On the Theoretical Limits of Learning Unconstrained Interatomic Potentials

Prajwal Ananth; Shuwen Yue

cs.LGOct 10, 2026

On the Theoretical Limits of Learning Unconstrained Interatomic Potentials

Authors: Prajwal Ananth, Shuwen Yue

Organizations: Center for Applied Mathematics, Cornell University, Ithaca, NY · R. F. Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY

Abstract

Machine-learned interatomic potentials (MLIPs) accelerate molecular dynamics (MD) simulations by learning the energies and forces that govern atomic motion. Recent work has shown that relaxing physical inductive biases can improve computational efficiency and predictive accuracy, raising the question of whether energy-and-force matching alone is sufficient to ensure accurate sampling. We show that, for unconstrained hypothesis classes, the relationship between prediction error and sampling accuracy is exponentially ill-conditioned in dimension. In particular, prediction errors that decay exponentially with dimension can coexist with nearly maximal total variation distance between the learned and target distributions. We identify two failure mechanisms. The first is kinetic trapping, where relaxation to an accurate learned equilibrium requires exponentially long times, even when simulations are initialized from training configurations at which energies and forces are matched exactly. The second is equilibrium mismatch, where the learned stationary distribution itself remains far from the target. Under explicit assumptions, we show that equilibrium mismatch can persist even in local, symmetry-preserving models, where small per-atom errors coexist with a distributional discrepancy that approaches its maximum as system size increases. Finally, we establish global energy-control conditions that guarantee equilibrium accuracy and identify additional conditions required for rapid relaxation.

Figures & tables

Result	Setting	Error / Control condition	Physical conclusion
Theorem 4.1	Harmonic target; displaced initialization	Exponentially small population error	Kinetic trapping for $e^{\Omega(d)}$ time
Theorem 4.2	Harmonic target; training-data initialization	Exponentially small population error + Exact interpolation	Kinetic trapping for $e^{\Omega(d)}$ time
Theorem 4.3	Broader class of confining targets; arbitrary initialization	Exponentially small population error	Incorrect stationary distribution at distance $1-e^{-\Omega(d)}$
Theorem 4.6	General targets with global energy control	Uniform or one-sided energy bounds	Stationary accuracy; uniform control also ensures relaxation.
Theorem H.5	Local, invariant site-energy models	Per-atom error $e^{-\Omega(\nu)}$	Global equilibrium error $1-e^{-\Omega(N)}$

Table 1: Error-to-sampling structure of the main results. “Population error” is measured with respect to the target distribution. The table distinguishes predictive accuracy, finite-time dynamics, and stationary-distribution accuracy.

Panel of Figure 1 — Figure 1: Illustration of kinetic trapping mechanism underlying Theorems 4.1 and 4.2 (details in Section A.1 ). (a) Escape time grows as $e^{\Omega(d)}$ with an $\mathcal{O}(1)$ rate per degree of freedom. The fully trapped plateau extends to arbitrarily large $d$ . (b) The true Einstein crystal shows a single sharp line at the analytic $\nu_{E}=6.25\,\mathrm{THz}$ ; the trapped surrogate’s line is at $12.5\,\mathrm{THz}$ .

Assumption	Provides	Does not control
Small $L^{p}$ energy/force error	Accuracy on typical configurations	Existence of equilibrium
Conservative force + Confinement	Scalar potential and Gibbs equilibrium	Equilibrium accuracy and mixing time
Smoothness + bounded Hessian	Lipschitz force and well-posed dynamics	Global wells, barriers, sampling
Locality, permutation and $E(3)$ symmetry	Exact transformation laws, extensivity and a potential receptive-field safeguard	Accumulation of residual local bias with system size
One-sided global control + on-target accuracy	Equilibrium accuracy	High barriers and kinetic trapping
Uniform two-sided control + functional inequality	Equilibrium accuracy and exponential relaxation	A strong global condition not supplied by ordinary matching

Table 2: Summary of guarantees provided by common assumptions and the aspects of sampling they leave uncontrolled.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Physical property	Analytic	True MD	Surrogate MD
Debye–Waller $\langle u^{2}\rangle$ (Å 2 )	0.0180	0.0180	0.0045
Structure error (Å)	0	0.004	1.06
Heat capacity $C_{v}$ ( $k_{B}$ /atom)	3.00	3.03	3.03

Appendix

Table 3: Analytic predictions and molecular-dynamics estimates for the Einstein-crystal construction. The surrogate trajectory is kinetically trapped ( Figure 1 ). Its heat capacity remains apparently correct even though its displacement statistics and mean structure are incorrect.

Panel of Figure 2 — Figure 2: Molecular dynamics of the harmonic argon crystal. Error bars show the standard deviation across four random seeds. (a) As the system size increases, the held-out force RMSE remains of order $10^{-3}$ or smaller, while the finite-trajectory Gaussian KL diagnostic increases with dimension. At $N=128$ , trajectories from all four seeds leave the crystal basin. (b) At fixed system size $N=128$ , increasing the training-set size does not substantially reduce the sampling discrepancy under the tested training protocol.

	$\langle u^{2}\rangle/\langle u^{2}\rangle_{\mathrm{t}}$			$\widehat{\mathrm{KL}}$			$C_{v}$ ( $k_{B}$ /atom)
$N$	Unconst.	C1	C2	Unconst.	C1	C2	Analytic	Unconst.	C1	C2
16	1.00	1.00	1.00	0.001	0.001	0.001	1.901	1.901	1.901	1.901
32	1.00	1.00	1.00	0.004	0.004	0.004	1.980	1.980	1.980	1.980
48	1.00	1.00	1.00	0.010	0.010	0.010	2.004	2.005	2.004	2.005
64	1.00	1.00	1.00	0.018	0.018	0.018	2.019	2.020	2.019	2.020
80	1.00	1.00	1.00	0.028	0.027	0.028	2.028	2.030	2.028	2.030

Appendix

Table 4: Debye–Waller inflation, finite-trajectory Gaussian KL diagnostic, and phonon-based quantum harmonic heat capacity as functions of system size. C1 reduces the large-

N

discrepancy, while C2 keeps the displacement statistics close to the target and substantially reduces the KL diagnostic.

C_{v}

is computed from the phonon spectrum and averaged over seeds that remain within the harmonic basin. A dash indicates that all seeds left the basin, so a basin-based phonon spectrum is not reported.

Figure 3: Figure adapted from Fu et al. [2023] (Figure 6) depicting simulation collapse when applying (a) NequIP to alanine dipeptide and (b) GemNet-T to water.

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Machine learning interatomic potentials (MLIPs) have become a hallmark of AI for scientific simulation. While efforts on new architectures and datasets have led to increasingly accurate and general models, the choice of optimizer for training has largely remained unexplored, defaulting to Adam and its variants in the community. Here, we implement and systematically compare a class of recently proposed matrix-structured optimizers, including Muon, SOAP, and the hybrid SOAP-Muon, for training NequIP and Allegro MLIP models. We find that these optimizers can substantially outperform Adam in both convergence speed and final accuracy. SOAP and SOAP-Muon emerge as robust and consistently strong methods, while Muon only provides partial gains relative to Adam. The improvements are particularly pronounced under partial force supervision. Our results indicate that optimizer choice is an overlooked yet impactful design axis for MLIPs.

Abstract

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