Machine Learning Force Fields with Data Cost Aware Training
Authors: Alexander Bukharin
Tianyi Liu
Shengjie Wang
Simiao Zuo
Weihao Gao
Wen Yan
Tuo Zhao
Venue: Submitted to ICLR 2023 (2023)
Keywords: Machine Learning Force Fields
Data-Cost Aware Training
AI for Science

Abstract: Machine learning force fields (MLFF) have been proposed to accelerate molecular dynamics (MD) simulation, which finds widespread applications in chemistry and biomedical research. Even for the most data-efficient MLFF models, reaching chemical accuracy can require hundreds of frames of force and energy labels generated by expensive quantum mechanical algorithms, which may scale as $O(n^3)$ to $O(n^7)$, with $n$ being the number of basis functions used and typically proportional to the number of atoms.
To address this issue, we propose a multi-stage computational framework -- ASTEROID, which enjoys low training data generation cost without significantly sacrificing MLFFs' accuracy. Specifically, ASTEROID leverages a combination of both large cheap inaccurate data and small expensive accurate data. The motivation behind ASTEROID is that inaccurate data, though incurring large bias, can help capture the sophisticated structures of the underlying force field. Therefore, we first train a MLFF model on a large amount of inaccurate training data, employing a bias-aware loss function to prevent the model from overfitting the potential bias of the inaccurate training data. We then fine-tune the obtained model using a small amount of accurate training data, which preserves the knowledge learned from the inaccurate training data while significantly improving the model's accuracy. Moreover, we propose a variant of ASTEROID based on score matching for the setting where the inaccurate training data are unlabelled. Extensive experiments on MD simulation datasets show that ASTEROID can significantly reduce data generation costs while improving the accuracy of MLFFs.

Under review as a conference paper at ICLR 2023
MACHINE LEARNING FORCE FIELDS WITH DATA COST AWARE TRAINING
Anonymous authors Paper under double-blind review
ABSTRACT
Machine learning force fields (MLFF) have been proposed accelerate molecu- lar dynamics (MD) simulation, which finds widespread applications in chemistry and biomedical research. Even for the most data-efficient MLLF models, reaching chemical accuracy can require hundreds of frames of force and energy labels gen- erated by expensive quantum mechanical algorithms, which may scale as O(n)) Onn)), with n being the number of basis functions used and typically proportional to the number of atoms. To address this issue, we propose multi-stage com- putational framework ASTEROID, which enjoys low training data generation cost without significantly sacrificing MLFFFs accuracy. Specifically, ASTEROID leverages combination of both large cheap inaccurate data and small expensive accurate data. The motivation behind ASTEROID is that inaccurate data, though incurring large bias, can help capture the sophisticated structures of the underlying force field. Therefore, we first train a MLFF model on a large amount of inaccu- rate training data, employing bias-aware loss function to prevent the model from overfitting the potential bias of the inaccurate training data. We then fine-tune the obtained model using a small amount of accurate training data, which preserves the knowledge learned from the inaccurate training data while significantly im- proving the model's accuracy. Moreover, we propose a variant of ASTEROID based on score matching for the setting where the inaccurate training data are unlabelled. Extensive experiments on MD simulation datasets show that ASTER- OID can significantly reduce data generation costs while improving the accuracy MLFFs.
INTRODUCTION
Molecular dynamics (MD) simulation is a key technology driving scientific discovery in fields such as chemistry, biophysics, and materials science (Alder & Wainwright, 1960, McCammon et al.) 1977j. By simulating the dynamics molecules, important macro statisties such as the folding probability of a protein [Tuckerman 2010) the density of new materials (Varshney et al., 2008 can be estimated. These macro statistics are an essential part of many important applications such structure-driven drug design (Hospital al., 2015) and battery development (Leung & Budzien, 2010). Most MD simulation techniques share a common iterative structure: MD simulations caleu- late the forces on each atom in the molecule, and use these forces move the molecule forward to the next state.
The fundamental challenge of MD simulation is how efficiently calculate the forces each it- eration. An exact calculation requires solving the Schrödinger equation, which is not feasible for many-body systems (Berezin Shubin 2012). Instead approximation methods such as the Lennard- Jones potential (Johnson et al. 1993), Density Functional Theory (DFT, Kohn (2019)) or Coupled Cluster Sogll--Doubl-Triipe
(CCSD(T), Scuseria et al. (1988)) are used. CCSD(T) is seen as the gold-standard for force calculation, but is computationally expensive. In particular, CCSD(T) has complexity O(n)) with respect to the number basis function used along with a huge storage requirement (Chen et al. 2020). To accelerate MD simulation while maintaining high accuracy, machine learning based force fields have been proposed. These machine learning models take a molecular configuration models have been successful, producing force fields with moderate accuracy while drastically re- ducing computation time (Chmiela al. 2017)
input and then predict the forces each atom in the molecule. These

