Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Authors: Xiang Fu
Brandon M Wood
Luis Barroso-Luque
Daniel S. Levine
Meng Gao
Misko Dzamba
C. Lawrence Zitnick
Venue: ICML 2025 oral (2025)
Keywords: Machine Learning Force Fields
Machine Learning Potentials
DFT
Computational Chemistry
Molecular Dynamics
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: A novel machine learning interatomic potential architecture achieving state-of-the-art performance on test error, Matbench-Discovery, phonon calculation, and thermal conductivity calculations, with detailed ablation studies and analysis.

Abstract: Machine learning interatomic potentials (MLIPs) have become increasingly effective at approximating quantum mechanical calculations at a fraction of the computational cost. However, lower errors on held out test sets do not always translate to improved results on downstream physical property prediction tasks. In this paper, we propose testing MLIPs on their practical ability to conserve energy during molecular dynamic simulations. If passed, improved correlations are found between test errors and their performance on physical property prediction tasks. We identify choices which may lead to models failing this test, and use these observations to improve upon highly-expressive models. The resulting model, eSEN, provides state-of-the-art results on a range of physical property prediction tasks, including materials stability prediction, thermal conductivity prediction, and phonon calculations.

Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Xiang Fu'
Brandon M. Wood Luis Barroso-Luque
Daniel S. Levine Meng Gao
Misko Dzamba
C. Lawrence Zitnick!
Abstract
Predicting physical properties in chemistry and materials sci- often requires complex workflows involving numerous evaluations of DFT MLIPs. For example, in molecu- lar dynamics (MD) simulations, forces are predicted over thousands to millions of time steps. However, the MLIP literature has mostly focused on assessing models based on energy and foree predictions over static DFT test sets rather than directly assessing their performance in complex simula- tions. This approach has limitations, as improved accuracy test sets does not always lead to better predictions of physical properties (Póta et al., 2024; Loew et al., 2024).
Machine learning interatomic potentials (MLIPs) have become increasingly effective at approximat- ing quantum mechanical calculations at fraction of the computational cost. However, lower errors on held out test sets do not always translate to im- proved results on downstream physical property prediction tasks. In this paper, we propose test- ing MI.IPs on their practical ability to conserve energy during molecular dynamic simulations. If passed, improved correlations are found between test errors and their performance on physical prop- erty prediction tasks. We identify choices which may lead to models failing this test, and use these observations to improve upon eighsv-expressive models. The resulting model, eSEN, provides state-of-the-art results on a range of physical prop- erty prediction tasks, including materials stability prediction, thermal conductivity prediction, and phonon calculations.
In this paper, we address two questions: Why does higher test accuracy sometimes fail to enhance a model's ability to predict physical properties, and how can we improve MLIPs to excel in this area? We first outline four critical property prediction tasks and identify the properties required for an MLIP to succeed in these tasks. These properties entail learning a conservative model with continuous and bounded energy derivatives, indicating a smoothly-varyin and phys- ically meaningful energy landscape. To test whether these properties hold, we propose testing the ability of MLIPs to practically conserve energy in MD simulations. We demon- strate models that pass this test have higher correlation between test errors and property prediction accuracy.
1. Introduction
Density Functional Theory (DFT), which models the elec- trons in materials and molecules, serves as the foundation for many modern drug and materials discovery workflows. Unfortunately, DFT calculations are notoriously compu- tationally intensive, scaling cubically with the number of electrons in the system: Onz)). Machine learning inter- atomic potentials (MLIPs) are promising in approximating and expediting DFT calculations. With increasing data set sizes and model innovations, MLIPs have shown substan- tial improvements in accuracy and generalization capabili- ties (Batatia et al., 2023; Merchant et al., 2023; Yang al., 2024; Barroso-L_qqee et al., 2024).
Building on these insights, we present novel MLIP called eSEN and training approach that achieves state-of- the-art (SOTA) performance on complex property predic- tion tasks. Specifically, our model is capable of running energy-conserving MD simulations for out-of-distribution systems (Figure 1 (a)). For materials stability prediction, eSEN achieves leading Fl score of 0.831 and K3RME of 0.340 on the compliant Mathench-Discovar mark (Riebesell et al., 2023; Póta et al., 2024). Previous models are only able to excel in one of these metrics Fiig ure I (b,c)). We also achieve a SOTA F1 score of 0.925 and KSRME of 0.170 on the non-compliant category. On the MDR Phonon benchmark (Loew et al., 2024). SOTA results are found (Figure (d)). Finally, eSEN achieves the highest test accuracy on the SPICE-MACE-OFF dataset (Kovács et al., 2023).
bench-
'Fundamental AI Research (FAIR) Meta. Correspondence to: Xiang Fu <xiangfu @ mcta.com>> Lawrence Zitnick <zit- nick @ meta.com>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
301
energies, forces, or stresses. To bridge the gap between these performance metrics and practical applicability, need to ensure they correlate with physical property prediction tasks, such as those described next.
Energy drift (meVittom))
0.60
scaro
101
0.70
20+
2.2. Physical property prediction tasks
Belter
Geometry optimizatioo//rlaattiio Many computational chemistry and materials science tasks rely on atomic sys- tems being in stable configurations, which correspond to minima of the PES. Stable states are found by minimizing the potential energy using an optimization procedure that iteratively updates atom positions based on the predicted forces (F F.). Given that many physical properties are evaluated at or near equilibrium states, geometry opti- mization (also referred to as "relaxation") is usually the first step in most computational workflows.
20
40
50
80
100
40
50
60
70
50
Time (ps)
Energy MAE (mee/stom)
140
Entropy MAE (/K/mol
300
0.8
120
Better
Bailer
40
50
60
20
80
40
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80
Energy MAE (meV/atom)
Enerry MAE (meV/stom)
MD simulations. Simulating the time evolution of atomic systems enables us to gain understanding of various chemi- cal and biological processes, as well as enabling the calcula- tion of macroscopic properties, such as liquid densities, that can be experimentally verified. For the task of molecular dynamics simulation, we typically use a potential to com- pute the per-atom forces which are then used to numerically integrate Newton's 's equations of motion. In this work, we will focus on the microcanonical ensemble (NVE), where the number of particles (N), the volume of the system (V), and the energy of the system (E) are kept constant.
(Durs) SevenNet 3/5
SevenNettO GRACE 2L (r6)
MACE-MP-D eqV2 S DeNS
B MPIr) CHGNet
30
Figure 1. (a) Energy conservation in MD simulations. Direct-force models (Orb, eq2 and CHGNet fail to conserve. (b) A higher Fl score on the Buubench--iscoveen strongly correlates with a lower test-set energy MAE. (c) Test-set energy MAE and KSRME on the Goutenchh-iiccoeen benchmark. (d) Test-set energy MAE and vibrational entropy MAE on the MDR Phonon benchmark. Our model (eSEN) achieves the best performance on all benchmarks. higher correlation between test-set energy MAE and physical property prediction performance can be observed among energy- conserving models. All models are trained on MPTrj.
Phonon and thermal conductivity calculations. Precise predictions of phonon band structures and vibrational modes are essential for understanding various material proper- ties, including dynamical stability, thermal stability (Bartel, 2022; Fultz, 2010), thermal conductivity (Razeghi, 2002), and optoelectronic behavior (Ganose al., 2021). The cal- culation of phonon band structures requires the MLIP to accurately predict higher-order derivatives and capture the subtle curvature of the true PES around critical points. Re- cent work (Póta et al., 2024) has demonstrated the usage of MLIPs in predicting thermal conductivity (k) by solving the Wigner transport equation (Simoncelli et al., 2022). In order to accurately predict K, MLIPs must reliably capture both harmonic and anharmonic phonon behavior, which ne- cessitates the calculation of second and third derivatives of the learned PES.
2. Preliminaries
2.1. Machine learning interatomic potentials
Under the Nom--Oppeneeimes approximation (Oppen- heimer, 1927) utilized by DFT (Parr et al., 1979), the Po- tential Energy Surface (PES) can be written as a function of positions, r. and atomic numbers, a: E(r, a). Per-atom forces be calculated by taking the negative gradient of the PES with respect the atom positions, F = F. For periodic systems such as inorganic materials, the lattice parameters I are also considered (E(r, a. L)), and the stress a may also be calculated, which can be understood as the gradient of the potential energy surface with respect to the lattice parameters.
3. Desideratum for physical property prediction
The goal of an MLIP (Unke et al., 2021b) is to predict the exact same properties DFT from training dataset of DFT calculations (Chanussot et al., 2021; Riebesell et al., 2023; Loew et al., 2024). The most straightforward benchmark for MLIPs is to evaluate the model on a held-out test set of DFT calculations, and compare models based on the mean absolute error (MAF) or root mean squared error (RMSF) of
We begin the section by defining what it means for an MLIP to be energy conserving, which is a fundamental principle for applications such as MD simulations (Tuckerman, 2023). For many physical property prediction tasks that probe the higher-order derivatives of the PES it also important that
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
the PES's derivatives are well-behaved (they exist and are bounded). To indicate whether PES meets these criteria, we discuss how MLIP's ability to conserve energy given fixed simulation settings may be used.
(a)
(b)
Node Embeddinng
Booksward
Inpeut
Stress
Input Embeadinng
Edgawise Convolution
Nas Masuel
Indesing Alignnng
Mazes L(L 1) Nauunee
3.1. Conservative forces
Naga x 2002 1)N Narned
Edgessise Convolution
Nago X N
For a force model to be conservative, the work done by moving in closed path must be zero, i.e., the integration of the forces along any path that starts and ends at the same point is zero:
Norm
X Maym
Non-linerrty
Edge Invariant Embedding
Nodowise Feed- Faad-Fodaadd
502 Convocution
Mate N
$ f
Add & Norm
(1)
Envelope
Edge distance
===0 - x X Naued
Scatter Sum
Nodowise Feed-
This property holds if the forces are calculated as the neg- ative derivative of the PES with respect to the atom po- sitions (Unke al., 2021b). However, predicting forces as derivatives requires an additional backpropagatton step through the network, which increases the computational cost of the MLIP. Alternatively, some networks (Liao et al., 2023; Neumann et al., 2024) directly predict forces using a sepa- rate force head to increase efficiency'. Although direct-force models can achieve high accuracy, their con-conservative nature leads to significantly larger errors in certain property prediction tasks (Fu et al., 2023; Loew al., 2024; Póta et al., 2024; Bigi et al., 2024).
Nas Masual
Sum
Node Ertberidnngg
Energy
Figure 2. (a) The cSEN architceture. The high-level architecture similar to Transforner/Eguffrmeer, while the edgewise/nodewise layers are simplifeed/enhanned.. The final-layer L features are used to predict nodewise energy, which is summed to get the total potential energy F.. Forces and stress are obtained through back propagration. (b) The Edgewise Convolution layer in eSEN.
which we refer interested readers to Hairer et al. 2003 and Hairer et al. 2006 for more details.
3.2. Bounded energy derivatives
Conservative forces is necessary but not sufficient con- dition for an MLIP to demonstrate energy conservation in MD. In practice, MD simulations use finite-order numer- ical integration algorithm and a finite time step Al, which introduces truncation errors. The most commonly used in- tegrator for the NVE ensemble is the Verlet algorithm-a second-order integrator. The Verlet integrator is known to approximately conserve the total energy of the system in long-time simulations. As shown by Theorem 5.1 of Hairer et al. 2003, the total energy drift of a simulation satisfies
In Equation (2), the At on the simulation time T I Ar-N implics that the PES must be continuously differentiable to high order for energy conservation in long-time simulations. The critical constant CN depends on the bounds of the derivatives of E up to the (N + )hh order. This implies that, given fixed time step size, E and its higher-orde order derivatives up to the (N)th or- der all need to be continuously differentiable with bounded derivatives to maintain long-time conservation. If the deriva- tives of a PES are more tightly bound, approximate energy conservation will be maintained even at larger step sizes At. Therefore, the magnitude of At for which the energy is sta- ble can be viewed as a proxy for the derivative bounds of the estimated PES. Alternatively, if a certain time step is known to be stable when using DFT, we can determine whether an MLIP has similar bounds on higher-order derivatives by testing whether is also stable using the same time step.
