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Risk-Controlled DFT Selection for Machine-Learned Interatomic Potentials

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

arXiv:2610.19630v1stat.MLcs.LG

TL;DR

DFT is costly, and pretrained MLIP errors vary across configurations, so deciding which predictions to trust remains a challenge. The paper uses a structure-only error predictor and conformal risk control to route higher-risk configurations to DFT. Across eight dataset–MLIP cells, its score captures more random-to-oracle headroom than committee scores.

  • Problem

    DFT evaluation is costly and pretrained MLIP errors vary across configurations, complicating decisions about which predictions to trust and which to send to DFT.

    Source: §1 · p. 1

  • Method

    A structure-only predictor ranks expected MLIP–DFT errors, and conformal risk control calibrates a threshold for DFT referral under a user-chosen risk budget.

    Source: §1 · p. 2, §2.4 · p. 4

  • Results

    Across eight dataset–MLIP cells, the structure-only score captures G = 0.81 of random-to-oracle headroom, versus 0.50 for raw committees and 0.34 for supervised committees.

    Source: §3.2 · p. 7

  • Takeaways & Limitations

    Without model internals, the structure-only score matches the target MLIP representation’s error ranking and outperforms SOAP and ACSF.

    Source: §3.2 · p. 9

    The guarantee is in expectation and depends on exchangeable identities; it does not guarantee risk control for a different distribution or per-configuration correctness.

    Source: §2.4 · p. 5, Ethics Statement · p. 9

Abstract

from arXiv · show

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 $\alpha$ of the expected error. Under an identical guarantee across eight dataset-MLIP pairs, our predictor routes 7% and 14% fewer configurations to DFT at $\alpha=0.3$ 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).

1 Introduction

DFT is costly and pretrained MLIPs can be inaccurate on particular configurations, making it difficult to decide which predictions to trust. The paper combines a structure-based error score with conformal risk control (CRC) to route configurations to DFT under a user-chosen risk budget.

1 Introduction

  • DFT is a major computational bottleneck, while pretrained MLIPs are cheaper approximations whose errors vary across molecules, geometries, and chemical environments.Conventional DFT scales approximately as O(N^3), whereas local MLIP architectures typically scale linearly in N.Source: §1 · p. 1
  • The routing problem is to avoid both the cost of evaluating every configuration with DFT and the large errors that can result from accepting every MLIP prediction.Estimating error alone is insufficient: the remaining MLIP error also needs a finite-sample guarantee.Source: §1 · pp. 1–2
  • The method treats MLIP–DFT routing as risk-controlled selection: a structural error predictor ranks configurations, and CRC calibrates a threshold for DFT referral.Calibration is at the molecule-identity level, while routing decisions are made for individual configurations.Source: §1 · p. 2, Figure 1 · p. 2
  • The predictor estimates energy, average-force, and maximum-force errors from a chemistry-informed structural representation, while CRC sets the threshold that satisfies the risk budget.The score ranks configurations for efficiency; CRC calibrates the threshold for risk control.Source: §1 · p. 2, Figure 1 · p. 2
  • The paper compares error scores by DFT-call cost under matched risk control across eight dataset–MLIP pairs and reports the largest gains when DFT references differ.Related precedents include ensemble-uncertainty thresholding by Liu et al. [2026] and CRC-based deferral by Overman and Bayati [2025]; this work uses a separate error predictor and molecular identity-level loss.Source: §1 · p. 2

2 Method

The method uses a structure-only error score to route configurations to DFT, then calibrates its threshold with conformal risk control to bound expected error on configurations retained by the MLIP. Its predictor estimates energy and force residuals, while the guarantee depends on the calibration framework rather than predictor quality.