---

Under review as a conference paper at ICLR 2023
5
CCSD(T)
DFT
Prediction Lirorr
Prediction Error
0.37
Aspirin
0.14
Benzene
2
Ethanol
0,05
Toluene
Malonaldchyde
0,02
200
400
600
800
950
Aspira
Rereane
Fihand Maloraddeiee Toluens Molecule
Number of Training Molecules
(a) Prediction Error vs. Training Set Size
(b) Prediction Error on CCSD(T)
Figure 1: (a) Log-log plot of the number of training point versus the prediction error for deep force fields (b) Prediction error on CCSD labelled molecules for force fields trained on large amounts of DFT reference forces (100,000 configurations) and moderate amounts of CCSD reference forces (1000 configurations). In both cases the model architecture used is GemNet (Gasteiger et al. 2021)
Built upon the success of machine learning force fields, deep learning techniques for force fields have been developed, resulting in highly accurate force fields parameterized by large neural networks (Gasteiger et al. 202T; Batzner et al. 2022). Despite this empirical success, a key draw back is rarely discussed existing literature: in order train state-of-the-ari machine learning force field models, a large amount of costly training data must be generated. For example, to train a model at the CCSD(T) level of accuracy, least a thousand CCSD(T) calculations must be done construct the training set. This is computationally expensive due to the method's O(n²) cost.
A natural solution this problem is to train less data points. However, if the number of training points is decreased, the accuracy of the learned force fields quickly deteriorates. In our experiments, empirically find that the prediction error and the number of training points roughly follow power law relationship, with prediction error Numher of Triining Points This can be seen in Figure la where prediction error and train set size are observed to have linear relationship with slope of when plotted on log scale.
Another option is to train the force field model on less accurate but computationally cheap reference forces calculated using DFT (Kohn 2019) or empirical force field methods (Johnson et al. 1993). However, these algorithms introduce undesirable bias into the force labels, meaning that the trained models will have poor performance. This phenomenon can be seen in Figure Tb] where models trained on large quantities of DFT reference forces shown to perform poorly relative force fields trained on moderate quantities of CCSD(T) reference forces. Therefore current methodologies sufficient for training force fields models in low amounts of accurate data (i.c. from CCSD(T)) or large amounts of inaccurate data (i.c. from DFT or empirical force fields) will result in inaccurate foree fields.
settings, training on either small
To address this issue, we propose to use both large amounts of inaccurate force field data (i.e. DFT) and small amounts of accurate data (i.e. CCSD(T)) to significantly reduce the cost of the data needed to achieve highly accurate force fields. Our motivation is that computationally cheap inaccurate data, though incurring large bias, can help capture the sophisticated structures of the underlying force field. Moreover, if treated properly, we can further reduce the bias of the obtained model by taking advantage of the accurate data.
More specifically, we propose a multi-stage computational framework datA cosST awarE. tRaining of FOrce flelDs (ASTEROID). In the first stage, small amounts of accurate data are used to identify the bias of force labels in large but inaccurate dataset. In the second stage, the model is trained on the large inaccurate dataset with a bias aware loss function. Specifically, the loss function generates smaller weights for data points with larger bias, suppressing the effect of label noise training. This inaccurately trained model serves as a warm start for the third stage, where the force field model fine-tuned the small and accurate dataset. Together, these stages allow the model learn
2

---

Under review as a conference paper at ICLR 2023
from many molecular configurations while incorporating highly accurate force data, significantly outperforming conventional methods trained with similar data generation budgets.
Beyond using cheap labelled data to boost model performance, also extend our method to the case where a large amount of unlabelled molecular configurations are cheaply available (Smith et al. 2017 Köhler et al. 2022). Without labels, we cannot adopt the supervised learning approach. In- stead, we draw a connection to score matching, which learns the gradient of the log density function with respect to each data point (called the score) (Hyvärinen & Dayan (005). In the context of molecular dynamics, we notice that if the log density function is proportional to the energy of each molecule, then the score function with respect to a molecule's position is equal the force on the molecule. Based on this insight, we show that the supervised force matching problem can be tackled in an unsupervised manner. This unsupervised approach can then be incorporated into the ASTEROID framework, improving performance when limited data is available.
We demonstrate the effectiveness of our framework with extensive experiments on different force field data sets. We use two popular model architectures, GemNet (Gasteiger et al.] (021) and EGNN Satooras et al., 2021), and verify the performance of our method in a variety of settings. In par- ticular, we show that our multi-stage training framework can lead to significant gains when either DFT reference forces or empirical force field forces are viewed as inaccurate data and CCSD(T) configurations are used accurate data. In addition, we show that we can learn accurate forces via the connection to score matching, and that using this objective in the second stage of training can improve performance on both DFT and CCSD(T) datasets. Finally, framework is backed by variety of ablation studies.
The rest of this paper is organized follows: Section 2 presents necessary background on machine learning force fields and training data generation, Section 3 details the ASTEROID framework, Section 4 extends ASTEROID to settings where unlabelled configurations are available, Section 5) presents our experimental results, and Section 6 compares our method to several related works (Ramakrishnan et al. 2015 Ruth et al. 2022) Nandi et al. 202T [Smith et al. 2019 Deringer al., 2020) and briefly concludes the paper.
BACKGROUND
Machine Learning Force Fields. Recent years have seen surge of interest in machine learning force fields. Much of this work has focused on developing large machine learning architectures that have physically correct equivariances, resulting in large graph neural networks that can gen- erate highly accurate force and energy predictions (Gasteiger et al., 202T; [Satorras al.) 202T] Batzner et al. 2022). Two popular architectures are EGNN and GemNet. Both models have graph neural network architectures that are translation invariant, rotationally equivariant, and permutation equivariant. EGNN is smaller model and is often used when limited resources are available. The GemNet architecture is significantly larger and more refined than the EGNN architecture, model- ing various types of inter-atom interactions. GemNet is therefore more powerful and can achieve state-of-the-ae thrt--rrttt--tttt- performance, but requires more resources.
It has been noted that in order to be reliable enough for MD simulations, molecular force field must attain an accuracy of kcal mol" A (Chmicla et al. 2018 Critically, the accuracy of deep force fields such as GemNet and EGNN is highly dependent on the size and quality of the training dataset. With limited training data, modern MLFFs cannot achieve the required accuracy for usefulness, preventing their application in settings where data is expensive to generate (e.g. large molecules). The amount of resources needed to train is therefore a key bottleneck preventing widespread use of modern MLFFs.
Data Generation Cost. The training data for MLFFs can be generated by a variety of force cal- culation methods. These methods exhibit an accuracy cost tradeoff: accurate reference forces from methods such as CCSD(T) require high computational costs to generate reference forces, while in- accurate reference forces from methods such as DFT and empirical force fields can be generated extremely quickly. Concretely, CCSD(T) is highly accurate but has (n²") complexity, DFT is less accurate with complexity (n))), and empirical force fields are inaccurate but have complexity ((n) (Lin et al. 2019 Rif et al. 2017). CCSD(T) is typically viewed as the gold standard for cal- culating reference forces, but its computational costs often make it impractical for MD simulation (it has been estimated that "a nanosecond-low MD trajectory for single ethanol molecule exe-
3