N in the second term and the bound
N
[F(T, a) FF(ro a)l S C&R + CNAINT, (2)
where 7,0 S is the total simulation time, N is positive integer representing the highest order for which the Nith-order derivative of E is continuously differentiable with a bounded derivative, and ro and TT are the starting and ending positions of the atoms in the simulation respec- tively. The constants C' and Cyy are independent of T and AL. The energy drift bound contains two terms: the first term represents a inme-independent fluctuation of O(At?), and the second term represents the long-term energy conser- vation. The proof for this theorem is long and technical, for
4. eSEN
We propose gquivariant Smooth Energy Network (eSEN), new MLIP architecture that improves upon architectures that demonstrate high test accuracies to achieve effective physical property predictions. eSEN is a message- message-passing neural network that conducts multiple blocks of edgewise
Sttictly speaking, direct-force models are not truly "poten- tials", but rather (non-conservative)) "force fields".
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
and nodewise neural processing. Initially, all nodes are em- bedded multi-channel spherical harmonic representations. Each eSEN layer block updates the node embedding by conducting an edgewise convolution, followed by node- wise ford-forward network with normalization layers and residual connections between all layers.
simulation tasks for both inorganic materials and organic molecules using models trained on the MPTr (Jain et al., 2013; Deng et al., 2023) and the SPICE-MACE-OFF (East- man et al., 2023; Kovács et al., 2023) datasets. For inorganic materials, we compute an average conservation error over 81 NVE MD simulations of 100 ps based on the TM23 dataset's simulation settings (Owen et al., 2024). For or- ganic molecules, we compute an average conservation er- ror over 7 NVE MD simulations of 100 ps based on the MD22 dataset's simulation settings (Chmiela et al., 2023), All eSEN models are 2-layer with 3.2M trainable param- eters. We include details regarding the task protocol in Appendix A.
model diagram is shown in Figure 2. eSEN utilizes the same SO2 convolution layer from the equivariant spherical channel network (eSCN) architecture (Passaro & Zitmick, 2023) inside the edgewise convolution block. Compared to eSCN, our edgewise convolution blocks first concatenate the source and target node embedding, then apply two SO2 convolution layers with an intermediate non-linearity. We also add an envelope function (details in Section 5) which not in eSCN. The nodewise feed-forward layer uses two equivariant linear layers and intermediate SiLU-based gated non-linearity (Weiler et al., 2018; Geiger & Smidt, 2022), which is the same as Equiformer (Liao & Smidt, 2022). Unlike eSCN and EquiformerV2 (Liao et al., 2023), spherical-harmonics which projects the spherical harmonics channels onto spa- tial grids for nodewise processing, the nodewise layers in eSEN do not discretize the node representations. As we demonstrate in Section 5, this design improves the ability of the model to conserve energy. Normalization is performed using the equivariant layer normalization (Ba, 2016) pro- posed by Equiformer (Liao & Smidt, 2022). In the next section, we conduct an in-depth analysis of the key design choices for energy conservation, which we argue is impor- tant for accurate physical property prediction.
5.1. Direct-force prediction
Models that directly predict forces F from the atomic con- figuration may produce forces that are inconsistent with the energy prediction, i.e., F + - V, and more impor- tantly are unlikely to be conservative. From the perspective of minimizing the test error, the direct-force approach has strong motivations: it avoids the backward pass for force prediction, which significantly improves model efficiency and enables fow-prccssoon training which further accelerates training. Empirically, current SOTA accuracy on the OC20, OC22, and natbench-Discovery (Chanussot et al., 2021; Tran et al., 2023; Riebesell et al., 2023) benchmarks are achieved by direct-force models. Despite this, the direct- force formulation results in significant energy drift in MD simulations, as shown in Figure 4 (al, a2). For this reason, we compute forces as the negative gradient of the PES with respect to the atom positions in eSEN.
5. Design choices for enhancing physical property prediction
4.0
4,0
Direct force Conserved
As discussed in Section 3, having conservative forces with continuous and bounded energy derivatives are properties an MLIP should obey for MD simulations. It can also be seen as prerequisite for the MLIP to accurately capture higher-order behavior of the PES and thus high accuracy in physical property prediction tasks such as phonon calcu- lations. Motivated by this observation, we identify design choices that impact a model's ability to conserve energy and whether its PES varies smoothly. These design choices can be categorized into three aspects: (1) conservative vs. direct-force prediction; (2) discretization of the representa- tion; and (3) obtaining a continuous and smoothly varying PES. For many of these design aspects, their impact on the desired properties is not well understood.
Validation loss
3.5
3.5
Finetuning
Rn Rnetuning starts
3.0
2.0
2.9
Epach
Time (hours)
Figure 3. Validation loss curves for epoch and wallclock time.
Direct-force pre-training. Although direct-force models are not suitable for certain physical property prediction tasks, they may still offer advantages (Bigi et al., 2024; Amin et al., 2025). We demonstrate their efficiency can offer significant benefit as a pre- pre-training strategy for conservative model. Figure 3 shows the validation loss of 2-layer eSEN models trained on the MPTrj dataset: direct-force, conservative, and conservative fine-tuning from a pre-trained direct-force backbone. We start from a direct-force model trained for 60 epochs, remove its direct-force prediction head, and fine-
To quantify whether an MLIP's PES is continuous and smoothly varying, we measure the ability of the resulting MLIP to conserve energy during MD simulations with a pre- determined fixed time step. We trained eSEN models under the same hyperparameters while ablating one design choice at a time. We construct out-of distribbiit (OOD) MD
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
(a
MPT] TM23, Direct force prediction
[61]
MPTí TM23, Representation discretization
(c1)
MPTr TM23, Smoothly varying PES
eSEN
eSEN
Energy orift (meViatom)
10'
10'
eSEN, discrete, resolution=s
10a
eSEN, neighbor limi
eSEN, discrete, resolution-10
eSEN, no enve ope
107
10a
discrete, resolution=14
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number of basis-512
eSEN
eSEN, Bessel radial basis
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SPICE-MD22, Smoothly varying PES
Energy drift (melzatom)
101
10a
eSEN, discrete, resolution-6 eSEN, discrete, resolution=10
eSEN
10a
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discrate, resolution-14
cSEN, neighbor
@SEN
eSEN, no envelope
direct-force
eSEN, number of basis=512
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Figure 4. Conservation error on the TM23 task (top row) and MDD22 task (bottom row) for ablating design choices of cSEN. Models that conserve energy are bolded in the legends.
tune using conservative force prediction. The conservative fine-tuned model achieves a lower validation loss after being trained for 40 epochs compared to the from-scratch conser- vative model being trained for 100 epochs. The fine-tuning strategy also reduces the wallelock time for model training by 40%. The strategy of combining direct-force pre-training and conservative fine-tuning is also shown be effective under different data/model settings (Bigi et al., 2024).
5.3. Smoothly varying PES
Subtle choices in the design of MLIPs can have a significant impact on whether a PES varies smoothly and can even lead to the presence of discontinuities. These include how neigh- boring atoms are chosen, whether envelope functions are used near atom distance cutoffs, and which basis functions are used to embed pairwise atom distances. We discuss each of these in turn.
5.2. Representation discretization
A maximum number of neighbors limit in graph construc- tion has been found to improve training efficiency without compromising test error (Liao et al., 2023; Qu & Krish- napriyan, 2024). However, results in a discontinuity in the learned PES as the nearest-K neighbors may change drastically under a small perturbation of the atom positions. As shown in Figure 4 (cl, c2), having maximum neighbor limit breaks energy conservation. In eSEN, instead of limit- ing the number of neighbors, we use the common approach of applying a distance cutoff (6A) under which all neighbors are kept.
As proposed by Cohen & Welling 2016 and later used in cSCN and EquiformerV2 (Zitnick al., 2022; Passaro & Zitnick, 2023; Liao et al., 2023), non-linearities may be per- formed by projecting the spherical harmonics to discrete grid. A 1 x convolution or pointwise non-linearit, may then be applied to this grid, which then get projected back to the spherical-han -arrficcs space. The non- non-linear step may introduce higher-frequency signals than cannot be properly represented by the spherical harmonics, i.e., they are beyond the Nyquist frequency. This can lead to sampling errors that break strict cquivariance and energy conservation. This problem can be mitigated by sampling the grid at higher res- olutions as shown in Figure 4 (b1, b2). eSEN, we instead use the based equivariant Gated non-linearity (Weiler al., 2018; Geiger & Smidt, 2022) that performs the non- linearity directly in the spherical harmonic representation. This does not require projection to discrete grid, so the model is perfectly cquivariant and conservative up to numerical accuracy.
Envelope functions were first introduced the DimeNet architecture (Gasteiger et al., 2020a) improve model smoothness. The radial basis function used in MLIPs is not twice continuously differentiable due to the use of a finite cutoff during graph construction. By applying a poly- nomial envelope function on the edge messages, the values in an edge message and its first/higher-order derivatives with respect to atom positions decays to when the edge distance approaches the cutoff distance. Figure 4 (cl, c2)
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
6.1. Matbench Discovery
Table 1. Test set MAF for design choices studied in Section 3. The conserved model significantly outperforms the dtrect-force model on SPICE-MACE- MACE-OFFE Nussis 512 performs slightly better on SPICE but slightly worse on MPTrj. Other eSEN variants Energy all have similar test errors on MPTHjSPPCTE-MACTOOI
The Matbench-Discoveer benchmark evaluates a model's s ability to predict ground-state (0 K) thermodynamic stability through geometry optimization and energy prediction. It is widely used benchmark for evaluating ML models in ma- terials discovery. The compliant benchmark only includes models trained on the MPTrj (Jain et al., 2013; Deng et al., 2023) dataset or its subset, which facilitate a fair comparison of model architectures. The Fl score the primary metric used to rank models. We train an eSEN with 30M parame- ters on MPTrj for 60 epochs of direct-force pre-training and 40 epochs of conservative fine-tuning, DeNS (Liao et al., 2024) is used during direct-force pre pre-training. As shown in Table 2, eSEN MPP achieves an F1 score of 0.831-the highest among all compliant models. eSEN-30M-MP also achieves the lowest root mean square deviation (RMSD) when comparing the relaxed structures to the ground truth DFT reference.
MAE is in me Force MAE is in meV/Á. Stress MAE is in
me V/A/atom.
MP'rj Force
SPICE
Model
Energy
Stress
Energy
Force
eSEN
17.02
43.96
0.14
0.23
6.36
eSE,, direct
18.66
43.62
0.16
0.56
10.98
eSEN, neighbor limit
17.30
44.11
0.14
0.24
cSEN, no cavelope
17.60
44.69
0.14
0.23
6.33
cSEN, Nhaes = 512
19.87
48.29
0.15
0.19
5.40
eSEN, Bessel
17.65
44.83
0.15
0.20
5.54
eSEN, discrete, res=6
17.05
43.10
0.14
0.26
6.34
eSEN, discrete, res= 10
17.11
43.13
0.14
0.33
6.57
eSEN, discrete, res= 14
17.12
43.09
0.14
0.33
6,51
shows a model fails to conserve energy without the envelope function.
The thermal conductivity prediction task requires ac- curate modeling of harmonic and anharmonic phonons in materials, which tests the accuracy of second and third or- der derivatives of the learned PES. The primary metric the symmetric relative mean error in predicting thermal conductivity (KSRME). We follow the protocol set forth in the Matbench-Discovery benchmark (Riebesell et al., 2023; Póta et al., 2024) to predict thermal conductivity k. After running a structural relaxation, k is computed using sec- ond and third order force constants obtained from phonon calculations using the supercell method.
