $\downarrow$ [meV]		Success change $\uparrow$ [pp]
		Raw	Sup.	Raw	Sup.	Raw	Sup.
COMP6	MACE-OFF23(M)	+5.85	+5.28	-16.12	-16.60	+2.19	+2.73
	AIMNet2	+5.01	+4.77	-14.13	-14.40	+2.10	+2.51
	AceFF-2	+6.47	+5.94	-14.63	-15.43	+2.67	+2.91
QCML	MACE-OFF23(M)	-2.51	+8.45	+16.13	-36.87	-1.19	+3.93

Appendix

Table 6: Matched resources over common attained support. (a) Comparisons with the raw and supervised committees; arrows mark the direction favoring ours. DFT saving is baseline minus ours; MAE and success changes are ours minus baseline. The bold mean weights the eight cells equally; pp denotes percentage points. (b) Common intervals used for the comparisons.

Figure 10: With a matched reference, committees edge ahead, and the structural score stays far above random. ANI-2x

\times

COMP6: (a) DFT calls with the random-to-oracle headroom shaded; (b) risk ratio

r(\alpha)

; (c)

G

and DFT calls at

\alpha=0.30

for ours, the native ANI committee (members

0

–

7

; target is their mean), and the supervised ANI committee. Means over

20

reallocations with

\pm 1

SD bands for the three selectors. ANI-2x training membership for these configurations is unverified, so this is not an unseen-data test.

Figure 11: An isolated D3(BJ) term does not explain the structural ranking. (a) Structural

G

against the

G

of a D3(BJ)-only score (dashed: equal

G

). (b) Structural

G

and (c) DFT calls at

\alpha=0.30

with the original reference and after adding D3(BJ) to the reference, joined per cell. Six COMP6 and QCML cells; cell statistics over

20

reallocations. Adjusting the reference changes the loss and refits the predictor, so before and after

G

use their own random and oracle anchors. A diagnostic, not proof that reference mismatch is absent.

Dataset	MLIP	Gini	Top 10%	$m_{\max}$	$B_{\rm cfg}$	$B$	Med. $\|\Delta E\|$	$p_{99}\|\Delta E\|$
COMP6	MACE-OFF23(M)	0.18	0.16	24	3.4	45.4	139	516
	AIMNet2	0.16	0.15	24	2.2	46.4	132	474
	AceFF-2	0.16	0.15	24	2.1	47.4	132	443
QCML	MACE-OFF23(M)	1.00	1.00	9	48.2	158.5	485	3,895
	AIMNet2	0.45	0.40	9	34.3	122.6	126	839
	AceFF-2	0.46	0.40	9	11.0	42.4	202	1,151

Appendix

Table 7: Residual concentration and frozen loss bounds. Resample-

0

test pool. Gini uses untruncated configuration scores

s

; “Top 10%” is the untruncated score mass in the highest-mass

10\%

of identities.

m_{\max}

is the maximum fitting-identity size,

B_{\mathrm{cfg}}

the configuration truncation level, and

B

the identity bound of Eq. ( 6 ). The last two columns are in meV.

Figure 12: Residual scales differ by orders of magnitude across cells, with catastrophic tails only on QCML. Aligned energy error, mean atom-force error, and maximum atom-force error of the target MLIP on the complete resample-

0

test pools. Thin lines span minimum to maximum, thick lines the

5

th–

95

th percentiles, markers the medians. Asinh axes, linear below

0.01

, so zero and the largest QCML

\times

MACE-OFF23(M) force error (

2.5\times 10^{7}

eV/Å) are both visible; nothing is omitted.

Figure 13: Target-model latents rival but do not dominate the structural score, and disagreement adds most on MACE-OFF23. (a) Per-cell

G

for ours, latent mean, latent max, and the hybrid (

y

axis zoomed to

0.55

–

1

). (b) Hybrid minus ours. (c) Native raw and supervised committees against ours in the five cells where they are defined; the native supervised value

-1.07

on QCML

\times

MACE-OFF23(M) is drawn clipped. Cell statistics over

20

reallocations. The small QCML hybrid gains are not claimed as improvements.

Figure 14: Every score source in every cell. (a) $G$ (diverging map centered at $0$ ) and (b) DFT calls at $\alpha=0.30$ (fixed $0$ – $100$ scale) for ours, latent mean and max, the hybrid, the raw and supervised committees, and the native raw and supervised committees. Cell statistics over $20$ reallocations. Hatched NA entries are undefined native committees, not zeros.