---

Under review as a conference paper at ICLR 2023
cuted with the CCSD(T) method would take roughly million CPU years modern hardware") (Chmiela et al. 2018). Due to this large expense. molecular configurations are typically generated first with MD simulations driven by DFT empirical force fields. These simulations generate large number of molecular configurations, and then CCSD(T) reference forces are computed for small portion of these configurations. Due this generation process, typically, large amount of inaccurately labelled molecular configurations are available along with the accurate CCSD(T) labelled data. Therefore large amounts of inaccurate force data can be obtained and used to train MLFF models, while incurring almost no extra data generation costs compared to CCSD(T).
ASTEROID
To reduce the amount of resources needed to train machine learning force fields, we propose a multi-stage training framework, ASTEROID, to learn from a combination of both cheaply available inaccurate data and more expensive accurate data.
Preliminaries. For a molecule with I atoms, we denote a configuration (the positions of its atoms in 3D) of this molecule as x E IR its respective energy as (z) E I. and its force as F(z) e
RR We denote the accurately labelled data as DA = {(0 ei ff).... (2N, ex.eN {x)] and the inaccurately labelled data as Dj {(aT. ef, f (2% At cy JM }. where (x,, J() represents the position, potential energy, and force of the ith accurately labelled molecule, using shorthand
notations e,e E(=;) and f F(af) (similarly (I7), e7. for the jth inaccurately labeled
data). Conventional methods train force field model ():)0) with parameters O on the accurate data by minimizing the loss
N min L(DA, - (1- (1- (1 Volect)) + + P Elee /
N
(1)
1=1
i=1
where E is the loss function for the force prediction, and le is the loss function for the energy prediction. Here the force is denoted by V, Ez; 0), i.e., the gradient of the energy E(z; 0) w.r.t. to the input z. We can use p to balance the losses of the energy prediction and the force prediction. In practice, most of the emphasis is placed on the force prediction, e.g. p 0.001.
3.1 BIAS IDENTIFICATION
The approximation algorithms used to generate cheap data D, introduce a large amount of bias into some force labels f", which may significantly hurt training. Motivated by this phenomenon, we aim to identify the most biased force labels so that we can avoid overfitting to the bias during training. In the first stage of our training framework, data DA to identify the levels of bias in the inaccurate dataset D,. Specifically, we train a force field model by minimizing C(DA, 0) (Eq. 1,. the loss over the accurate data, get parameters 0g Although the resulting model E(; 0o) will not necessarily have good prediction performance because of the limited amount of training data, it can still help estimate the bias of the inaccurate data. For every configuration x" in the inaccurate dataset D,. we suspect it to have large bias if there a large discrepancy between its force label f and the force label predicted by the accurately trained model V=E(=;;; fo). We can therefore use this discrepancy as a surrogate for bias, i.e. B(x))) = V[V 0o) f7111
small amounts of accurately labelled
3.2 BIAS-AWARE TRAINING WITH INACCURATE DATA
In the second stage of our framework, we train a force field model E': 0nit) from scratch on large amounts of inaccurately labelled data Dj. To avoid over-fitting to the biased force labels, a bias aware loss function that weighs the inaccurate data according to their bias. In particular, we use the weights wy exp(-B(x")/r) for configuration zy, where 7 is hyperparameter to be tuned. In this way, low-bias points are given higher importance and high bias points are treated more carefully. We then minimize the bias aware loss function
M min (D,, (1 Σ wi CG V.E(T + Σ
w. La E(x;;
(2)
4

---

Under review as a conference paper at ICLR 2023
to get parameters mitt resulting in the initial estimate of the force field model (1;8m)).
3.3 FINE-TUNING OVER ACCURATE DATA
The model E(: (iitt) contains information useful to the force prediction problem, but may still contain bias because it is trained on inaccurately labelled data Dr. Therefore, further refine it using accurately labelled data DA- Specifically, we use ( mi)) as an initialization for our final stage, in which we fine-tune the model over the accurate data by minimizing C(DA, ffiaal) (Eq. I
4 ASTEROID FOR UNLABELLED DATA
In several settings molecular configurations are generated without force labels, either because they are not generated via MD simulation (e.g. normal mode sampling, [Smith et al.] (2017)) or because the forces are not stored during the simulation (Köhler et al.] 2022). Although these unlabelled configurations may be cheaply available, they are not generated for the purpose of learning force fields and have not been used in existing literature. Here, we show that the unlabelled configurations can be used to obtain an initial estimate of the force field, which can then be further fine-tune on accurate data. More specifically, we consider a molecular system where the number of particles, volume, and temperature are constant (NVT ensemble). Let refer to the molecule's configuration and Fxr refer to the corresponding potential energy. It is known that follows Boltzmann distribution, i.e.
1 7ex exp
1 T B(x)
p(x) =
where Z is a normalizing constant, T is the temperature, and k3 is the Boltzmann constant. prac- tice, configurations generated using normal mode sampling (Unke et al.) (021) or via sufficiently long NVT MD simulation follow Boltzmann distribution.
Recall that we model the energy F(x:) as Fla;; O), and the force can be calculated as F(x; 0) = VE(x;; f). It follows from Hyvärinen & Dayan (2005) that we can learn the score function of the Boltzmann distribution using score matching, where the score function is defined the gradient of the log density function Vzlog p(x). In our case, we observe that the force on a configuration æ is score matching
proportional to the score function, i.e., F(x) x V,log p(x). Therefore, to learn the forces by minimizing the unsupervised loss
TrVV 0)) + 111=000)
(( Ep(e)
KTT
Although this allows solve the force matching problem in an unsupervised manner, the unsu- pervised loss is difficult to optimize in practice. To reduce the cost of solving Eq. we adopt sliced score matching (Song al. 2020). Sliced score matching takes advantage of random projections to significantly reduce the cost of solving Eq. 3 allowing us to apply score matching to large neural models such GemNet.
In our experiments, we find that score matching does not match the accuracy of CCSD(T) force labels. Instead, we can think of score-matching as form of inaccurate training. We therefore use score matching as an alternative to stages one and two of the ASTEROID framework. That is, we minimize Eq. 33oo to get fmit after which the model is fine-tuned on the accurate data.
EXPERIMENTS
5.1
DATA AND MODELS
For our main experiments, we consider three settings: using DFT data to enhance CCSD(T) training. using empirical force field data to enhance CCSD(T) training, and using unlabelled configurations to enhance CCSD(T) training. For the CCSD(T) data, we use MDI7@CCSD which contains 1,000 configurations labelled at the CCSD(T) and CCSD level of accuracy for molecules (Chmiela et al.) 2017). For DFT data, we use the MD17 dataset which contains over 90,000 configurations labelled at the DFT level of accuracy for 10 molecules (Chmiela et al.) 2017). For the empirical force field data, we generate 100,000 configurations using the OpenMM empirical force field software (Fastman et al. 2017). For the unlabelled datasets, we use MD17 with the force labels removed.
5