Radial basis functions are commonly used to embed inter- atomic (Bsttnc et al., 2013). A larger number of basis functions (512 in Passaro & Zitnick 2023, as opposed to 10 in eSEN's default setting) allows higher-frequency sig- nals to pass through the network. This can lead to the PES being more sensitive to small shifts in the atom positions. In our experiments, using large number of basis functions breaks conservation for the TM23 tasks, but is able to con- serve energy for the MD22 tasks. Using a Bessel radial basis function (as opposed to Gaussian radial basis in the default setting) does not impact conservation properties in both tasks.
As shown in Table 2, our model achieves a KSRME of 0.340 under the default evaluation protocol proposed by Póta et al. 2024. Notably, our model excels in both the Fl score and KSRME, while all previous models only achieve SOTA per- formance on one or the other of these metrics.
5.4. Ablation studies
The non-compliant Matbench-Diseoveer benchmark in- cludes models trained on datasets other than MPTrj. eSEN- 30M-0AM is eSEN model with 30 million param- eters pre-trained on the OMat24 (Barroso- 2024) dataset then fine-tuned on the subsampled Alexan- dria (sAlex) dataset (Barroso- Luque et al., 2024; Schmidt et al., 2024) and MPTri dataset. As shown in Table 3, eSEN- 30M-0AM achieves the best performance among all non- compliant models with an Fl score of 0.925, a KSRME of 0.170. and an RMSD of 0.0608, significantly advancing tfate-of-the-art.
Many of the architecture choices described above have neg- ligible impact on the test set errors as shown in Table However, as shown in Figure 4, they can have a dramatic impact on whether a model is conservative in practice. If model is found to be conservative, stronger correlations are found between test errors and property prediction tasks
al.,
(Figure and Figure 6).
6. Experiments
In the previous section, we demonstrated the design of eSEN results in its ability to be energy-conserving in MD simula- tions. In this section, we evaluate eSEN in physical property prediction tasks: (1) materials stability prediction based on geometry optimization; (2) thermal conductivity prediction: and (3) phonon calculation. We also demonstrate the cor- relation between test energy MAE and physical property prediction tasks for eSEN.
6.2. MDR phonon benchmark
The MDR Phonon benchmark (Loew et al., 2024) assesses the performance of MLIPs in predicting key phonon proper- ties, including maximum phonon frequency (wmax), entropy (S), free energy (F) and heat capacity at constant volume
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Table Mathench-Discovery benchmark results of compliant models (trained only on MPtrj or its subset) with results on the unique
prototype split. MAE is in units of eVVatom. (+/J) stands for higher/lower the better.
Metric
eSPN-30M-MP
ogV2.5 DeNS
MiRRIS-MP
AlphaNes-MP
DPA3--2MMP
QR MPtj
SevenNer-B35
GRACE-2I-MPy
MACE-MP-0
CHCNN
M3GiNer
FL
0.831
0,815
0.809
0.799
0.786
0.765
0.760
0.591
0.669
0.613
0.569
DAPT
5.260
5,042
5.049
4,863
4.822
4.702
4,629
4,163
3.777
3,361
2.882
Precision 7
0.504
0.771
0.772
0,743
0.737
0.719
0.708
0.636
0,577
0.514
0.441
Accuracy
0,946
0
0.938
0.333
0.929
0.922
0.920
0.896
0.878
0.85]
0.813
MAE 4
0,036
0,037
0.041
0.045
0.044
0.057
R2 1
0.822
0.788
0.803
0.745
0.804
0.756
0.776
0,741
0.697
0.689
0.585
0.340
1.676
0.861
1.31
0.959
1.725
0.550
0.525
0.647
1.317
11422
RMSD
0.0752
0,0757
0.0773
0.1067
0.0823
0.1007
0.0847
0.0897
0/0915
0.0949
0.1117
Table 3. Matbench- Mathench-Discovery benchmark results of non-compliant models with results on the unique prototype split.
Model
cSEN-30M-DAM
cgV2-M-OAM
v3
SevenNet-FF-omp.
DBA3-v2-OpenLAM
CRACE-2L-OAM
MaterSim-v1-5M
MACE-MPA-0
GNoME
F1
0.925
0.917
0.905
0.901
0.890
01880
0.862
0.852
0.829
DAFT f
6.069
6.047
5.912
5.825
5.747
5,774
5.852
5,582
5.523
Precision
0.928
0.924
0,904
0,879
0.879
0.883
0.895
0.853
0,844
Accuracy 1
0.977
0.975
0.971
0.969
0.966
0,963
0.959
0.954
0.955
MAE
0,018
0.020
0,024
0.021
0.022
0.023
0.024
0.028
0.035
R21 1
0.866
0.848
0.821
0.867
0.869
0.862
0.863
0.842
0.785
*BRME
0.170
1,771
0.210
0.317
0.687
0.294
0.574
0,412
N/A
RMSD 4
0.0608
0.0691
0,0750
0,0639
0.0679
0.0666
0,0733
0,0731
N/A
(Gv), for around 10,000 materials. The evaluation follows the testing protocol outlined by Loew et al. 2024. Sec- tion 6.2 shows the resulting MAE of our model and those of several other models cSEN achieves SOTA results in both compliant and non-compliant categories.
Table 4. Summary model performance on the MDR Phonon benchmark. Metries include maximum phonon frequency (MAB(wmax), in K), the vibrational entropy (MAE(S), in J/K/mol), the Helmholtz free energy (MAB(F)), in kJ/mol), and the heat ca- pacity at constant volume (MAB(Cv)), ), in J/K/mol). "The OMat24 dataset uses a slightly different DFT setting from the DFT setting of the MDR Phonon benchmark.
Our results are consistent with those reported by Loew et al. 2024, showing that conservative MLIPs significantly out- perform direct-force models in terms of prediction accuracy when tested using phonon calculations with a displacement of 0.01 A. The high error of direct-force models be largely attributed to high-frequency prediction errors at small displacements (Loew et al., 2024). Increasing the displacement used in the finite-difference phonon calcula- tions to 0.2 can considerably improve prediction accuracy of direct-force models (with caveats). We include a more detailed analysis of the relationship between atom displace- ment and phonon prediction in Appendix B.
Compliant models
MAB(slm)
MAE(S)
MAE(F)
MAE(Cу)
M3GNet CHGNet
98
150
56
22
89
114
45
21
MACE
61
60
24
13
(IAACE-2L((1)))
25
9
5
SevenNet-O
40
48
19
9
SevenNet-1255
26
28
10
cSEN-30M-MP
4
Direct-force models
Orb MPTj [0.01 A]
309
476
181
Orb MPTi (0.2 Åj
34
11
8
eqV2-S-DeNS (0.01 A]
280
224
54
94
eqV2-S-DeNS (0.2 Á]
58
26
8
8
In physical phonon calculations, we expect the results to converge as the displacement goes to zero. By examining the resulting phonon band structure, we can gain insight into this behavior. Figure 5 presents the predicted phonon band structure and density of states for three representative materials using eSEN. The predicted phonon bands exhibit convergence the displacement decreases. In contrast, Fig- ures Figure C.11 and Figure 12 display the phonon bands for the same three materials predicted using VqV2-S-DeNS (direct-forces), which not only fail to demonstrate conver- gence but also exhibit significant errors, including missing
Non-compliant models
eq 122-M-AAN [0.01 A]
780
403
241
100
cqV2-M-OAM (0.2 A]
50
25
7
9
MatterSim
17
15
5
3
GRACE-2L-OAM
19
14
5
4
SevenNet-MF-onpp
3
3SEN-30M-OAM
3
acoustic branches and spurious imaginary frequencies.
While we find that the OMat-trained models (without sAlex/MPTrj finetuning) may provide lower error on phonon prediction (7/7/2/2 for //nx/S/F/Cy MAE), we refrain from direct comparison due to mismatch in level of theory. We attribute this result to the softening issue of the sAlex and MPTrj dataset (Deng et al., 2025; Barroso- et al., 2024), which OMar24 addresses. We refer interested
In addition to our model, we also run the evaluation for GRACE (Bochkarev et al., 2024), SevenNet (Park al., 2024), Orb (Neumann al., 2024), and V2-S-DDeNS (Liao et al., 2023; Barroso-Luug Luque et al., 2024), which were not included in the work by Loew et al. 2024.
7

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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
AIN
Frequency (TH2)
Frequency (!Hz)
Frequency (1H2)
20
20
XR
14
10
AL MH K0
DOS (States/THiz)
DOS (States/THi)
DOS
(States/THz)
Figure 5. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCI (CsCI structure), AIN (wurtzite structure) using eSEN at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using displacement of 0.01 A.
Table Test set MAE for SPICE-MACE-OFE Energy (E) MAE in me Force (F) MAE is in meV/Á. ASscAIP-45M is
(UKmal)
direct-force model.
47M
ESSAIP-45M
6.2M
6.EN-6.5M
Dataset
E
F
E
F
PuhChem
0.88
14.75
0.53
5.86
0.22
6.10
0.15
4.21
0.5
DES370K M.
0.59
6.58
0.41
3.48
0.17
DES370K
6.62
0.4
Dipeptides
0.42
10.19
0.31
5.21
0.10
3.04
2.00
20
Sol. AA
0.98
19.43
0.61
11.52
0.30
5.76
0.25
3.68
60
70
Water
0.83
13:57
0.72
10.31
0.24
3.88
0.15
2.50
Energy MAE (meV/stom)
Energy MAE (ime (imeWatom))
QMugs
0.45
16.93
0.41
8.74
0.16
5.70
0.12
3.78
eSEN-30M
eSEN, Bessel
eSEN, neighbar limit
eSEN 6.5
eSEN, direct-force
eSEN. envelope
eSEN-3.2M
30
eSEN, discrete
eSEN, Newa m512
readers to Deng et 2025 and Barroso-Luqqe al. 2024 for detailed discussion on the softening issue of some DFT datasets and its relation to phonon properties.
Figure 6. Test error correlation across several property prediction tasks for eSEN variants. Conservative models are shown boxes and those found to not conserve as crosses. Note metrics for conservative models have stronger correlation with test errors.
6.3. SPICE-MACE-OFF
We train and evaluate eSEN models on the SPICE-MACE- OFF dataset (Kováes et al., 2023), which is built upon the SPICE dataset (Eastman et al., 2023). As shown in Ta- ble 5, eSEN with 6.5M parameters outperforms MACE- OFF-L (4.7M parameters) and EscAIP (45M parameters, direct-force) on all test-set splits for both energy and force MAE. We also include results for eSEN with 3.2M parame- ters, which has inference efficiency similar to MACE-4.7M, while achieving lower energy/force MAE. More de- tails the inference efficiency benchmark are included in Appendix C.
correlation between test error and ksRmp//ibrational en- tropy MAE can be observed. We include experimental details about Figure and Figure 6 in Appendix and addi- tional results for other phonon properties Appendix
7. Related works
MLIP architectures have made significant progress since their initial proposal (Behler & Parrinello, 2007). These architectures are usually symmetry-preserving (Smith et al., 2017: Schütt al., 2017: Gilmer et al., 2017: Chmiela et al., 2017; Artrith et al., 2017; Unke & Meuwly, 2018; Zhang et al., 2018; Zubatyuk al., 2019; Smith al., 2020; Kovács et al., 2021), with increasingly expressive atom environment embeddings and message- tions (Gasteiger et al., 2020b; 2021; Schütt et al., 2021; Liu et al., 2021; Unke et al., 202 Chen & Ong. 2022; Deng et al., 2023; Cheng, 2024; Yin al., 2025). Notably, equiv-
6.4. Test-set error for model development
In Figure 1, showed the correlation between test enror and physical property prediction tasks for different architec- tures. Figure 6 demonstrates this correlation for different variants of eSEN (with a k-materials subset the MDR Phonon benchmark for efficiency). In particular, among models that pass the MD energy conservation test, strong
opera-
8

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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
ariant architectures based spherical harmonics represen- tations (Thomas al., 2018; Thölke & De Fabritiis, 2021; Batzner et al., 2022; Musaelian et al., 2022; Batatia et al., 2022; Passaro & Zitnick, 2023; Liao et al., 2023; Bochkarev et al., 2024; Park et al., 2024; Batatia et al., 2025) have shown strong performance on large-scale datasets. Mean- while, the high computational cost of these architectures has sparked significant interest in scalable architectures that may not respect physical principles such energy conser- vation (Langer et al., 2024; Brehmer et al., 2024; Hu et al., 2021; Yang et al., 2024; Qu & Krishnapriyan, 2024; Neu- mann et al., 2024; Rhodes et al., 2025). These models have demonstrated strong performance in accuracy, scalability, and relaxation tasks (Chanussot al., 2021; Riebesell et al., 2023). While their non-physical nature may make them unsuitable for direct usage some physical property pre- diction tasks, they may still provide benefit by using the pre-training strategy proposed in this paper and Bigi et al. 2024, distilling them to conservative models (Amin et al., 2025), or combining them with a conservative model using multiple-time-stee integration (Bigi et al., 2024).
knowledge and is usually time-consumingg whereas evaluat-
ing test set error is straightforward and efficient.