2.1 Setup and notation

  • The workflow trains a label-free score on Dfit and calibrates it on Dcalib before selecting configurations from unlabeled Dtest, using aligned DFT–MLIP residuals as targets.Composition-based energy alignment corrects isolated-atom convention differences while leaving force residuals and same-composition energy differences unchanged.Source: §2.1 · p. 3

2.2 Structural vector

  • The structure-only descriptor uses atomic numbers and coordinates to encode chemistry-informed geometric features in nine fixed feature families spanning 350 dimensions.Its features cover composition, coordination, distances, crowding, angular and radial structure, and neighboring-element identity.Source: §2.2 · p. 3

2.3 A joint energy–force error predictor

  • A lightweight predictor estimates normalized energy, average-force, and maximum-force residuals, combining them into one score whose weights encode application priorities.The three targets represent distinct error modes, including broadly inaccurate forces and a single badly mispredicted atom.Source: §2.3 · pp. 3–4

2.4 Configuration selection by conformal risk control

  • CRC calibrates a score threshold to bound expected capped error among configurations retained by the MLIP, with α setting the error budget.The budget is α times the fitting set’s mean per-identity loss; the guarantee is an expectation over exchangeable calibration and test identities, not a high-probability bound for one calibration draw.Source: §2.4 · pp. 4–5

3 Experiments

The experiments compare error-ranking scores under a shared conformal risk-control procedure across eight dataset–MLIP cells. The structure-only score lowers DFT cost most when dataset and MLIP training references differ, while its ranking performance also matches the target model’s latent representation.

3.1 Setup

  • The baselines are random selection, raw and supervised three-MLIP committees, and an oracle; all use identical CRC, while committees require three MLIP evaluations per configuration.The study reports DFT calls, risk ratio, N^3-weighted DFT cost as a size-scaling proxy rather than wall time, and normalized gain relative to random-to-oracle headroom.Source: §3.1 · pp. 5–6
  • The eight cells use three datasets and three pretrained MLIPs; each dataset is split by molecule identity into fitting, calibration, and test sets at 40/20/40.The study draws 20 calibration/test resamples per cell and sweeps α from 0.05 to 0.50.Source: §3.1 · p. 6

3.2 Results

  • At α = 0.30, ours routes 57.8% of the pool, versus 61.9% for the raw committee, 67.5% for the supervised committee, and 71.6% for random selection.Our DFT call was comparable with oracle (54.4%). Under N 3-weighted cost and α ≥ 0.15, ours was cheaper than the raw committee and random in all eight cells, and cheaper than the supervised committee in seven.Source: §3.2 · p. 7
  • On COMP6, ours reaches G = 0.82–0.89 versus 0.14–0.27 for committees; on QCML, it reaches 0.75–0.84 versus 0.32–0.53.On more reference-aligned MACE-OFF23 cells, the supervised committee leads; the authors caution that reference alignment and molecule size are confounded.Source: §3.2 · pp. 7–8, Figure 3 · p. 8
  • Across eight cells, the structure-only score captures G = 0.81 of random-to-oracle headroom, versus 0.50 for raw and 0.34 for supervised committees.All scores use the same CRC rule and guarantee.Source: §3.2 · pp. 6–7, Figure 3 · p. 8
  • At matched realized loss, ours saves 3.0 percentage points of DFT versus raw committee and 4.0 versus supervised committee, with positive savings in six of eight cells each.The reference relation is known before calibration, so the score can be chosen without changing the guarantee.Source: §3.2 · p. 8
  • Ours scores G = 0.81, near the target MLIP’s latent representation at 0.82, and exceeds ACSF at 0.56 and SOAP at 0.48.At α = 0.30 it reduces relative-energy MAE by at least 36.4% in every cell, but lowest-energy selection is less successful than using no DFT on three COMP6 cells.Source: §3.2 · p. 9

4 Conclusion

The paper frames MLIP–DFT routing as risk-controlled selection and evaluates a structure-only error score against committee scores. Its results show that score choice affects DFT cost most when training references differ, while the guarantee is unchanged.