---

Under review as a conference paper at ICLR 2023
In each setting use 200, 400, 600, 800, 950 CCSD(T) training samples accurately labelled data. A validation set of size 50 and a test set of size 500 is used in all experiments, except for ethanol, where test set of size 1000 is used. Recall that inaccurately labelled datasets can be generated at a tiny of the cost CCSD(T) data.
the same model parameters as [Gatteiger et al. (202T). For EGNN, we use a 5-layer model and an embedding size of 128. We also compare ASTEROID sGDML (Chmiela al. 2019), kernel-based method that has been shown to perform well when limited training data is available. For training with inaccurate data, we train with batch size of 16 and stop training when the validation loss stabilizes. In the fine-tuning stage, we use batch size of 10 and train for a maximum of 2000 epochs. To tune the bias aware loss parameter 7, we search the set {0.5, 0.75, 1.0, 2.0, 3.0, 4.0, 5.0] and select the model with the lowest validation loss. Comprehensive experimental details are deferred to Appendix A.1
We implement method on GemNet and EGNN. For GemNet
5.2 ENHANCING FORCE FIELDS WITH DFT
We display the results for using DFT data to enhance CCSD(T) training in Figure 2 for GemNet and Figure 3 for EGNN. From these figures, can see that ASTEROID can outperform standard train- ing for all amounts of data. In particular, the performance gain from using ASTEROID is strongest when limited amounts of accurate data are available and decreases the amount of accurate data increases. When applied to GemNet in low resource settings, ASTEROID on average improves pre- diction accuracy by 52% and sample efficiency by factor of 2.4. For EGNN, ASTEROID improves prediction accuracy by 66% and increases sample efficiency by more than 5 times. The large perfor- mance increase for EGNN may be due to the fact that the EGNN architecture has less inductive bias than GemNet, and therefore may struggle to learn the structures of the underlying force field with only a small amount of data.
0.10
Standard Training
0.08
sGDML. ASTEROID
Test MAB
Test MA
20 9.6
0.06
1.0
0.5
0.6
0.02
0.2
400
950
700
400
600
800
950
0.1
200
400
Number of Points
Number Points
Nomber Points
(a) Benzene
(b) Aspirin
(c) Malonaldehyde
0.6
0.8
0.4
0.6
0.4
0.2
0.1
600
350
200
400
600
800
950
Points
Number of Points
(d) Toluene
(c) Ethanol
Figure 2: Main results for GemNet when DFT data is viewed as inaccurate.
5.3 ENHANCING FORCE FIELDS WITH EMPIRICAL FORCE CALCULATION
We present the results for empirical force field in Table Due to limited space, we only display the results for the case where 200 accurate data points are available. The remaining results for GemNet can be found in Appendix A.2 Again we find that ASTEROID significantly outperforms the supervised baseline, improving prediction accuracy by 36% for GemNet and by 17% for EGNN. We note that empirical force fields are typically much less accurate than DFT (see Figure but also much faster compute. Despite this accuracy gap, the ASTEROID framework is able utilize the empirical force field data to improve performance, which allows ASTEROID to benefit from even cheaper labels than DFT and obtain better efficiency.

---

Under review as a conference paper at ICLR 2023
1.8
1.5
1.2
Tour MAT
Text MAT
1.4
1.0
1.2
Stundurd Training
1.0
sGDML
0.4
ASTEROLD
200
660 Points
Number
(a) Benzene
(b) Aspirin
(c) Malonaldehyde
0.8
0.3
Test MAR
Tos MAtS
0.5
0.2
0.2
200
950
200
400
600
800
950
Number
Nomber Points
(d) Toluene
(e) Ethanol
Figure 3: Main results for EGNN when DFT data is viewed as inaccurate.
Table Accuracy of ASTEROID with empirical force field data. The training set for the fine-tuning stage contains 200 CCSD(T) labelled molecules.
Aspirin
Benzene
Malonaldehyde
Toluene
Ethanol
GemNet
Standard Training ASTEROID
1.554
0.083
0.801 0.516
0.591 0.337
0.348 0.301
0.843
0.048
EGNN Standard Training ASTEROID
1.897
0.297
1.466
0.777 0.664
0.840
1.314
0.268
1.341
0.637
5.4 ENHANCING FORCE FIELDS WITH UNLABELLED MOLECULES
We first verify that score matching can learn the forces on unlabelled molecules by comparing its prediction accuracy CCSD(T) data with models trained on supervised data (DFT and empirical force fields). We show the results in Figure 4 Surprisingly, we find that the prediction error of score matching is between that of DFT and empirical force fields. This indicates that relatively accurate force predictions can be obtained only by solving Eq. 3 Next we extend ASTEROID to settings where unlabelled data is available by fine-tuning the model obtained from We present the results for using ASTEROID on unsupervised configurations in Table 2 where we find that ASTEROID can improve prediction accuracy by 18% for GemNet and 4% for EGNN.
matching.
5.5 ABLATION STUDIES
We study the effectiveness of each component of ASTEROID. Specifically, we investigate the im- portance of bias- bias-aware training (BAT) and fine-tuning (FT) when compared with standard training. As can be seen in Table each of ASTEROID's components is effective and complementary to one another. We find that bias-aware training most helpful with GemNet, which has more capability to overfit harmful data points than EGNN.
5.6 ANALYSIS
Size of inaccurate data. To demonstrate that ASTEROID can exploit varying amounts of accurate data, plot the performance of ASTEROID with randomly sub-sampled inaccurately