Software and Data
Code is available at: https://github: com/ facebookresarch/fatti wwwerrhhcaaarrem.... Pretrained check- points are available at https: httpg//hggggggg.eee/ee//// co/ facabook/oMAT24.
A Acknowledgements
We acknowledge Zachary W. Ulissi (FAIR at Meta), Aditi S. Krishnapriyan (UC Berkeley), Samuel M. Blau (LBNL) and other members of the FAIR Chemistry team for helpful discussions, and Ammar Rizvi (FAIR at Meta) for project support.
Impact Statement
This paper presents work whose goal is to advance the field of machine learning for atomistic modeling, with the po- tential accelerate scientific discovery chemistry and biological sciences. As the field of ML-baaed atomistic modeling continues evolve, we emphasize the importance of ongoing discussions ethical considerations and best practices to ensure that these technologies are developed and applied in ways that benefit society as a whole.
MLIPs and physical observables. While MLIPs con- tinue to improve, it is necessary to evaluate them in realis- tic tasks that are relevant scientific discovery. Physical property prediction benchmarks that involve geometry opti- mization (Riebesell et al., 2023; Lan et al., 2023; Wander et al., 2024), MD simulations (Fu et al., 2023; Kovács et al., 2023; Moore et al., 2024; Sabanes Zariquiey et al., 2024; Eastman al., 2024), vibrational analysis and phonon cal- culations (Póta et al., 2024; Loew al., 2024; Wines & Choudhary, 2024), and others are increasing in scale with broader applications and wider adoption. Training strategies for learning from physical observables (Wang et al., 2020: Greener, 2024; Röcken et al., 2024; Raja et al., 2024) and the higher-order derivatives of the PES (Fang et al., 2024; Williams al., 2025) are promising directions to further improve MLIPs for predicting physical properties.
References
Ackland, G., Warren, M., and Clark, S. Practical methods in initio lattice dynamics. Journal of Physics: Condensed Matter, 9(37):7661, 1997.
Amin, I., Raja, S., and Krishnapriyan, A. Towards fast, specialized machine learning force fields: Distilling foundation models via energy hessians. arXiv preprint arXiv:2501.0000, 2025.
8. Diseussion
Artrith, N., Urban, A., and Ceder, G. Efficient and accu- rate machine-learning interpolation of atomic energies in compositions with many species. Physical Review B, 96 (1):014112,000 2017.
We identify conservative forces and a smoothly-varying PES as two important properties for MLIPs to consistently per- form well in physical property prediction tasks. We offer an analysis of design choices enhance these two properties. The resulting eSEN architecture bridges the gap between the test-set еигог and downstream applications, achieving SOTA performance in force/energy prediction, geometry optimization, phonon calculations, and thermal conductivity prediction. This implies it may be possible to use test error as a proxy metric for evaluating model performance during development, if model passes energy conservation tests. This can accelerate innovations in MIIPs, since benchmark- ing physical properties usually requires significant domain
Ba, J. L. Layer normalization. 1.Xiv:1007.066 2016
arXiv preprint
Barroso-Luque, L., Shuaibi, M., Fu, X., Wood, B. M., Dzamba, M., Gao, M., Rizvi, A., Zitnick, C. L., and Ulissi, Z. W. Open materials 2024 (omat24) inor- ganic materials dataset and models. arXiv preprint arX1v:2410.12771, 2024.
Bartel, C. J. Review of computational approaches to predict

---

Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
the thermodynamic stability of inorganic solids. Journal of Materials Science, 57(23):10455-10499 2022.
ACS Catalysis, 11(10):059-6072, 2021. doi: 10.
1021 MW21aascaaall.0004555... StRthttps::// http::::///...///1./.1 org/10. 1021/acscatall 0c04525.
Bartók, P., Kondor, R., and Csányi, G. On representing chemical environments. Physical Review B Condensed Matter and Materials Physics, 87(18) 184115, 2013.
Chen, C. and Ong, S. A universal graph deep learning interatomic potential for the periodic table. Nature Com- putattioall Science, 2(11):718-72) 2022.
Batatia, I. Kovacs, D. P., Simm, G. N. C., Ortner, C., and Csanyi, G. MACE: Higher order equivariant mes- sage passing neural networks for fast and accurate force fields. In Oh, A. H., Agarwal, A., Belgrave, D., and Cho, K. (eds.), Advances in Neural Information Processing Systems, 2022. URL https::////ccccc. forum? Siamm2ddePPpSgPee-
Cheng, B. Cartesian atomic cluster expansion for machine learning interatomic potentials. npj Computational Mate- rials, 10(1):157, 2024.
httpps//ppenreriee..cc http://pppeeeeeeeeeeeeee net/
Chmiela, S., Tkatchenko, A., Sauceda, H. E.,, Poltavsky, I., Schüit, K. T., and Müller, K.-R. Machine learning of ac- curate energy-conserving molecular force fields. Science advances, 3(5):c1603015, 2017.
Batatia, I., Benner, P., Chiang, Y., Elena, A. M., Kovács, D. P., Riebesell, J., Advincula, X. R., Asta, M., Avaylon, M., Baldwin, W. J., et al. A foundation model for atomistic materials chemistry, arXiv preprint arXiv:2401.00096 2023.
Chmiela, S., Vassilev-Galindo, V., Unke, O. T., Kabylda, A., Sauceda, H. E., Tkatchenko, A., and Müller, K.- R. Accurate global machine learning force fields for molecules with hundreds atoms. Science Advances, 9 (2)eadf00873, 2023.
Batatia, I., Batzner, S., Kovács, D. P.. Musaclian, A., Simm, N., Drautz, R., Ortner, C., Kozinsky, B., and Csányi, G. The design space of e3)equuaari atom-centred interatomic potentials. Nature Machine Intelligence, pp- 1-12, 2025.
Cohen, T. and Welling, M. Group equivariant convolutional networks. In International conference on machine learn- ing, pp. 2990-2999. PMLR, 2016.
Batzner, S., Musaclian, A., Sun, L., Geiger, M., Mailoa, P., Kornbluth, M., Molinari, N., Smidt, E., and Kozinsky, B. E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature communications, 13(1):1-11, 2022.
Deng, B., Zhong, P., Jun, K., Riebesell, J., Han, K., Bartel, C. J., and Ceder, G. Chgnet as pretrained universal neural network potential for charge-informed atomistic modelling, Nature Machine Intelligence, 5(9):1031-1041, 2023.
Behler, J. and Parrinello, M. Generalized
representation of high-dimensional potential-energy sur- faces. Physical review letters, 98(14) 146401, 2007.
Deng, B., Choi, Y., Zhong, P., Riehesell, J., Anand, S., Li, Z., Jun, K., Persson, K. A., and Ceder, G. Systematic softening in universal machine learning interatomic po- tentials. npj Computational Materials, 11(1):1-9, 2025.
Bigi, F., Langer, M., and Ceriotti, M. The dark side of the forces: assessing non-conservative force models for atom- istic machine learning, arXiv preprint 2rX1v2241..11569, 2024.
Eastman, P., Behara, K., Dotson, D. L., Galvelis, R., Herr, J. E.. Horton, J. T., Mao. Y., Chodera, J. D., Pritchard, B. P., Wang. Y., et al. Spice, a dataset of drug-like molecules and peptides for training machine learning potentials. Scientific Data, 10(1):11, 2023.
Bochkarev, A. Lysogorskiy, Y., and Drautz, R. Graph atomic cluster expansion for semilocal interactions be- yond equivariant message passing. Phys. Rev. X, 14:021036, Jun 2024. doi: 10.1103/PhysRevX.ik 021036. URL //ltns::/link.... ottppooo./o///..///////////////. 1103/PhysRevae 14. 021036.
Eastman, P., Pritchard, B. P., Chodera, D., and Markland, T. E. Nutmeg and spice: models and data for biomolec- ular machine learning. Journal of chemical theory and computation, 20:19)385,,,8599 2024.
Brehmer, J., Behrends, S., de Haan, P., and Cohen, T. Does equivariance matter at scale? arXiv preprint 1rXiv:2410.2317,, 2024.
Fang. S., Geiger, M., Checkelsky, J. G., and Smidt, T. Phonon predictions with (3)-equivariant graph neural networks, arXiv preprint 4rXiv:2403.11347,20 2024.
Chanussot, L., Das, A., Goyal, S., Lavril, T., Shuaibi, M., Riviere, M., Tran, K., Heras-Domingo J., Ho, C., Hu, W., Palizhati, A., Sriram, A., Wood, B., Yoon, J., Parikh, D., Zitnick, C. L., and Ulissi, Z. Open cat- alyst 2020 (oc20) dataset and community challenges.
Fu, X., Wu, Z., Wang, W., Xie, T., Keten, S., Gomez- Bombarelli, R., and Jaakkola, T. Forces are not enough:
10

---

Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Benchmark and critical evaluation for machine learn- ing force fields with molecular simulations. Transac- tions Machine Learning Research, 2023. ISSN 2835- 8856. URL https::///.. http://///pp..eeeeeeeeeeom.. i id=A8pqQi pwkt. Survey Certification.
Kovács, D. P., Oord, C. d., Kucera, J., Allen, A. E.,, Cole, D. J., Ortner, C., and Csányi, G. Linear atomic cluster expansion force fields for organic molecules: beyond rmse. Journal of chemical theory and computation, 17 (12):766-7711,, 2021.
forum?
Fultz, B. Vibrational thermodynamics of materials. Progress in Materials Science, 55(4):247-332, 2010.
Kovács, D. P., Moore, J. H., Browning, N. J., Batatia, I., Horton, J. T., Kapil, V., Witt, W. C., Magdäu, I.-B., Cole, D. J., and Csányi, G. Mace-off23; Transferable machine learning force fields for organic molecules. arXiv preprint PrXiv:2312.1525 2023.
Ganose, A. M., Park, J., Faghaninia, A.,
R., Persson, K. A., and Jain, A. Efficient calculation of carrier scattering rates from first principles. Nature communications, 12(1):2222, 2021.
Lan, J., Palizhati, A., Shuaibi, M., Wood, B. M., Wander, B., Das, A., Uyttendaele, M., Zitnick, C. L.,, and Ulissi, Z. W. Adsorbml: a leap in efficiency for adsorption energy calculations using generalizable machine learning potentials. npj Computational Materials, 9(1):172, 2023,
Gasteiger, J., GroB, J., and Günnemann, S. Directional message passing for molecular graphs. arXiv preprint arXiv:2003.03123. 2020a.
Gasteiger, J., GroB, J., and Günnemann, Directional message passing for molecular graphs. In International Conference on Learning Representations, 2020b.