  • CRC calibrates a routing threshold so the expected capped error retained by the MLIP stays within a user-chosen fraction of the fitting-set mean, under exchangeability of molecule identities.The score determines DFT cost, not the guarantee.Source: §4 · p. 9
  • Across eight dataset–MLIP cells, the structure-only score captures 0.81 of random-to-oracle headroom, versus 0.50 and 0.34 for three-MLIP committees, and matches the target MLIP’s latent features without model access.Committees recover when DFT and MLIP training references match.Source: §4 · p. 9
  • The score choice should follow the known reference relation; the method is intended for repeated screening with fitting and calibration labels amortized over a population.Partial DFT replacement can break error cancellation in relative energies, which the framework can certify as a separate objective.Source: §4 · p. 9

AI Use Statement

The authors report AI assistance in writing, methodology feedback, experiments, implementation, and manuscript preparation, while retaining responsibility for the paper’s claims and final content.

  • AI tools assisted with proof writing, methodological and experimental feedback, implementation, result interpretation, code editing, figure production, manuscript editing, and literature searches.Source: AI Use Statement · p. 9
  • The authors state that AI tools were not used to generate synthetic data, develop the theoretical framework, formulate mathematical claims, propose hypotheses, or design the experimental methodology.Source: AI Use Statement · p. 9
  • The authors say they independently verified proofs, reviewed AI-assisted code and reproduced its outputs, checked references, and take responsibility for the final content.Source: AI Use Statement · p. 9

Ethics Statement

The ethics statement describes the data and model use and states limitations on the guarantee. The related-work review situates the approach among uncertainty, error-prediction, screening, and conformal methods, while the descriptor section outlines its geometry-based feature construction.

Ethics Statement

  • The work uses public datasets and pretrained potentials under their licenses, with no human subjects, personal information, or new data collection; its expected-risk guarantee assumes exchangeability and does not transfer across distributions.It controls average rather than per-configuration risk, and capped loss may reduce reliability for extreme MLIP residuals.Source: Ethics Statement · p. 9

A Related Works

  • Related methods include committee or latent-space uncertainty, post-hoc error predictors, active-learning DFT selection, screen-then-verify workflows, and conformal risk control.The paper distinguishes its single-pass routing of a frozen MLIP and identity-level loss guarantee from empirically chosen cutoffs and prediction-level calibration.Source: A Related Works · pp. 13–14

B The 350-dimensional structural descriptor

  • The 350-dimensional descriptor is a deterministic function of atomic numbers and coordinates, using nine chemistry-motivated geometric feature families without labels, MLIP predictions, embeddings, or fitted parameters.Source: B The 350-dimensional structural… · p. 14

B.1 Geometric quantities and summary conventions

  • The construction uses two graphs: G0 includes candidate pairs with q_jk ≤ 1.20, while G restricts hydrogen and fluorine to at most one connection; fixed summary conventions maintain dimension for small structures.Coordination and pair-state counts use G0; local strain and angular summaries use G.Source: §B.1 · p. 14

B.2 Composition, coordination, and pair states: 218 coordinates

  • Composition, coordination, and pair-state features provide 218 inputs: element counts, element-specific coordination counts, and element-pair distance-region counts relative to covalent-radius sums.The pair-state bins do not assign bond order or bond energy.Source: §B.2 · p. 15

B.3 Short-range and extreme local environments: 33 coordinates

  • ShortRange contributes 21 summary coordinates, while the extreme-environment block contributes 12 angular coordinates based on degree-dependent reference angles.The angular reference is not an orbital-hybridization assignment; distance features still record close contacts.Source: §B.3 · p. 15

B.4 Radial and inverse-distance summaries: 54 coordinates

  • The inverse-distance block contributes 36 coordinates from unweighted proximity summaries across six powers, separating distance effects from atomic-number weighting.Source: §B.4 · p. 15

B.5 Soft crowding and neighboring elements: 45 coordinates

  • SoftCrowding contributes 24 coordinates using heuristic distance scales, and ZPotential contributes 21 coordinates retaining neighbor-element identity; together the feature-family counts yield 350 dimensions.Source: §B.5 · p. 15

C Proof of Proposition 1

The proof uses exchangeability and monotonicity of the capped loss to establish an expected risk bound for the calibrated threshold. The guarantee is finite-sample, but controls expected capped identity-level retained loss rather than realized pool loss or uncapped physical error.