Figure 15: No two-member committee approaches the structural score on average; membership effects are cell-specific. (a) $G$ for ours, the full raw and supervised committees ( $K{=}3$ ), and each two-member subset ( $K{=}2$ ), per cell and as the equal-cell mean. A black corner triangle marks subsets that exclude the target MLIP; these still need three distinct models, so they are not cheaper than $K{=}3$ . (b) $\Delta G=G_{K=2}-G_{K=3}$ per cell and subset, raw (green) and supervised (blue), with equal-cell means at right.

Figure 16: The structural score beats every simple model-agnostic control in every cell. Per-cell

G

for ours and six controls under the same loss and CRC rule: kNN-1 and kNN-5 descriptor-space distance, the target MLIP’s maximum predicted force, learned composition + atom count, learned atom count, and raw atom count. Gray lines join the same cell (marker: dataset; color: MLIP); bars are equal-cell means. Negative

G

(worse than random) is shown unclipped and does not imply that size is irrelevant.

Figure 17: Scores that rank configurations better capture more of the random-to-oracle headroom.

G

against the mean test Spearman correlation between score and true error, per cell (marker: dataset) and score (color): the five main scores plus latent mean, latent max, hybrid, SOAP, and ACSF (hollow). Spearman values are means over

20

test reallocations;

G

uses the mean DFT-call curve. Oracle and random sit near

(1,1)

and

(0,0)

Figure 18: The predictor ranks energy residuals even where it misestimates their size. True against predicted aligned $|\Delta E|$ (energy channel of the frozen predictor) for every resample- $0$ test configuration; asinh axes, linear below $0.01$ eV; dashed line $y=x$ . Boxes give $n$ , Spearman correlation, MAE, and $R^{2}$ on raw residuals with outliers kept. The QCML $\times$ MACE-OFF23(M) axis reaches about $10^{6}$ eV because of a few catastrophic residuals, which is also why its $R^{2}$ is near $0$ .

Figure 19: The full descriptor outperforms SOAP and ACSF, and no single feature family carries the signal. (a) Equal-cell mean

G

(bars) and per-cell values (dots) for the full 350-dimensional descriptor, SOAP, and ACSF. (b)

\Delta G

(drop minus full) for each single-family drop, ordered by mean effect. (c) DFT calls at

\alpha=0.30

(

y

axis zoomed to

40

–

75\%

). All use the frozen fitting protocol and

20

reallocations; small positive

\Delta G

values are not significant improvements.

Figure 20: Descriptor effects are consistent across cells. (a) $G$ (diverging scale centered at $0$ ) and (b) DFT calls at $\alpha=0.30$ (fixed $0$ – $100$ scale) for every representation and single-family drop in every cell; the last column is the equal-cell mean. Rows below the rule are drops from the full descriptor, ordered by mean $\Delta G$ .

Figure 21: More fitting labels improve ranking; the ranking-loss weight matters only when large. (a)

G

against fitting-set size (

2.5

–

40\%

of population identities, nested subsets of the original fitting pool), one thin line per cell and the bold equal-cell mean. (b)

G

against the ranking-loss weight

\lambda\in\{0,0.1,1\text{ (frozen)},10\}

for the three-seed ensemble (solid) and seed

0

(dashed). Lines join cell values and are not uncertainty intervals.

Figure 22: At matched deployment DFT, ours lowers relative-energy error and raises selection success on average, but not in every cell. Per-cell difference, ours minus committee, against the raw (green) and supervised (blue) committees: (a) relative-energy MAE (lower is better); (b) selection success (higher is better). The right group is the equal-cell mean. Each value averages the difference of the

20

-reallocation mean curves over the cell’s common attained deployment-DFT interval. Descriptive, not a new guarantee.

Figure 23: On COMP6, ours reaches a given relative-energy error and selection success with less deployment DFT than either committee; elsewhere the curves largely overlap. Per cell, relative-energy MAE (asinh axis, linear below

10

meV) and selection success against deployment DFT, with all-MLIP (square) and all-DFT (star) endpoints and a ring at ours,

\alpha=0.30

. Means over

20

reallocations at

5\%

DFT steps, each split at its nearest attainable point; bands

\pm 1

SD. The saved sweep has no oracle. Success axes are zoomed per cell.