---

Under review as a conference paper at ICLR 2023
DFT
Preciction Eiror (MAE)
14
Score Match
12
Empirical Force Field
Aspinn
Biannl Molecule
Figure 4: Prediction errors of models tested on CCSD(T) data. Models trained with different inaccurately labelled data but are not fine-tuned on the accurate CCSD(T) data.
Table 2: Accuracy of ASTEROID with unlabelled molecular configurations. The training set for the fine-tuning stage contains 200 CCSD(T) labelled molecules.
Aspirin
Benzene
Malonaldehyde
Toluene
Ethanol
GemNet
Standard Training 0.928
1,554
0.083
0.801 0.629
0.591 0.475
0.348
ASTEROID (Unsupervised)
0.093
0.314
EGNN
Standard Training
1.897
0.297
1.466 1.382
0.777
0.840
ASTEROID (Unsupervised)
1.756
0.305
0.740
0.823
Table Ablation study on ASTEROID. The inaccurate data is DFT labelled configurations and the accurate dataset contains 200 CCSD(T) labelled configurations.
Aspirin
Benzene
Malonaldehyde
Toluene
Ethanol
GemNet
Standard Training
1.554
0.083
0.801
0.591
0.348
ASTEROID w/o FT
4.670
3.252
2.726
3.342
5.107
ASTEROID w/o BAT
0.788
0.062
0.311
0.360
0.112
ASTEROID
0.738
0.057
0.312
0.325
0.109
EGNN
Standard Training
1.897
0.297
1.466
0.777
0.840
ASTEROID w/o FT
4.617
3.260
2.765
3.304
5.120
ASTEROID w/o BAT
0.577
0.225
0.213
0.272
0.009
ASTEROID
0.563
0.223
0.219
0.268
0.102
labelled data. ASTEROID performs best when large amounts inaccurate data are available, but still increases the accuracy by 32% when trained using small amounts of inaccurately labelled data.
We also investigate the performance score matching when varying amounts of unlabelled data available. We find that score matching is fairly robust to the number of available configurations, beating empirical force fields with only 100 configurations and only requiring 5000 configurations for optimal performance.
Runtime Comparison. Although there no official documentaion on the time needed to create the DFT and CCSD(T) data in the MD17 dataset, do create own empirical force field dataset and measure the time needed to create it. Generating 100,000 force molecular conformations and their respective force labels takes roughly two hours with single V100 32GB GPU. This is much
8

---

Under review as a conference paper at ICLR 2023
Prediction Error (MAT)
1.5
Prediction Error (MAF)
18
Score Match
1.4
16
Empirical Force Field
1.3
14
DFT
1.2
ASTEROID
12
1.1
Standard Training
1.0
0.9
0.8
0.7
1000
10000
50000
100000
1010
1000
10000
100000
Number of Inaccurate Points
Number of Points
(a) ASTEROID
(b) Score Matching
Figure 5: Prediction error of models trained on with varying amounts of inaccurate data. The molecule is aspirin, the inaccurate data is labelled with DFT, and the accurate data is made up of 200 CCSD(T) labelled molecules. For score matching we do not fine-tune the model.
faster than DFT or CCSD(T), which can take hundreds or thousands of seconds for single compu- tation (Bhattacharya et al. 2021 Datta Gordon 2021b. In general, it is difficult to get an exact time comparison between DFT and CCSD(T) because the runtime depends the molecule, the number basis functions used, and the implementation of each algorithm.
Hyperparameter Sensitivity. We investigate the sensitivity of ASTEROID the hyperparame- ters y and (note that is only used in GemNet). From the Figure we can see that ASTEROID is robust the choice of hyperparameters, outperforming standard training in every setting.
Prediction Error (MAE)
Prediction Error (MAE)
Prediction Error (MAE)
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.1
0.1
0.3
0.0
0.75
1.0
0.0
0.75
1.0
2.0
0.0
0.01
0.001
0.0001
(a) 7 (GemNet)
(b) (EGNN)
(c) P
Figure 6: Ablation study on the toluene molecule. We set 1.0 and 0.001 by default. The red line represents standard training.
DISCUSSION AND CONCLUSION
One related area of work A A-machine learning (Ramakrishnan et al. 2015 Ruth et al. 2022 Nandi al., 2021), which also uses both inaccurate force caleulations and accurate force calculations. Specifically, A-machine learning learns the difference between inaccurate and accurate force pre- dictions, therefore speeding up MD simulation while maintaining high accuracy. However, such an approach requires an equal amount of accurate and inaccurate training data, which therefore makes the training data expensive to generate. Another related area of work training general purpose MLFFs over many molecules (Smith et al. 2019 Deringer et al. 2020). Different from our work, these methods require lots of data from multiple molecules and generated in cost-aware manner. Additionally, the method from Smith et al. (2017) only trains on equilibrium states and may not work well for MD trajectory data. Numerical comparisons can be found in Appendix A.7
on huge dataset that is not
Different from previous works on machine learning force fields, we propose to learn MLFFs data cost aware manner. Specifically, propose new training framework, ASTEROID, to boost prediction accuracy with cheap inaccurate data. In addition, we extend ASTEROID to the unsuper- vised setting. By learning force field structures the cheap data, ASTEROID only requires small amount expensive and accurate datapoints to achieve good performance. Extensive experiments validate the efficacy of ASTEROID.