C Proof of Proposition 1

  • Exchangeability makes the oracle threshold symmetric across calibration and test identities, while monotonicity lets the calibrated threshold inherit its expected-loss bound.The proof fixes the fitting-derived score and loss quantities before applying exchangeability.Source: C Proof of Proposition 1 · p. 16
  • The calibrated threshold gives expected capped loss at most ϵ0 per test identity and at most mϵ0 across the test set, without requiring test-molecule independence.The aggregate bound follows by exchangeability and linearity of expectation.Source: C Proof of Proposition 1 · p. 16
  • The finite argument needs neither continuity nor tie-free scores, and infeasibility triggers the all-DFT fallback when ϵ0 < B/(n + 1).The guarantee is for expected capped identity-level retained loss, not realized ρ(α) or uncapped physical energy or force error.Source: C Proof of Proposition 1 · p. 16
  • The proof also permits other fixed, bounded, monotone per-configuration losses that vanish at λ = −∞.Monotonicity-preserving bounded aggregations over molecules are also allowed.Source: C Proof of Proposition 1 · p. 16

D Datasets, experimental protocol, and baselines

The experiments split data by molecule identity, freeze fitting-derived quantities before calibration, and compare structural and committee-based scores under a shared protocol. Energy alignment, residual normalization, and reference conventions define what the error scores measure.

D Datasets, experimental protocol, and baselines

  • Datasets are split by stored molecule identity into 40/20/40 fitting, calibration, and test sets, with 20 random calibration/test reallocations per combination.The protocol sweeps α from 0.05 to 0.50; dataset sizes and group counts differ across COMP6, QCML, and MACE-OFF23.Source: D Datasets, experimental protocol, and… · p. 16
  • Three structural scorers use identical two-layer architectures and independent seeds, while all fitting-derived quantities and checkpoints remain frozen across reallocations.The arithmetic mean of the three scores is used directly; calibration data is spent on choosing the threshold.Source: D Datasets, experimental protocol, and… · p. 17
  • Composition-linear energy alignment removes per-element convention offsets, while trimmed least squares limits the influence of extreme MLIP blow-ups without absorbing geometry-dependent error.The correction can absorb composition-linear systematic bias but does not make different electronic-structure methods equivalent.Source: D Datasets, experimental protocol, and… · p. 17
  • Residual targets are normalized for atom-count scaling using fitting-set exponents and scales, which are frozen before calibration and inference.The predictor’s normalized outputs can be converted back to physical residual units using the same frozen quantities.Source: D Datasets, experimental protocol, and… · pp. 17–18
  • Committee baselines measure energy and force disagreement among three pretrained MLIPs; the supervised committee learns to combine those disagreement features.The committee energies are composition-aligned before measuring spread, and both committees require three MLIP evaluations per configuration.Source: D Datasets, experimental protocol, and… · p. 18
  • Every dataset–MLIP pairing differs in at least one electronic-structure reference component, so residuals include reference-level differences as well as model approximation error.The MACE-OFF23(M) target is excluded on MACE-OFF23 because of shared training-source provenance; complete training membership is not established for every model.Source: D Datasets, experimental protocol, and… · p. 18, Table 3 · p. 19

E The shared guarantee

Across the evaluated cells and scores, the shared CRC procedure meets its expected-risk guarantee, while realized retained loss tracks the target. The finite-sample correction protects the budget but can increase DFT use when calibration data are scarce.