Langer, M. F., Pozdnyakov, S. N., and Ceriotti, M. Probing the effects of broken symmetries in machine learning. Ma- chine Learning: Science and Technology, 5(4):04LT01, 2024.
Gasteiger, J., Becker, F., and Günnemann, S. Gemnet: Uni- versal directional graph neural networks for molecules. Advances in Neural Information Processing Systems, 34: 6790-6802, 2021.
Larsen, A. H., Mortensen, J. J., Blomqvist, J., Castelli, I. E., Christensen, R., Dutak, M., Friis, J., Groves, M. N., Hammer, B., Hargus, C.,, Hermes, E. D., Jennings, P. C., Jensen, P. B., Kermode, J., Kitchin, J. R., Kolshjerg, E. L., Kubal, J., Kaasbjerg, K., Lysgaard, S., Marons- son, J. B., Maxson, T., Olsen, T., Pastewka, L., Peter- son, A., Rostgaard, C., Schiøtz, J., Schütt, O., Strange, M., Thygesen, K. S., Vegge, T., Vilhelmsen, L., Walter, M., Zeng, Z., and Jacobsen, K. W. The atomic simu- lation environment-4 python library for working with atoms. Journal of Physics: Condensed Matter, 29(27); 273002, 2017. URL /ttt:://stacle iop.org/ 00//////2/1122/=27300000
Geiger, M. and Smidt, T. e3nn: Euclidean neural networks. arXiv preprint arXiv:2207.0m 00945,, 2022.
Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chem- istry. In International conference on machine learning, pp. 1263-1272. PMLR, 2017.
Greener, J. G. Differentiable simulation to develop molecu- lar dynamics force fields for disordered proteins. Chemi- cal Science, 15(13):4897-490, 2024.
Liao, Y.-L. and Smidt, T. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs. arXiv 2022.
Hairer, E., Lubich, C., and Wanner, G. Geometric numerical integration illustrated by the störmer-verlet method. Acta numerica, 12:399 450. 2003.
preprint arXiv:2206.
Liao, Y.-L., Wood, B., Das, A., and Smidt, T. Equiformerv2: Improved equivariant transformer for scaling to higher- degree representations. arXiv preprint arXiv:2306. 12059, 2023.
Hairer, E., Hochbruck, M., Iserles, A., and Lubich, C. Geo- metric numerical integration. Oberwolfach Reports, 3(1): 805-882, 2006.
Liao, Y.-L.. Smidt, T., and Das, A. Generalizing denoising to non-equilibrium structures improves equivariant force fields. arXiv preprint 24X3:22403.09549 2024.
Hu, W., Shuaibi, M., Das, A., Goyal, S., Sriram, A., Leskovec, J., Parikh, D., and Zimick, C. L. Forcenet: A graph neural network for large-scale quantum calcula- tions. arxiv preprint 2003v:2103.011 2021.
Liu, Y.. Wang, L., Liu, M., Lin, Y., Zhang, X., Oztekin, B., and Ji, S. Spherical message passing for 3d molec- ular graphs. In International Conference Learning Representations, 2021.
Jain, A., Ong, S., Hautier, G., Chen, W., Richards, W., Dacck, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., et al. The materials project: a materials genome approach to accelerating materials innovation. apl mater 1: 01000, 2013.
Loew, A., Sun, D., Wang, H.-C., Botti, S., and Marques, M. A. Universal machine learning interatomic potentials

---

Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
ready for phonons. arXiv preprint arXiv:2412.16551, 2024.
Póta, B., Ahlawat, P., Csányi, G., and Simoncelli, M. Ther- mal conductivity predictions with foundation atomistic models. arXiv preprint 240.0024 2024.
Merchant, A., Batzner, S., Schoenholz, S. S., Aykol, M., Cheon, G., and Cubuk, E. D. Scaling deep learning for materials discovery. Nature, 624(7990):80-85,, 2023.
Qu, E. and Krishnapriyan, A. S. The importance of be- ing scalable: Improving the speed and accuracy of neu- ral network interatomic potentials across chemical do- mains. In The Thirty-eighth Annual Conference on Neu- ral Information Processing Systems, 2024. URL https: / //opprreviees net. www.rrmmmmxAAaaaaauu d=Y4mBazu4yy.
Moore, H., Cole, D. J., and Csanyi, G. Computing hydra- tion free energies of small molecules with first principles accuracy. arXiv preprint arXiv:2405. 18171, 2024.
d=
Musaelian, A., Batzner, S., Johansson, A., Sun, L., Owen, C. J., Kornbluth, M., and Kozinsky, B. Learning local equivariant representations for large-scale atomistic dy- namics. arXiv preprint arXiv:2204.05229, 2022.
Raja, S., Amin, I., Pedregosa, F., and Krishnapriyan, A. Stability-awar aware training of neural network interatomic potentials with differentiable boltzmann estimators. arXiv preprint arXiv:2402. 2024.
Neumann, M., Gin, J., Rhodes, B., Bennett, S., Li, Z., Choubisa, H., Hussey, A., and Godwin, Orb: A fast, scalable neural network potential. arXiv preprint arXiv:2410.22570 2024.
Razeghi, M. Thermal Properties of Crystals. In Razeghi, M. (ed.). Fundamentals of Solid State Engineering, pp. 197- .7.
220. Springer US, 2002. doi: 10.1007/0-336-477555
Rhodes, B., Vandenhaute, S., Šimkus, V., Gin, J., Godwin, J., Duignan, T., and Neumann, M. Orb-v3: atomistic simulation at scale. arXiv preprint arXiv:2504.0623/ 2025.
Nickolls, J., Buck, I., Garland, M., and Skadron, K. Scalable parallel programming with cuda: cuda the parallel pro- gramming model that application developers have been waiting for? Queue, 6(2):40-53, 2008
Riebesell, J., Goodall, R. E., Jain, A., Benner, P., Persson, K. A., and Lee, A. A. Matbench discovery-an evalu- ation framework for machine learning crystal stability prediction. arXiv preprint arXiv:2308. 0aa::3338..444 14920, 2023.
Oppenheimer, M. Zur quantentheorie der molekeln [on the quantum theory of molecules). Annalen der Physik, 389 (20):457-484, 1927.
Owen, J., Torrisi, S. B., Xie, Y., Batzner, S., Bystrom, K., Coulter, J., Musaelian, A., Sun, L., and Kozinsky, B. Complexity of many-body interactions transition metals via machine-learee force fields from the tm23 data set. npj Computational Materials, 10(1):92, 2024.
Röcken, S., Burnet, A. F., and Zavadlav, Predicting solva- tion free energies with an implicit solvent machine learn- ing potential. arXiv preprint arXiv: arXtv:2406.00183 2024.
Sabanes Zariquiey, F,, Galvelis, R., Gallicchio, E., Chodera, J. D., Markland, T. E., and De Fabritüs, G. Enhancing -igein binding affinity predictions using neural network potentials. Journal of Chemical Information and Modeling, 64(5):1481-1488 2024.
Park, Y., Kim, J., Hwang, S., and Han, S. Scalable parallel algorithm for graph neural network interatomic potentials in molecular dynamics simulations. J. Chem. Theory Comput., 20(11):4857-4868 2024. doi: 10. 10.1021/acs.jctc. 4c00190.
Schmidt, J., Cerqueira, T. F., Romero, A. H., Loew, A., Jäger, F., Wang, H.-C., Botti, S., and Marques, M. A. Improving machine-learning models materials science through large datasets. Materials Today Physics, 48: 101560, 2024.
Parr, R. G., Gadre, S. R., and Bartolotti, I.. J. Local density functional theory of atoms and molecules. Proceedings of the National Academy of Sciences, 76(6):2522-2520 1979.
Passaro, S. and Zitnick, C. L. Reducing so (3) convolutions to so (2) for efficient equivariant gnns. International Conference on Machine Learning, pp. 27420-27438. Pro- ceedings of Machine Learning Research, 2023.
Schütt, K., Kindermans, P.-J., Sauceda Felix, H. E., Chmiela, S., Tkatchenko, A., and Müller, K.-R. Schnet: A continuous- convolutional neural network for model- ing quantum interactions. Advances in neural information processing systems, 30, 2017.
Paszke, A., Gross, S., Massa, F. Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019.
Schütt, K., Unke, O., and Gastegger, M. Equivariant mes- sage passing for the prediction of tensorial properties and molecular spectra. In International Conference on Machine Learning, pp. 9377-9388. PMLR, 2021.
12

---

Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Simoncelli, M., Marzari, N., and Mauri, F. Wigner formula- tion of thermal transport in solids. Physical Review X, 12 (4):041011, 2022.
Van De Walle, A. and Ceder, G. The effect of lattice vibra- tions substitutional alloy thermodynamics. Reviews of Modern Physics, 74(1):11, 2002.
Smith, J. S., Isayev, O., and Roitberg, A. E. Ani-l: an extensible neural network potential with dft accuracy at force field computational cost. Chemical science, 8(4): 3192-3200 2017.
Wander, B., Shuaibi, M., Kitchin, J. R., Ulissi, Z. W., and Zitnick, C. L. Cattsunami: Accelerating transition state energy calculations with pre-trained graph neural net- works. arXiv preprint arXiv:2405.02073 2024.
Smith, J. S., Zubatyuk, R., Nebgen, B., Lubbers, N., Barros, K., Roitberg, A. E., Isayev, O., and Tretiak, S. The ani- 1cex and ani-1x data sets, coupled-cluster and density functional theory properties for molecules. Scientific data, 7(1):334, 20200.
Wang, W., Axelrod, S., and Goonez-Bombarllly Differ- entiable molecular simulations for control and learning, arXiv preprint arXiv:2003. 00868, 2020.
Weiler, M., Geiger, M., Welling, M., Boomsma, W., and Cohen, T. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. Advances in Neu- ral Information Processing Systems, 31, 2018.
Thölke, P. and De Fabritiis, G. Equivariant transformers for neural network based molecular potentials. In Interna- fional Conference on Learning Representations, 2021.
Williams, N. J., Kabalan, L., Stojanovic, L.., Zólyomi, V., and Pyzer-Knapp E. O. Hessian qm9: A quantum chem- istry database of molecular hessians in implicit solvents. Scientific Data, 12(1):9, 2025.
Thomas, N., Smidt, T., Kearnes, S., Yang, L., Li, L., Kohlhoff, K., and Riley, P. Tensor field networks: Rotation-and mnnislatioo-eqivariiin neural networks for 3d point clouds. arXiv preprint 1rXiv:1802.06219 2018.
Wines, D. and Choudhary, K. Chips-ff: Evaluating univer- sal machine learning force fields for material properties. arXiv preprint arXiv:2412.105166 2024.
Togo, A., Chaput, L., and Tanaka, I. Distributions of phonon lifetimes in brillouin zones. Phys. Rev. B, 91:094306, Mar 2015. doi: 10.1 10.1103/PhysRevB..o 91094306..
Yang, H., Hu, C., Zhou, Y., Liu, X., Shi, Y., Li, J., Li, G., Chen, Z.,, Chen, S., Zeni, C., et al. Mattersim: A deep learning atomistic model across elements, temperatures and pressures. arXiv preprint arXiv:2405.04990 04967, 2024.
Togo, A., Chaput, I.., Tadano, T., and Tanaka, I. Im- plementation strategies in phonopy and phono3py. J. Phys. Condens. Matter, 35(35):353001, 2023. doi:
10. 14.088/1311-664XXaadB333
Yin, B., Wang. J., Du, W., Wang, P., Ying, P., Jia, H., Zhang, Z., Du, Y., Gomes, C. P., Duan, C., et al. Alphanet: Scaling up local frame-based atomistic foundation model. arXiv preprint arXiv:2501. 05155, 2025.