---

Under review as a conference paper at ICLR 2023
REFERENCES
Berni Julian Alder and Thomas Everett Wainwright. Studies in molecular dynamics. ii behavior of a small numher of elastic spheres. The Journal of Chemical Physics, 33(55:1439-1433 1960.
Simon Batzner, Albert Musaelian, Lixin Sun, Mario Geiger, Jonathan P Mailoa, Mordechai Ko- rnbluth, Nicola Molinari, Tess Smidt, and Boris Kozinsky. E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nature communications, 13(1):
1-11, 2022.
Feliks Aleksandrovich Berezin and Mikhail Shubin. The Schrödinger Equation, volume 66. Springer Science & Business Media, 2012.
Kaushik Bhattacharya, Vikram Gavini, Michael Ortiz, Mauricio Ponga, and Phanish Suryanarayana. Accurate approximations of density functional theory for large systems with applications to de- 2021.
fects in crystalline solids. arXiv preprint
Jiu-Li Chen, Tao Sun, Yi-Bo Wang, and Weizhou Wang. Toward a less costly but accurate calcula- tion of the cesd (t)/cbs noncovalent interaction energy. Journal of Computational Chemistry, (13):1252-1260, 2020.
Stefan Chmiela, Alexandre Tkatchenko, Huziel E Sauceda, Igor Poltavsky, Kristof T Schütt, and Klaus-Robert Müller. Machine learning of accurate energy-conserving molecular force fields. Science advances, 365):e1603015, 2017.
Stefan Chmiela, Huziel E Sauceda, Klaus-Robert Müller, and Alexandre Tkatchenko. Towards exact molecular dynamics simulations with machine-learned force fields. Nature communications, 9(1): 1-10, 2018
Stefan Chmiela, Huziel Sauceda. Igor Poltavsky, Klaus-Rober Müller, and Alexandre Tkatchenko. sgdml: Constructing accurate and data efficient molecular force fields using ma- chine learning. Computer Physics Communications, 240:38-45, 2019.
Dipayan Datta and Mark S Gordon. A massively parallel implementation of the cesd (1) method using the esooution-of-the-idee lelization model. Journal of Chemical Theory and Computation, 17(8):4799-492 2021.
approximation and hybrid distributed/shared memory paral-
Volker L Deringer, Miguel Caro, and Gábor Csányi. general-purpose machine-learning force field for bulk and nanostructured phosphorus. Nature communications, 11())1-11,,00
Peter Eastman, Jason Swails, John D Chodera, Robert T McGibbon, Yutong Zhao, Kyle A Beauchamp, Lee-Ping Wang, Andrew C. Simmonett, Matthew P Harrigan, Chaya Stern, et al. Openmm 7: Rapid development of high performance algorithms for molecular dynamics. PLoS computational biology, 13(7)ee005665) 2017.
Johannes Gasteiger, Florian Becker, and Stephan Günnemann. Gemnet: Universal directional graph neural networks for molecules. Advances Neural Information Processing Systems, 34:6790- 6802, 2021.
Adam Hospital, Josep Ramon Goñi, Modesto Orozco, and Josep L Gelpi. Molecular dynamics sim- ulations: advances and applications. Advances and applications in bioinformatics and chemistry: AABC, 2015.
Aapo Hyvärinen and Peter Dayan. Estimation of non-normalized statistical models by score match- ing. Journal of Machine Learning Research, 6(4), 2005.
J Karl Johnson, John A Zollweg, and Keith E Gubbins. The lennard jones equation of state revisited. Molecular Physics, 78(3):591-618, 1993.
Jonas Köhler, Yaoyi Chen, Andreas Krämer, Cecilia Clementi, and Frank Noé. Force-matching coarse-graining without forces. arXiv preprint arXiv:2203. 1116,, 2022.
Walter Kohn. Density functional theory. Introductory Quantum Mechanics with MATLAB: For Atoms, Molecules, Clusters, and Nanocrystals, 2019.
10

---

Under review as a conference paper at ICLR 2023
Ask Hjorth Larsen, Jens Jørgen Mortensen, Jakob Blomqvist, Ivano E Castelli, Rune Christensen, Marcin Dulak, Jesper Friis, Michael N Groves, Bjørk Hammer, Cory Hargus, et al. The atomic simulation environment- python library for working with atoms. Journal of Physics: Con- densed Matter, 29(27))273002, 2017.
Kevin Leung and Joanne Budzien. Ab initio molecular dynamics simulations of the initial stages of solid-clectrolyte interphase formation lithium ion battery graphitic anodes. Physical Chemistry Chemical Physics, 12(25):6583-6586 2010.
Lin Lin, Jianfeng Lu, and Lexing Ying. Numerical methods for kohn-sham density functional theory. Acta Numerica, 28:405-5555 (0005-539,2000 2019.
J Andrew McCammon, Bruce Gelin, and Martin Karplus. Dynamics of folded proteins. nature, 267(5612):585-590 1977.
Apurba Nandi, Chen Qu, Paul L Houston, Riccardo Conte, and Joel M Bowman, 8-machine learning for potential energy surfaces: A pip approach to bring dft-based pes to cesd (t) level theory. The Journal of Chemical Physics, 1.4(5))051 102, 2021.
Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Big data meets quantum chemistry approximations: the 6-machine learning approach. Journal of chemical theory and computation, 11(5):2087-2096 2015.
Laura E Ratcliff, Stephan Mohr, Georg Huhs, Thierry Deutsch, Michel Masella, and Luigi Gen- ovese. Challenges in large scale quantum mechanical calculations. Wiley Interdisciplinary Re- views: Computational Molecular Science, 7(1): 2017.
Marcel Ruth, Dennis Gerbig, and Peter Schreiner. Machine learning of coupled cluster (t)-eneryy corrections via delta (6)-learning Journal of Chemical Theory and Computation, 18(8):4846- 2022.
Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E (n) equivariant graph neural net- works. In International conference on machine learning, pp.
PMLR, 2021.
Gustavo Scuseria, Curtis I. Janssen, and Henry F Schaefer Iii. An efficient reformulation of the closed-shell coupled cluster single and double excitation (cesd) equations. The Journal of Chemical Physics, 89(12):7382-7387, 1988.
Justin S Smith, Olexandr Isayev, and Adrian E Roitberg. Ani-l, a data set of 20 million calculated off-equilibrium conformations for organic molecules. Scientific data, 4(1):1-8, 2017.
Justin S Smith, Benjamin T Nebgen, Roman Zubatyuk, Nicholas Lubbers, Christian Devereux, Kipton Barros, Sergei Tretiak, Olexandr Isayev, and Adrian E Roitberg. Approaching coupled cluster accuracy with a ceneral-purposs neural network potential through transfer learning. Nature communications, 10(1):1-8, 2019.
Yang Song, Sahaj Garg, Jiaxin Shi, and Stefano Ermon. Sliced score matching: A scalable approach to density and score estimation. In Uncertainty in Artificial Intelligence, pp. 574-584. PMLR, 2020.
Mark Tuckerman. Statistical mechanics: theory and molecular simulation. Oxford university press, 2010.
Oliver Unke, Stefan Chmiela, Huziel Sauceda, Michael Gastegger, Igor Poltavsky, Kristof T Schutt, Alexandre Tkatchenko, and Klaus-Robert Muller. Machine learning force fields. Chemi- cal Reviews, 121(16):0142210188 2021.
Vikas Varshney, Soumya S Patnaik, Ajit K Roy, and Barry L Farmer. A molecular dynamics study of epoxy-based networks: cross-linking procedure and prediction of molecular and material prop- erties. Macromolecules, 41(18):6837-6842, 2008