E The shared guarantee

  • At n = 10, finite CRC has mean risk/budget 0.17 and 0.9% observed violations; scarce calibration increases DFT use, and violation frequencies are not a high-probability guarantee.The finite correction keeps the budget on average, whereas the plug-in cutoff without B/(n+1) exceeds it in about 52% of calibration draws.Source: E The shared guarantee · p. 19, Figure 6 · p. 22
  • 1.0359 was the largest mean risk ratio and 1.333 the largest single draw across cells, scores, targets, and reallocations; the bound is on expectation.Figure 4 displays risk ratios for all eight cells and five scores; individual draws above 1 are compatible with the expectation guarantee.Source: E The shared guarantee · p. 19, Figure 4 · p. 20
  • The realized retained fraction tracks α for every score, while points below the reference diagonal indicate that more DFT was spent than the target required.The diagonal is a visual reference, not a guarantee.Source: E The shared guarantee · p. 19, Figure 5 · p. 21

F DFT cost in every cell

Across the eight cells, the structural score reduces DFT calls relative to both committees in five cells at α = 0.30, with exceptions concentrated in specified MACE-OFF23 comparisons. Matched-resource analyses compare deployment cost and retained loss over common support rather than treating equal α as equal realized loss.

F.1 Matched-resource comparisons

  • At α = 0.30, the supervised committee retains ρ = 0.251 versus 0.289 for ours, so equal-α comparisons can include lower retained loss purchased with extra DFT.Source: §F.1 · p. 22

F DFT cost in every cell

  • At α = 0.30, ours routes less DFT than both committees in five of eight cells; the raw committee wins on QCML × MACE-OFF23(M), and committees match or beat ours on both MACE-OFF23 cells.The oracle comparison is retrospective and not deployable.Source: F DFT cost in every cell · p. 22, Figure 7 · p. 23
  • With N^3-weighted DFT cost, ours beats the raw committee and random in all eight cells for α ≥ 0.15, and the supervised committee in seven, except MACE-OFF23 × AceFF-2.The weighting is a size-scaling proxy, not measured wall time.Source: F DFT cost in every cell · p. 22, Figure 8 · p. 24

F.1 Matched-resource comparisons

  • At equal realized loss, ours saves DFT against each committee in six of eight cells.The figure plots DFT calls against realized retained loss, with comparisons restricted to the common attained range.Source: F DFT cost in every cell · p. 22, Figure 9 · p. 25
  • Matched-resource comparisons use piecewise-linear curves over common attained support, with no extrapolation; they are descriptive comparisons, not new CRC guarantees.The accompanying table reports matched retained-loss and deployment-DFT comparisons over that support.Source: §F.1 · pp. 22, 26, Table 6 · p. 27

G Why the gap opens and closes

Committee disagreement becomes competitive when the reference is matched, while the structural score is strongest against shared errors. The residual patterns and model comparisons also expose a concentrated-tail exception and modest gains from combining structural and disagreement signals.

Matched reference

  • At α = 0.30, ours reaches G = 0.66 and routes 57.9%, versus G = 0.69 and 57.2% for both ANI committees, and 70.7% for random.At matched ρ, ours needs 0.71 and 0.75 pp more DFT; training membership is unverified, so this is not an unseen-data test.Source: G Why the gap opens and closes · p. 27, Figure 10 · p. 28

Isolated dispersion control

  • A D3(BJ)-only score reaches G = 0.05–0.12, while adding D3(BJ) to the reference changes structural G by at most 0.02 and DFT calls by under 1 pp.This diagnostic does not rule out mismatch from functional form, basis set, or other effects.Source: G Why the gap opens and closes · p. 27, Figure 11 · p. 29

Residual concentration

Shared and target-specific components

  • Ours correlates more with consensus than target-specific residual magnitudes, while raw committees correlate more with target-specific force components; these are descriptive associations, not causal decomposition.For energy, mean-force, and maximum-force consensus residuals, correlations are 0.47/0.54/0.45 versus 0.28/0.45/0.36 for target-specific residuals.Source: G Why the gap opens and closes · p. 28

Model internals and hybrids

H Are the committees and controls strong enough?