Tran, R., Lan, J., Shuaibi, M., Wood, B. M., Goyal, S., Das, J., Kolluru, A., Rizvi, A., Shoghi, N., et al. The open catalyst 2022 (oc22) dataset and challenges for oxide clectrocatalysts. ACS Catalysis, 13 (5):3066-3084, 2023.
A.,
Zhang. L.., Han, J., Wang. H., Saidi, W., Car, R., et al. End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems. Advances Neural Information Processing Systems, 31, 2018.
Tuckerman, M. E. Statistical mechanics: theory and molec- ular simulation. Oxford university press, 2023.
Zitnick, L.,, Das, A., Kolluru, A., Lan, J., Shuaibi, M., Sri- ram, A., Ulissi, Z., and Wood, B. Spherical channels for modeling atomic interactions. Advances in Neural Information Processing Systems, 15:8054-8067, 2022.
Unke, O. T. and Meuwly, M. A reactive, scalable, and transferable model for molecular energies from a neu- ral network approach based on local information. The Journal of chemical physics, (48224):241708, 2018.
Zubatyuk, R., Smith, J. S., Leszezynski. J., and Isayev, O. Accurate and transferable multitask prediction of chemical properties with an atoms-in-molecule neural network. Science advances, 5(8)eaav66490, 2019.
Unke, O. T., Chmiela, S., Gastegger, M.. Schütt, K. T., Sauceda, H. E., and Müller, K.-R. Spookynet: Learn- ing force fields with electronic degrees of freedom and nonlocal effects. Nature communications, 12(1):1-14, 2021a.
Unke, O. T., Chmiela, S., Sauceda, H. E., Gastegger, M., Poltavsky, I., Sch utt, K. T., Tkatchenko, A., and Müller, K.-R. Machine learning force fields. Chemical Reviews, 121(16)10142-1018 2021b.
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
A. Experimental details
A.2. Phonon calculation protocols
A.1. MD simulation protocol
Harmonic and anharmonic phonon calculations and solu- tions to the Wigner transport equation (Simoncelli et al., 2022) used for the thermal conductivity benchmark (ksrme) values given in Table 2 were carried out using the supercell method with finite differences implemented in PHONO0PP (Togo et al., 2015; 2023). The calculations followed the protocol described in the Matbench- Matbench-Discovery bench- mark (Riebesell et al., 2023; Póta et al., 2024). For eSEN- 30M, an even lower KSRME of 0.298 is obtained when we adjust the evaluation parameter atom displacement from 0.03 Å to 0.05 A.
Simulating molccular systems not seen during training is a key capability of MLIPs. Therefore, we construct chal- lenging MD simulation tasks featuring out-of-distribtiio data to test model's conservation capability. We conduct experiments on two chemical domains: inorganic crystals and organic molecules.
Inorganic materials. For training models on inorganic ma- terials we utilize the MPTrj (Deng et al., 2023) dataset, and we establish a suite of simulation tasks based on the TM23 dataset (Owen et al., 2024). TM23 contains MD samples of 27 single vacancy defect transition metal systems at cold, warm, and melt temperatures using an NVT ensemble, in total& 81 combinations of different metals and temperature. These defected systems are out-of-distributtin for a model trained on the MPTrj datasets, which only contains relax- ation trajectories of non-defected systems. Additionally, some of the metals in the TM23 dataset is very rare in the MPTrj dataset. We initialize the simulation by sampling frame from the TM23 dataset, run a relaxation using the LBFGS algorithm, randomly initialize the atom velocities at cold/warm/melt temperatures using a Maxwell-Boltznan distribution, then run MD simulations under the NVE en- semble for 100 ps using a time step of fs (same as the time step used in the TM23 ab initio MD protocol).
Harmonic phonon calculations for the MDR benchmark re- sults listed in Section 6.2 were carried out following the calculation protocol used by (Loew et al., 2024) which em- ploys phonon calculations using the supercell method with finite differences with displacement of 0.01 A. Calcula- tions were done using the PHIONOPY software (Togo et al., 2023).
A.3. Test-set error for MP'rrj-trained models
Since MPTrj lacks an official test split and various models are typically trained on distinct subsets of the data, we ran- domly selected 5000 samples from the subsampled Alexan- dria (sAlex) dataset (Schmidt et al., 2024; Barroso-Luque et al., 2024) for a fair comparison. This subset was used to calculate the test-set energy mean absolute errors (MAEs) presented in Figures 1, 6, B.7, and B.8.
Organic molecules. For training models on organic molecules we use the SPICE-MACE-OFF dataset (Kovács et al., 2023), which is mainly based on the SPICE-1.0 dataset (Fastman et al., 2023), and we establish a suite of simulation tasks from the MD22 dataset (Chmiela et al., 2023) that contains seven large molecules. Molecules in MD22 are -iitr-idistiitttinn for a model trained on the SPICE-MACE-OFF dataset as they are considerably larger than all molecules in the SPICE-MACE-OFF training dataset. We initialize the simulation by sampling a frame from the MD22 dataset, run a relaxation using the LBFGS algorithm, randomly initialize the atom velocities at a tem- peratures of 400/500 using a Maxwell-Boltee Boltzmann dis- tribution (400 K for Buckyball catcher and Double-walled nanotube and 500 K for other molecules, which are the same as the MD22 protocol), then run MD simulations under the NVE ensemble for 100 ps using time step of fs (same as the time step used in the MD22 ab initio MD protocol).
In the ablation study reported in Table 1, we use a random test split of the MPTrj dataset. We find the performance of eSEN to be highly stable with different random seeds. We trained 2-layer eSENmodels on the MPTrj dataset using 3 different seeds for 50 epochs, with loss coefficients the same loss coefficients of models in Table 1. The validation set errors and standard deviations are 19.67 = 0.23 for energy (me V/tomm, 43.85 = 0.058 for forces (me and 0.16 + 0.00038 for stress V/(3. atom)).
B. Phonon calculations and vibrational properties
B.1. Correlation of test-set energy errors and vibrational property errors
Figure B.7 presents the correlation between test-set energy MAE and the other three phonon calculation tasks, evalu- ated across various model architectures. The corresponding correlations for different variants of eSEN are displayed in Figure B.8. In both figures, improved correlation can be observed among
All ML-based MD simulations use a Velocity- Verlet inte- grator and are conducted with ASE (Larsen et al., 2017). We measure the energy conservation error (extent of energy drift) across the 100-ps simulations. All eSEN models are 2-layer with Imax = 2 and Mnax = 2 (3.2M trainable parameters). Detailed model hyperparameters are included in Appendix D.
models.
Figure B.8 illustrates that failing the conservation test can
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Max Max frequency MAE (I)
Free energy MAE (k//mol)
Heat Capacity (JK/mmll)
250
50
150
eSEN (Ours)
SevenNet-IBi5
200
40
SevenNet-0
100
GRACE-2L (r6)
75
MACE-MP-O
eqV2 DeNS
Orb MPTrj
50
28
CHGNet
60
Energy MAE (meV/atom)
Energy MAE (meV/atom)
Energy MAE (meV/atom)
Figure B.7. The correlation between sett-sct energy error and maximum frequency, free energy and heat capacity across different model architectures.
35
Max frequency MAE (K)
Free energy MAE (k//mo))
Heat Capacity (IK(mmol)
30
eSEN-30M
-SEN-6.5M
100
25
eSEN-3.2M
eSEN, Bessel
80
20
eSEN, direct-force
eSEN, discrete
60
15
eSEN, neighbor limit
20
10
eSEN, no envelope
eSEN, Nows 5
50
60
70
40
50
60
70
40
50
60
Energy MAE (meV/atom)
Energy MAE (meV/atom)
Energy MAE (meV/atom)
Figure B.8. The correlation hetween test-set energy error and maximum frequency, free energy and heat capacity for different variants of eSEN.
result in varying degrees of impact on different properties, depending on the specific design choices made. Although the neighbor limit, envelope function, and number basis functions substantially affect AşrMe. (see Figure 6), their influence on the properties evaluated in the MDR Phonon benchmark is relatively minor. Representation discretization impacts vibrational entropy and heat capacity but not other properties. While model might still be able to get good performance in some physical property task when the energy conservation test is failed, when the conservation test is passed, the model performs very well robustly across all metries.
benchmark MAE values for eq S DeNS become compa- rable to those of conservative force models (Section 6.2).
While the prediction accuracy of thermodynamic properties such as maximum frequency, entropy, free energy, and heat capacity improves with increasing displacement for direct- force models, this improvement deceptive and does not translate to accurate predictions of the underlying phonon band structure and density of states (DOS). As illustrated in Figure C.11, the predicted phonon bands and DOS for three selected materials exhibit significant errors, particu- larly in capturing the correct dispersion relations. Moreover, imaginary frequencies are commonly predicted small dis- placement values, suggesting rough energy landscape (i.e. the learned PES is not truly convex when it's very close to the minima). The eq S DeNS model also fails to accu- rately capture acoustic modes-thone that go to zero linearly at the structure. In contrast, non-zero net force at energy local min- ima does not occur for conservative models by definition (Figure 5).
B.2. Displacement values and their relation to phonon band structure predictions
Figure B.9 presents the MAEs on phonon calculations for eSEN, MACE, and V2 S DeNS function of increasing displacement values. As anticipated, both eSEN and MACE exhibit constant or slightly increasing MAE with respect to displacement. In contrast, eqV2 S DeNS displays notable decrease in MAE with increasing displacement. Notably, when using a displacement of 0.2 A. the resulting phonon
which is due to non- zero net force on the
By enforcing net zero force prediction, as proposed by Neumann et al. 2024, direct-force models can be modified
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Max frequency (K)
Entropy U///mol)
Heat capacity (kj/mol)
Free energy (J/K/ool)
100
250
eSEN
eqV2 DeNS
200
eqV2 S DeNS
eSEN
50
<SEN
eqV2 DeNS
80
eqV2 DeNS
eSEN
MACE-MP-0
150
MACE-MP-0
40
MACE-MP-0
MACE-MP-O
MAE
150
30
100
100
20
50
20
50
10
0.01
0.04
0.08
0.12
0.16
0.20
0.01
0.04
0.08
0.12
0.16
0.20
0.01
0.04
0.08
0.12
0.16
0.20
0.01
0.04
0.08
0.12
0.16
0.20
Atom displacement
Atom displacement
Atom displacement
Atom displacement
Figure B.9. Error in a randomly sampled subset (1000 samples) of the MDR Phonon benchmark when the atom displacement is adjusted.
to accurately capture acoustic phonon modes. As demon- strated in Figure C.12, incorporating this constraint allows the model to predict acoustic modes correctly. However, despite this improvement, the model continues to struggle with accurately reproducing the phonon band structure, and the issue of predicting imaginary frequencies at small dis- placement values persists.
convergence at larger displacements and the absence of acoustic modes, these issues are less pronounced compared to eqV2 S DeNS. Moreover, the predicted phonon bands and DOS are significantly improved, providing more accu- rate representation of the DFT reference values. The better approximation of phonon bands and a lower tendency to predict imaginary frequencies highlight the importance of a smoothly-varying model, even without being conservative.
The apparent paradox of direct-force models like eq S DeNS failing accurately capture phonon band structures while still achieving competitive accuracy for thermody- namic properties such as entropy, free energy, and heat capacity can be resolved by examining the underlying cal- culation methodology. These properties are computed using weighted integrals of the DOS. Additionally the metries in the MDR performed room temperature (300 K). The predicted DOS in Figure C. and Figure C.12 at larger dis- placement values adequately captures the overall features of the DFT DOS, but does not reproduce finer details such as high density areas and fluctuations. This level of agreement sufficient for accurate predictions because the Boltzmann- weighted integrals used in calculating thermal properties help to mitigate the impact of point-wiss errors, making it less crucial to precisely capture fine details in the band struc- ture and DOS (Ackland et al., 1997; Van De Walle & Ceder, 2002). Moreover, since properties are estimated at 300 K, models can achieve accurate predictions by prioritizing prediction accuracy of lower-frequency modes, which are more relevant for thermal property calculations, rather than attempting capture higher-frequency modes. Although v2 S DeNS without net-zero force contraint may not accurately capture acoustic phonon branches, its ability to predict vibrational thermodynamics at room temperature remains unaffected due to the relatively small number of acoustic phonon states at low frequencies.