---

Under review as a conference paper at ICLR 2023
A APPENDIX
A.1 EXPERIMENTAL DETAILS
In this section go over the experimental details.
GemNet Training Details. To train the bias identification method, we train freshly initialized model with batch size of 10 the accurate dataset for 2000 epochs. To train the inaccurate model, we train a freshly initialized model with the bias aware loss function and batch size 16 over the inaccurate dataset. Finally to finetune the inaccurately trained model, we train model with a batch size of 10 on the accurate dataset for 2000 epochs. In each stage of training we use the following hyperparamers:
Evaluation Interval:
epoch
Decay steps: 1200000
Warmup steps: 10000
Decay patience: 50000
Decay cooldown: 50000
The rest of the parameters the same used in Gasteiger al. 2021
EGNN Training Details. The EGNN training setup is similar to GemNet. To train the bias identi- fication method, train freshly initialized model with batch size of 10 the accurate dataset for 2000 epochs. To train the inaccurate model, we train a freshly initialized model with the bias aware loss function and batch size 32 over the inaccurate dataset. Finally to finetune the inaccurately trained model, we train with batch size of 10 on the accurate dataset for 2000 epochs. In each stage of training we use the following hyperparamers:
Evaluation Interval: epoch
Learning rate: 10-4 for inaccurate training, 10-5 for finetuning
nam_layers: 5
embedding.sizee 128
A.2 ADDITIONAL RESULTS
Here we include additional results for ASTEROID when empirical force field data is viewed as inaccurate. For the baseline model we use GemNet. The ASTEROID framework again leads to consistent gains across all amounts of data.
A.3
DERIVATION OF SCORE MATCHING FOR FORCES
For given molecule with conformations 21, Ene let denote energy E(x). Then the the Boltzmam/Equllibriun distribution for the molecule is given by
1 pxx) 1 eexp(-3E(x))
where Z is normalizing constant, T Kg is the Boltzmann constant, and T is the temperature under which the simulation run. Then we can see that the force on a conformation is equivalent to the score, i.c. F(z) = ,(E()) V p(r). Therefore learning the force F(r) is equiva- lent to learning the score log p(x). Suppose we parameterize the MLFF to directly predict the force as (z) Then the force matching loss can be written as
C(0) = Ex FF F(x) = Eampte) | 12 Ex-p(z) (())) Exmptey Fo Fo 2
12

---

Under review as a conference paper at ICLR 2023
0.10
0.5
0.00
Standard Training
0.7
sGDML.
Tey MAE
0.07
ASTEROID
Teat MAB
Texl MAF
0.05
0.03
0.8
0.3
0.03
0.02
0.4
Number of Points
Number Points
Number of Points
(a) Benzene
(b) Aspirin
(c) Malonaldehyde
0.35
0,30
Test MAR
0.25
0.3
0.20
0.2
0.15
0.1
0.10
200
400
600
500
950
Number of Points
Number of Points
(d) Toluene
(e) Ethanol
Figure 7: Main results for GemNet when DFT data viewed as inaccurate.
The middle term can then be expanded
Ez-p(>) [(F(z),
(()))
1 dlog p(x) dx;
p(x)
Fo(a)))
dlog p(x) dx
dz)
3
do(2),d
Σ
p(=)dFo(x)) de.
dFF()); dx;
Σ
p(x)
da,do.-
3
1=1 J.-
1
d
[dFo(x)) dx
Expp((n)
Eupp(s)
Tr [v.
3
1=1
Therefore we have the loss
(()) Exwp(u)
-r
A.4 ASTEROID DIAGRAM
In order to to make the ASTEROID framework more clear, provide workflow diagram.
A.5 ASTEROID TOY EXAMPLE
We have added a new result using a two-layer MLP with 128 hidden units each and synthetic data. This experiment shows that ASTEROID can significantly improve generalization error in variety of
13

---

Under review as a conference paper at ICLR 2023
Stage
Stage 2
Stage
Train MLFF small accurately labelled data and estimate bias on inaccurate data
Train MLFF large inaccurately labelled data with Bias-Aware Training
Finetune MLFF small accurately labelled data
Estimate
Small Accurate Data
MLFF B(Ga)
MLFF E 660
MLFF E(O
Inacourate Data
E' (Oinie)
Large Inaocurate Data
Small
Figure 8: Asteroid workflow diagram.
settings. In this experiment, generate biased dataset of 2000 points according to Y = AX + b, where where X ~ [(, 1) has dimension 16, is the bias, and A is a randomly generated Gaussian matrix of dimension 16 x 16. The bias is chosen uniformly from the set 20 4, 8, 16). We also generate varying levels of accurate data according to Y AX, where ~ N(0, 1). We then evaluate the test MAE of ASTEROID and standard training over variety of accurate data sizes. We find that ASTEROID significantly outperforms standard training.
6
ASTEROID
5
Standard Training
4
Test MAE
3
2
1
0
20
30
40
50
60
70
Size of Accurate Data
Figure 9: Asteroid toy example.
A.6 MOLECULAR DYNAMICS SIMULATION
We also evaluate the performance of MLFFs trained by ASTEROID in downstream MD simulation tasks. First, demonstrate that ASTEROID trained MIFFfs produce stable dynamics, while
14

---

Under review as a conference paper at ICLR 2023
100
Trained Lennard-jones ASTEROID
1.0
Maximum Pairwise Distance
Converged
60
Standard Training
ASTEROID
0.20 0.25 0.30
0.45
Time Step (ps)
Simulation Step Size
(a) Benzene molecule during MD simula- tion.
(b) Proportion of converged simulations (Aspirin).
both MLFFs trained on on DFT data alone and empirical force field diverge. Using the Atomic Sim- ulation Environment (ASE) (Larsen et al., 2017), we simulate the behavior of a Benzene molecule using forces calculated by MLFF trained with ASTEROID, a MLFF trained on DFT data only, empirical force field. We simulate the molecule with Langevin dynamics,
and the Lennard-Jone where the steps size is 0.5 femptoseconds, the temperature 500K, the friction coefficient 0.002, and the maximum number of time steps is 10000. The results of these simulations be seen in I0a where ASTEROID is able to produce stable dynamics, while the error compounding effect of the DFT trained force field and the Lennard-Jones potential is too heavy, resulting in diverged simulation.
Next we compare the simulation ability of ASTEROID with that of standard training, where both MLFFs are trained on 200 points of CCSD(T) data. To evaluate the robustness of the MLFFs, we run MD simulations with varying step sizes on the Aspirin molecule. The setting is the same as in Figure T0a In Figure we plot the number of simulations that do not converge out of 20 random runs. The ASTEROID framework is able maintain steady performance across step sizes and almost all the simulations converge. In contrast, the simulations powered by standard MLFFs fail with larger step size.
A.7 COMPARISON WITH ANI-
We compare the performance of the method from Smith al. (2017) with in Table 4 in both the case when the ANI-1 model fine tuned on MD17 and the zero-shot setting. Despite having much larger data generation budget, the open-domain pre-training of Smith et al. (2017) is not as powerful as ASTEROID.
Table 4: Accuracy of ASTEROID compared with ANI-Iccx. The training set for the fine-tuning stage contains 200 CCSD(T) labelled molecules. FT refers to fine-tuning ANI-Icex on MD17.
Aspirin
Benzene
Malonaldehyde
Toluene
Ethanol
ASTEROID (GemNet)
0.843
0.048
0.516
0.337
0.301
ANI-
1.897
0.297
1.466
0.777
0.840
ANI- (FT)
1.314
0.268
1.341
0.664
0.637
15