Figure 24: Routing at

\alpha=0.30

removes most physical error, but rare large force errors remain. (a) Full-pool mean aligned energy error and (b) mean atom-force error before routing and after routing with ours; numbers give the relative reduction (%), triangles the retained 99th percentile. (c) Retained maximum-force error: 99th percentile, per-pool maximum (mean

\pm

SD), and largest over

20

pools. (d) Share of the pool truncated at

B_{\rm cfg}

, as configurations and as raw score mass. Log axes. Uncapped tails are outside the guarantee.

Figure 25: Signed residuals of one molecule’s configurations are strongly correlated on COMP6 and MACE-OFF23 $\times$ AceFF-2, so energy differences benefit from cancellation that partial routing removes. Weighted density of signed aligned residual pairs $(\Delta E_{i},\Delta E_{j})$ within an identity (up to $64$ pairs per eligible identity, both orientations), in $\operatorname{asinh}(\Delta E/0.1\,\mathrm{eV})$ coordinates; the dashed diagonal is exact cancellation. Annotations give the paired Pearson $r$ before routing $\to$ after ours at $\alpha=0.30$ , averaged over $20$ reallocations. One shared log color scale of pair fraction per bin.

Figure 26: Partial routing can raise a molecule’s relative-energy error before full DFT drives it to zero, and such paths are common. (a) One saved COMP6

\times

AceFF-2 identity: Loss A (identity MAE of relative energies) and its running-maximum envelope against the number of configurations routed; an illustration, not a prevalence estimate. (b) Per cell, the percentage of identities whose routing path is non-monotone for Loss A and for Loss B (selection failure).

Figure 27: Strict physical-unit budgets cost almost all of the DFT savings. (a–c) DFT calls against an absolute budget for raw and aligned energy (kcal/mol/atom) and maximum force error (eV/Å), with the same structural score and CRC rule; thin lines are cells, the bold line the equal-cell mean of

20

-reallocation means. (d) A strict binary loss (aligned

|\Delta E|>1

kcal/mol or maximum force

>0.10

eV/Å; budget

0.10

violations per identity) at each cell’s calibration size. Thresholds are examples, not universal tolerances.

Figure 28: The proxy guarantee does not carry over to downstream losses, and direct control has its own cost. (a, b) Eligible DFT against the budget for Loss A and Loss B under M, W-max, and L; the dashed line is the uncertified proxy rule P at

\alpha=0.30

. Equal-cell means of

20

-repeat means. (c) Realized risk/budget of P per cell (log axis); the tiny QCML

\times

MACE-OFF23(M) Loss-A ratio reflects its extreme fitting anchor. (d, e) Per-cell realized risk/budget of every method at the main budgets, mean

\pm

SD over

20

repeats. L is high-probability; M and W are expectation-level.

Figure 29: Downstream certification cost depends strongly on the cell and the objective. Eligible DFT against the budget in every cell, Loss A (top two rows) and Loss B (bottom two), for M, W-max, W-mean, L, and N (mean

\pm 1

SD over

20

repeats); P is a horizontal reference. ‡ Data issue, plotted without alteration: the saved QCML

\times

MACE-OFF23(M) Loss-A fitting mean is

14{,}761

eV (main budget

4{,}428.5

eV), and on all three QCML Loss-A panels every certified method routes

100\%

. Budgets are relative to fitting anchors, not a common absolute accuracy.

Figure 30: Up-front labels pay back after 2–4 reused pools. Total DFT in full-pool equivalents (fitting and calibration labels up front, plus routed deployment labels) against the number

P

of comparable reused pools, for fitting at

2.5\%

10\%

, and

40\%

of identities and for all-DFT (

y=P

). Equal-cell means, each cell normalized by one deployment pool; rings mark the first saved integer

P

at or below all-DFT. Cost-model projections in configuration counts, not measured wall time.

Figure 31: A frozen structural score transfers poorly across populations. (a–c) $G$ of the frozen score from each source population (rows) on each target population (columns), per target MLIP, after target recalibration; boxed diagonals are in-domain, NA marks non-cells. (d) Transfer $G$ against the target’s in-domain $G$ for all $14$ directed transfers (dashed: equality). (e) Anchored QCML $\to$ COMP6: $G$ against the number of target-anchor identities for the target-trained (solid) and frozen QCML (dashed) predictors. Target anchors and calibration labels are charged, so this is not zero-label deployment.

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Shams Mehdi, Ilkwon Cho, Olexandr Isayev

Department of Chemistry, Mellon College of Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA. · Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA. · Ray and Stephanie Lane Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213, USA.

Machine Learning Interatomic Potentials Selective Prediction Interpretable ML

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Abstract

Selective Prediction

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