Extending the existing metrics proposed by Loew et al. 2024 with additional evaluations would provide a more compre- hensive assessment of MI.IP performance. Specifically, new metries could be developed to assess phonon dispersion across all modes and frequencies at commensurate points; and computing vibrational thermodynamic properties at a range of temperatures.
eSEN-3.2K 2M
eSEN 6.5M
0.5
MACE-OFF-L (4.7M)
Millions steps per
0.3
0.1
216
1000 Number of atoms
512
1728
2774
Figure 10. Inference efficiency of MACE-OFF-L and eSENs of similar scale.
C. Inference efficiency
We benchmarked the inference speed of our models against the similar sized MACE-OFF-L (Kováes et al., 2023) (4.7M) single 80GB Nvidia A-100 GPU. For MACE-OFF-L we used the exact benchmark code found in https: main
As comparison with V2 DeNS, Figures Figure C.13 and Figure 14 shows the predicted phonon dispersion and DOS for the same three materials using eSEN with direct- force prediction. Although the results still exhibit some of the characteristic artifacts of direct-force models, such
wwwpeoo///ttteeaaaeeeeeeee /blob/ py with mace-torch v0.3.6 (PyPi). To create a fair comparison, we replicated the identical benchmark environment as MACE benchmarks
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
using the same diamond system with variable number of supercells (carbon atoms) input. All models are bench- marked using the standard Python (3.12) runtime with Py- torch v2.4.0 (Paszke et al., 2019) and CUDA 12.1 (Nickolls et al., 2008). No compilettocchsrrit was used for standard- ization of runtime. Across all system sizes, cSEN-3.2M has comparable inference efficiency to MACE-OFF-LE For 216 atoms (Figure B.1 our models (3.2M, 6.4M) can run approximately (0.4, 0.8) million steps per day comparable to MACE-OFF-L (0.7 million steps per day).
D. Hyper-parameters
Hyper-parameters used for model training are shown in Table 6 We train all models using per-atom energy MAE loss, force lz loss, and stress MAE loss. For direct- force models direct-force pre-training, decomposed loss as described in Barroso-Luque et al. 2024. The eSEN-30M-MP model trained on MPTrj uses Denoising Non-equilibrium Structures (DeNS) (Liao et al., 2024) with a noising probablity of 0.3, a standard deviation 0.1 À for the added Gaussian noise, and 10 for the DeNS loss coefficient during direct-force pre-training. DeNS is not used during conservative fine-tuning. In our ablation study for maximum neighbor limit we used 30 as the limit. The main effect of DeNS is reducing overfitting in the pretraining stage. Such benefit is found to carry over to the conservative finetuning stage. The eSEN-30M-OMat and the OAM Fine- tuned models do not use DeNS.
the same
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
Frequency (TH2)
Frequency (1Hz)
20
15
3.00
10
(States/THz)
(States/THz))
(States/THz)
Figure C.II. Predicted phonon band structure and density of states (DOS) of Si (diamond structure). CsCI (CsCI structure), AIN (wurtzite structure) using eqV2 S DeNS (direct-fore prediction) at different displacement values. DFT baseline taken from the PBE MDR dataset (Loew al., 2024) calculated using displacement of 0.01 Á
25
Frequency (TH2)
Frequency (THz)
20
Frequency
25
10
DOS (States/THHr)
(States/THz))
(States/THz)
Figure C2 Predicted phonon hand structure and density of states (DOS) of Si (diamond structure), CsCI (CsCI structure), AIN (wurtzite structure) using vqV2 S DeNS (direct-fore prediction with zero net force constraint) at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew al., 2024) calculated using a displacement 0.01
Si
CsCi
AIN
Frequency (TH2)
Prequency (1Hz)
Frequency (1H2)
20
10
15
10
UK
X
DOS
10
XR
M
DO5 (States/THz)
10
K
AL MH K
(States/THz)
DOS
(States/THz)
Figure C.1.. Predicted phonon hand structure and density of states (DOS) of Si (diamond structure), CsCI (CsCl structure), AIN (wurtzite structure) using eSEN with direct-force prediction at different displacement values. DFT baseline taken from MDR dataset (Loew et al., 2024) calculated using displacement of 0.01
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Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
CsCl
AIN
17.5
15.0
25
Frequency TR2
Frequency (THzl
Frequency (THZ)
20
10.0
7.5
155
3.02
10
2.5
0.0
M
MH
(States/THz))
(States/THz)
(States/THz)
Figure C. 14. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCI (CsCI structure), AIN (wurtzite structure) using cSEN with direct-force prediction and a zero-net net force constraint different displacement values, DFT baseline taken from the PBE MDR datasct (Loew al., 2024) calculated using displacement of 0.01 Á
Table 6. Hyper-parameters parameters for eSEN variants reported in this paper. -SEN-30M on MPTrj was trained for 60 epochs using direct- force pre-training and epochs of conserved fine-tuning. eSEN-30M-OMat was trained for epochs using direct-force pre-training and epochs of conserved fine-tuning. 'The eSEN-30M-OAM model starts from the eSEN-30M-OMa model, and was finetuned for epoch on dataset constructed by combining the sAlex training datasct and copics of the MPTr training dataset.
Hyper parameters
SPICE-3.2M
SPICB66.M
MPTrj-3.2
MPTrj-6.5M
MPTT--UM
30M
OAM Fine Fine-tuning
Number of layer blocks
10
10
Maximum degree Lenx
3
3
Maximum order Manc
2
2
Number of channels Nehammll
128
128
128
128
128
128
128
Radial busis function
Bessel
Bessel
Gaussian
Gaussian
Gaussian
Gaussian
Gaussian
Number of radial basis functions
10
64
Cutoff radius (A)
Batch
128
128
$12
512
512
256
Optimizer
AdamW
Adamm
AdamW Cosine
AdamW
Adam W
Adam Cosine
Adam
Learning rate scheduling
Cosine
Cosine
Cosine
Cosine
Cosine
Warnup epochs
0.1
0.01
0.1
0.1
0.1
0.1
0.1
Warmup factor
0.2
0.2
0.2
0.2
0.2
0.2
Maximum learning rate
10-1
10-1
10-1
10-4
10-1
x 10-4
x 10-1
Number of epochs
100
100
100
100
60 10"
2-22"
Gradient clipping
100
100
100
100
100
100
100
Model EMA decay
0.999
0.999
0.999
0.999
0.999
0.999
0.989
Weight docay
x 10-3
1 10-3
x 10-3
x 10-3
x 10-3
x 10-3
1 x 10-3
Energy loss coefficient
10
20
20
20
coefficient
20
20
Stress loss coefficient
100
100
5
5
19


Reviews:
[{"review_id": "0eFUflZHNy", "reviewer": "Reviewer_XYRs", "summary": "This paper presents eSEN, a machine learning interatomic potential (MLIP) model designed for accurate and energy-conserving molecular dynamics (MD) simulations and physical property predictions. The study identifies key factors that impact an MLIP’s ability to generalize well to physical property prediction tasks, such as ensuring conservative force predictions and maintaining a smoothly varying potential energy surface (PES). The proposed eSEN model achieves state-of-the-art results on a range of benchmarks, including materials stability prediction, thermal conductivity prediction, and phonon calculations. By establishing a correlation between test errors and physical property prediction performance in energy-conserving models, the authors offer insights into improving the reliability of MLIPs.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer XYRs for the helpful feedback. We address each of the reviewer’s comments below.\n\n> The design choices of the eSEN model are all from existing works.\n\nRegarding the originality of our technical contributions, we would like to highlight that while energy-conservation-related design choices have been explored in previous studies, our work systematically investigates the impact of each individual design choice. This, to our knowledge, has not been comprehensively addressed before. Our paper's originality lies in (1) elucidating these effects; (2) demonstrating state-of-the-art results when these choices are integrated with the novel eSEN architecture and the direct-force pretraining strategy; (3) we present a novel finding on the correlation between test-set error and physical property prediction performance for models that satisfy the conservation test; (4) we propose the MD conservation test that is critical in establishing the correlation between test errors and downstream predictions. \n\nWe believe these findings represent a significant contribution to the community and that they will impact model development practices in the field moving forward.\n\n> The authors do not provide an empirical study from the perspective of efficiency.\n\nWe would like to respectfully point out that we provided a brief empirical study on the efficiency of eSEN in Appendix C. We found eSEN to have comparable efficiency to existing equivariant models while being more accurate. A 2-layer eSEN model with 3.2M parameters can simulate around 0.8 million steps per day for a periodic system of 216 atoms on a single NVIDIA A100 GPU. We will highlight this result in the main paper in the next revision.\n\n> How is the proposed eSEN evaluated in Table 2? Is it submitted to the Matbench Leaderboard (https://matbench-discovery.materialsproject.org/) for evaluation?\n\n\nWe confirm that eSEN was evaluated using the same dataset and metrics as those provided in the Matbench Leaderboard, as referenced in the link you shared. The results have been submitted and verified by the benchmark maintainer, ensuring their accuracy and reliability. The slight difference in the $\\kappa_{\\mathrm{SRME}}$ metric at the benchmark site is due to a very recent minor update to the Matbench-Discovery evaluation protocol.\n\nWhile the results of eSEN are on the Matbench-Discovery Leaderboard, we have refrained from directly linking our submission to maintain the integrity of the double-blind review process. We appreciate your understanding in this matter.\n\n\nWe look forward to further discussions if you have additional questions or suggestions. Thank you again for your valuable input.", "claims_and_evidence": "The paper claims that energy conservation in MD simulations leads to improved correlation between test errors and downstream physical property predictions. This claim is well-supported by experimental results.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria align well with the problem of developing reliable MLIPs. The authors test their models on a representative range of material property prediction benchmarks including Matbench-Discovery, MDR Phonon benchmark, and SPICE-MACE-OFF.", "theoretical_claims": "The theoretical claim of the submission refers to Hairer et al. 2003.", "experimental_designs_or_analyses": "The experimental design is well-structured, with clear comparisons between eSEN and existing MLIPs on a representative set of benchmarks. The ablation studies systematically evaluate the impact of various architectural decisions, such as representation discretization, neighbor selection, and envelope functions. The results support the paper’s hypotheses, with energy-conserving models consistently outperforming non-conservative alternatives. However, further exploration of the impact of different hyperparameter choices could strengthen the robustness of these conclusions.", "relation_to_broader_scientific_literature": "In the field of MLIP, there has been a debate about training conservative (physical but computationally expensive) or non-conservative force fields (computationally efficient). This submission provides great evidence for conservative force fields from the perspective of generalization performance, i.e., the test-set energy error of conservative force fields correlates better with other physical properties.", "essential_references_not_discussed": "The paper provides a comprehensive literature review.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The experiments in this paper are comprehensive. \n\n2. The proposed eSEN achieves SOTA results on multiple benchmarks, demonstrating strong empirical performance.\n\nWeaknesses:\n\n1. The design choices of the eSEN model are all from existing works. \n\n2. The authors do not provide an empirical study from the perspective of efficiency.", "other_comments_or_suggestions": "1. I would suggest the authors provide some discussion about computational efficiency.", "questions_for_authors": "1. How is the proposed eSEN evaluated in Table 2? Is it submitted to the Matbench Leaderboard (https://matbench-discovery.materialsproject.org/) for evaluation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CC5o8lUsFR", "reviewer": "Reviewer_kXc9", "summary": "This work investigates failure cases of machine learning interatomic potentials (MLIPs) in actual MD simulations. From these insights, the authors draw actionable improvements to MLIP that they implement in their eSEN model. eSEN shows promise in being more accurate on hold-out test sets as well as in preserving energy in MD simulations. This work questions many common design choices that led to reduced test set MAEs but unstable MD simulations.\n\n## After the rebuttal\nI stay with my initial judgment. I find this work a pleasant read and well executed. I recommend acceptance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer kXc9 for the helpful feedback. We address each of the reviewer’s comments below.\n\n> Its main downside is the limited originality in its technical and theoretical contribution. \n\nRegarding the originality of our technical and theoretical contributions, we would like to highlight that while energy-conservation-related design choices have been explored in previous studies, our work systematically investigates the impact of each individual design choice. This, to our knowledge, has not been comprehensively addressed before. Our paper's originality lies in (1) elucidating these effects; (2) demonstrating state-of-the-art results when these choices are integrated with the novel eSEN architecture and the direct-force pretraining strategy; (3) we present a novel finding on the correlation between test-set error and physical property prediction performance for models that satisfy the conservation test; (4) we propose the MD conservation test that is critical in establishing the correlation between test errors and downstream predictions. \n\nWe believe these findings represent a significant contribution to the community and that they will impact model development practices in the field moving forward.