Committee subsets and simple model-agnostic controls do not match the structural score on average, though committee membership effects vary by cell.

Committee subsets

  • No two-member committee subset approaches ours on average: raw subsets reach mean G = 0.445–0.494 and supervised subsets 0.244–0.509, versus 0.808 for ours.Membership effects are cell-specific, and some subsets beat ours in individual QCML × MACE-OFF23(M) and MACE-OFF23 cells; excluding the target still requires three distinct MLIPs.Source: H Are the committees and controls… · p. 33, Figure 15 · p. 33

Model-agnostic controls

I What gives the structural score its ranking

Across these sections, routing improves physical observables at matched DFT in some settings, but downstream objectives need their own certification and costs. Ranking, descriptor design, fitting choices, reuse, and transfer each constrain when the gains apply.

I What gives the structural score its ranking

J What routing does to chemical observables

  • At matched deployment DFT, ours lowers relative-energy MAE by 2.0 and 9.7 meV versus raw and supervised committees, and raises selection success by 0.7 and 1.5 pp.At α = 0.30, relative-energy MAE falls at least 36.4% in every cell, but selection success falls below all-MLIP on three COMP6 cells; rare retained maximum-force errors reach 50.300 eV/Å.Source: J What routing does to chemical… · p. 39, Figure 22 · p. 39, Figure 23 · p. 40

J.1 Signed-residual cancellation

  • Same-sign residuals cancel within energy differences, but routing reduces their correlation from 0.92–0.96 to 0.55–0.60 on COMP6 and MACE-OFF23 × AceFF-2.Routing paths are non-monotone for relative-energy MAE in 45–95% of eligible identities and for selection failure in 1–46%.Source: §J.1 · p. 42, Figure 25 · p. 42, Figure 26 · p. 43

K Certifying other losses directly

  • Strict physical-unit certification costs most savings: mean DFT is 55.0% for aligned energy, 85.3% for raw energy, and 94.3% for maximum force at the smallest budgets.The proxy rule exceeds the Loss-A budget in five of eight cells; direct certification costs vary by objective, and all certified methods route 100% in the three QCML Loss-A panels.Source: K Certifying other losses directly · p. 43, Figure 27 · p. 44, Figure 28 · p. 45

L Label cost and transfer

  • Charging fitting and calibration labels up front, the selector breaks even after two comparable reused pools with 2.5% or 10% fitting, or four with 40%.A frozen score transfers poorly: mean G is 0.17 across directed transfers versus 0.80 in-domain, and target anchors and calibration labels are charged.Source: L Label cost and transfer · p. 47, Figure 30 · p. 47, Figure 31 · p. 48

M Limitations

The guarantee is scoped to a fitted dataset–MLIP cell and a capped loss, while downstream objectives and deployment populations require separate consideration. The paper also cautions that repeated-screening savings do not guarantee lower total compute or safe retraining.

M Limitations

  • The CRC bound is marginal and expectation-level within the fitted dataset–MLIP cell, applies to capped loss, and requires refitting and recalibration for a new population or target MLIP.High-probability or group-conditional variants may protect rare families or single calibration draws at additional DFT cost.Source: M Limitations · p. 48

N Broader Impacts

  • Savings target repeated screening of comparable pools, but cheaper screening can enlarge campaigns, and retraining on routed labels requires refitting and recalibration.Reports should state the controlled loss, its caps, and the calibration population; the guarantee does not cover stable trajectories, correct conformer energies, or absence of rare large errors.Source: M Limitations · p. 48, N Broader Impacts · p. 49

M Limitations

  • The proxy guarantee does not certify relative energies or conformer selection; those objectives can instead be certified directly using bounded monotone losses.The descriptor covers isolated molecules built from ten elements, and periodic systems require extending its distance definitions.Source: M Limitations · p. 48

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