\n\n> Further, error bars for different trainings would greatly help in indicating the stability of results.\n\nWe appreciate your suggestion regarding error bars for different training runs. To quantify the variation, we trained 2-layer eSEN models on the MPTrj dataset using 3 different seeds for 50 epochs, with loss coefficients E: 1/F: 10/S: 100. The validation set errors and standard deviations are as below:\n| Task | MAE |\n| -- | -- |\n| Energy (meV/atom)      | 19.67 ± 0.23        |\n| Forces (meV/Å)         | 43.85 ± 0.058       |\n| Stress (meV/Å³ atom)   | 0.16 ± 0.00038      |\n\nWe find the results to be highly stable across random seeds. We will incorporate this in our revised manuscript.\n\nWe look forward to further discussions if you have additional questions or suggestions. Thank you again for your valuable input.", "claims_and_evidence": "The authors claim that MLIP should be\n1. conservative vector fields.\n2. bounded in their derivatives.\n3. smooth.\n\nThe authors support these claims very well with the implementation of MD simulators, various ablation studies, and empirical evidence.", "methods_and_evaluation_criteria": "The authors offer a broad range of benchmark datasets and include various metrics beyond simple MAE on energies and force to accurately paint a picture. Further, MD simulations are performed with each model to judge its practical usefulness.", "theoretical_claims": "The paper makes no theoretical claims.", "experimental_designs_or_analyses": "The experiments are sound and well-analyzed, with meaningful conclusions. However, statistical fluctuations would, by repeating experiments multiple times, benefit communication.", "relation_to_broader_scientific_literature": "This work finds that common choices for improving hold-out test set accuracies in machine-learning force fields lead to unphysical behavior in MD simulations. Further, the authors provide clear and concrete guidelines on what properties MLFF should fulfill to yield accurate MD simulations and low test set errors. This work enriches the literature with a fresh evaluation metrics through MD simulation that also correlate well with existing metrics.", "essential_references_not_discussed": "-", "other_strengths_and_weaknesses": "The paper is very well written, thoroughly investigates failure cases of modern force fields, and draws reasonable conclusions. I enjoyed reviewing this work and am confident that it greatly aids the current field of molecular force fields and the debate on the impact of Euclidean symmetries.\n\nIts main downside is the limited originality in its technical and theoretical contribution. Further, error bars for different trainings would greatly help in indicating the stability of results.", "other_comments_or_suggestions": "-", "questions_for_authors": "-", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qi9nR1vf93", "reviewer": "Reviewer_8LdR", "summary": "This paper draws attention to the inability of energy conservation, and thereby instability of simulation, common in many popular machine learning interatomic potentials (MLIPs). Next, it proposes a novel architecture addressign this problem, while showing state-of-the-art performance on a wide range of tasks.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer 8LdR for the helpful feedback. We address each of the reviewer’s comments below.\n\n> Can the claims made in Section 5 be supported by theoretical arguments?\n\nWe refer to (1) Hairer et al. 2003 for theoretical arguments on the relationship between potential energy surface (PES) smoothness/bounds on derivatives and energy conservation in simulations; and (2) Molecular simulation textbooks (e.g., Tuckerman, 2023) for theoretical arguments on conservative forces. Section 5 is then organized to discuss practical implementations of machine learning potentials that impact conservative forces, smoothness and bounds on derivatives. Some arguments on the design choices can be made from a theoretical perspective:\n\n- **Direct-force:** a direct-force model the direct-force prediction framework imposes no constraint on the output forces being a conservative force field. Using the derivative of the energy with respect to the atomic positions is a requirement for ensuring the predicted forces are conservative.\n- **Representation discretization:** The discretization of spherical harmonics representations breaks the conservative forces requirement because it introduces discretization errors to the computation of energy gradients. Increasing the grid resolution theoretically reduces such discretization errors and helps conservation. \n- **Max neighbor limit:** From a theoretical perspective, we can show examples where the K-NN graph introduces PES discontinuity. Consider a model with a cutoff of 6 A, and a node having 3 neighbors at distance 3 A and a fourth neighbor at (3 + epsilon) A. a small perturbation to the atomic positions will introduce discontinue change in the predicted energy, if a max neighbor limit of 3 is enforced. \n- **Envelope functions:** From a theoretical perspective, the radial basis functions used in graph neural network machine learning potentials are not twice continuously differentiable due to the finite radius cutoff in graph construction, which causes a step change at the cutoff radius. The envelope function theoretically eliminates this issue.\n- **Number of radial basis functions:** While empirically reducing the number of radial basis functions helps the PES to vary smoothly and improves model conservation properties, due to the flexibility of neural networks, it does not theoretically enforce these properties. \n\n\nWe look forward to further discussions if you have additional questions or suggestions. Thank you again for your valuable input.", "claims_and_evidence": "Yes. The claims were all well supported experimentally. The discussion around the force conservation is particularly exhilarating and well supported.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are very reasonable. Apart from the conservation errors, the ensemble property experiments proposed in this paper also throw new light on the discussion around the desired properties of MLIP.", "theoretical_claims": "NA. There is little theoretical claims.", "experimental_designs_or_analyses": "The experiments are designed properly and convincingly. In particular, they refreshingly used larger, more realistic datasets rather than the toy datasets the field has been using.", "relation_to_broader_scientific_literature": "The relevant papers have been properly cited and discussed.", "essential_references_not_discussed": "NA.", "other_strengths_and_weaknesses": "NA.", "other_comments_or_suggestions": "NA.", "questions_for_authors": "Can the claims made in Section 5 be supported by theoretical arguments?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uIuns0yuuV", "reviewer": "Reviewer_yR9i", "summary": "This paper argues that the test MAE, when energy conservation is guaranteed in MD simulations, demonstrates the practicality of machine learning potentials. The authors provide empirical evidence indicating that, within established model designs, specific designs uphold energy conservation principles while others do not. Furthermore, they demonstrate a correlation between the Energy/Force MAE on the test dataset and the predictive performance of physical properties utilizing MD simulations, specifically for those models that maintain energy conservation. The resulting model, eSEN, achieves state-of-the-art results across various physical property prediction tasks based on phonon calculations.\n\n## update after rebuttal\nThank you for the author's responses. This paper is valuable to share with the community, and I support its publication.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer yR9i for the helpful feedback. We address each of the reviewer’s comments below.\n\n> It should be noted somewhere in the text that the physical properties considered in this study are limited to those requiring higher-order derivatives of the PES and that the applications of machine learning potentials have not been examined comprehensively.\n\nThank you for your suggestion. While other properties such as formation energy in the Matbench-Discovery benchmark were studied in the paper, the properties that require higher-order derivatives of the PES are indeed more significantly impacted by the MD-energy-conservation properties. We will note this in the revised manuscript.\n\n> I am wondering about the contribution of DeNS to thermodynamics property predictions, and I would like to see DeNS's ablation study.\n\nOnly conservative models are well-suited for thermodynamics property prediction tasks, which require accurate modeling of higher-order PES derivatives. DeNS is only used during direct pre-training on the MPTrj dataset to alleviate overfitting. Conservative models do not use DeNS during conservative training. We present an ablation study over two 2-layer direct-force eSEN models (with loss coefficients E: 1/F: 10/S: 100), with and without DeNS:\n\n| Metric                          | With DeNS | Without DeNS |\n|-----------------------------------|-----------|--------------|\n| Energy MAE (meV/atom)         | 18.0      | 19.4         |\n| Force MAE (meV/Å)             | 43.7      | 43.7         |\n| Stress MAE (meV/Å³ atom)      | 0.14      | 0.16         |\n\nThe higher error of the model without DeNS is due to overfitting. We will incorporate the validation error curves in the revised manuscript to reflect that. \n\nWe look forward to further discussions if you have additional questions or suggestions. Thank you again for your valuable input.", "claims_and_evidence": "It has been experimentally demonstrated that the energy conservation law, which is a property that the potential must satisfy, affects the estimation accuracy of physical quantities that require phonon calculations.\nThe model design, which is well-known and of interest to readers, has been experimentally verified and supported by evidence. However, it should be noted somewhere in the text that the physical properties considered in this study are limited to those requiring higher-order derivatives of the PES and that the applications of machine learning potentials have not been examined comprehensively.", "methods_and_evaluation_criteria": "It is reasonable to evaluate the performance of a potential that satisfies the energy conservation law using physical properties that require differentiation.", "theoretical_claims": "The results are empirical, so there is no theoretical claim.\n\nAdditionally, I have briefly reviewed Hairer et al. 2003 and believe that it aligns with the claims in Section 3.2. However, in my understanding, Section 3.2 does not present any novel theoretical claims.", "experimental_designs_or_analyses": "I have checked the NVE MD simulations to verify the energy conservation law, as well as the experimental settings for Matbench Discovery and the MDR phonon benchmark. There are no issues with these experimental settings.", "relation_to_broader_scientific_literature": "The machine learning community has proposed designs that differ from conventional potential designs. However, some of these machine learning potentials, despite having good MAE for energy and forces, are not practical for real-world use, and it has been repeatedly pointed out that Energy/Force MAE does not necessarily indicate the practicality of machine learning potentials. \nIssues with the practicality of these potentials can be observed during actual simulations, such as structural failures, but a simple and broadly applicable method to identify these problems was not previously known.", "essential_references_not_discussed": "I do not find any issue with the references.", "other_strengths_and_weaknesses": "Strengths\n- Conservative fine-tuning could be a popular method in the community since it speeds up the training without sacrificing energy conservation.", "other_comments_or_suggestions": "I am wondering about the contribution of DeNS to thermodynamics property predictions, and I would like to see DeNS's ablation study.", "questions_for_authors": "No question.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The submission kicks off a serious endeavor to push forward the known but insufficiently attended issue of the quality of MLIP for practical downstream utilities, e.g., MD simulation. The authors noticed that for energy conservation, it is not sufficient to only modeling a gradient field, but also requires smooth operations in an MLIP model, such as discretization of spherical harmonics representations, maximum neighbor limit, and envelope functions. The work conducted a comprehensive study on the effect of these design choices under the consideration of stable MD simulation and downstream physical property prediction. The eSEN model is then proposed to fulfill the requirements based on the insights, which surpasses competitive models on physical property prediction in public benchmarks. Overall, the submission makes a timely and informative contribution to the community, in terms of the emphasis on the problem, approaches to investigate it, observations from the systematic empirical study, and the resulting model useful for challenging downstream tasks.

All reviewers expressed the appreciation of the contribution, and rated quite supportive scores. Some reviewers brought up further queries (empirical efficiency study by Reviewer XYRs, statistical significance by Reviewer kXc9, effect of DeNS by Reviewer yR9i). The authors replied to the questions and queries responsibly.
Decision: Accept (oral)