Reviews:
[{"review_id": "yhMxDRDsM8", "reviewer": "Reviewer_VP9R", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors in this paper, propose a framework which enables training data generation with is computationally inexpensive while being accurate for molecular dynamics simulation. The authors argue that the current state of the art for machine learning force fields are computationally very expensive when it comes to achieving high accuracy, including generating a large amount of costly training data. While there are options to reduce computational complexity by training on fewer data points or cheaper reference forces, these options come with a tradeoff of loss in accuracy/poor performances. \n\nThe authors propose that ASTEROID works in a multi stage training framework where step 1: small amounts of accurate data is first identified in a large inaccurate dataset, step 2: train the models on large amounts of training data which is inaccurate and cheaply available, followed by step 3: where small amounts of accurate training data which is more expensive to obtain. step 1 is achieved via a bias aware loss function to prevent overfitting on inaccurate training data. The authors demonstrate that such a method significantly outperforms conventional methods such as standard training and sGDML on networks such as GEMNet and EGNN\n", "summary_of_the_review": "I propose this work is conditionally accepted with few more experiments such as time component comparison with other methods to demonstrate cost awareness and efficiencies to drive home the point of why the method is superior over prior work of art. While the authors argue that the method is superior in terms of accuracy over other current work, additional metrics are needed to really drive home the point without which the work seems less convincing. Also even though there is no availability of materials to test reproducibility, it is expected that a workflow to guide experimental setup and analysis would be necessary to validate some of the results. ", "strength_and_weaknesses": "\nStrength:\n1. Several performance comparisons around accuracy are method with different methods (state of the art) under various conditions to really drive home the point around performance improvement and superiority over existing methods\n2. Sufficient background provided to identify gaps and opportunities for development, although there could be more clarity around datasets and overall learning workflows. \n\n\nWeaknesses: \n1. Description of datasets is complex, and not simplified eg. what does accurate, inaccurate data look like along with labels. \n2. Method to estimate bias, metrics around bias measurement/identification could be clearer, eg. why is there a surrogate for bias measurement and how would ideal bias measurement look like prior to introducing surrogate method. \n3. Need additional commentary on any limitations of this method eg. cases where such an approach would not work and what could be alternatives. \n4. Calculations/stats/graphs around time comparison with other methods is not available which could be a key result. \n5. No clear explanation why the 5 particular chemicals were chosen. \n6. Also why test size is different for ethanol vs others? Do other molecules also need to be compared with the same test set size? \n7. For the hyper parameter tuning setting, I dont see the comparison between ASTEROID and standard training, even though its mentioned that way. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "vgyH7DhBt0", "reviewer": "Reviewer_Y3t9", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper address NNP training using an expensive data generation method, CCSD(T).\nThe proposed method is based on the pretraining of NNP using the training data collected by relatively low-cost data generation methods, such as DFT.\nThe authors extended it to a three-stage method, in which an importance-weighting method is used for the pretraining of NNP.\nAn NNP is trained using CCSD(T) at the first stage, and the instance weights for the pretraining are estimated by the discrepancy between the force of low-cost data generation methods and the force of the NNP trained using CCSD(T).   In addition, the authors propose unsupervised pretraining using score matching for the pretraining.\nThe proposed method outperforms the baseline methods on the MD17 dataset.", "summary_of_the_review": "The fatal problem of the paper is no evaluation result on its applications, such as MD simulations or structural optimization.  \nIn addition, the baseline algorithm in the experiments is also not appropriate.\nI think a major revision is required for acceptance.", "strength_and_weaknesses": "Strength\n* It is interesting to see the pretraining results using a dataset collected by not only DFT but also the empirical force field.\n* The use of score-matching for unsupervised pretraining is reasonable approach.\n* Unsupervised pretraining of NNP might be useful since it might be possible to aggregate multiple data sets collected by different methods.\nThe proposed method down-weight structures with higher energy, but we also need the energy/force estimation for disordered structures, i.e., structures of higher energy, in real-world applications, such as MD simulations or structural optimization.  \n\nWeaknesses\n* There is no evaluation result on its applications, such as MD simulations or structural optimization.  Since force accuracy is not sufficient for obtaining stable dynamics [1], I am not confident about the effectiveness of the proposed approach for its applications.\n* I think ANI- 1ccx  (Smith et al., 2019), a simple pretraining-finetuning method, should be a baseline method.  In my understanding, ASTEROID w/o BAT is the proposed transfer learning in Smith et al., 2019.  However, the results of it are not available in Figures 2, 3, 4, and Table 2. \n* If the unsupervised learning method is used for the second stage, the first stage is meaningless.  Thus, the unsupervised learning method does not extend ASTEROID in this case, and this contradicts the section title of Section 4.   In addition, the experiments conducted in this case.\n\n\nQuestions:\n1.  Can you add the simple pretraining-finetuning results, i.e., ASTEROID w/o BAT, in Figures 2, 3, 4, and Table 2?   Since ANI- 1ccx  (Smith et al., 2019) is a suitable baseline method, I think you can omit \"Standard Training\" results if there is no space available.\n\n2.  I miss the meaning of \"Next we extend ASTEROID to settings where unlabelled data is available by fine-tuning the model obtained from score matching\".\nIs this mean that the unlabeled data is collected by MD simulations using a fine-tuned model?  Or does it simply mean you pre-trained an NNP using unlabelled data and fine-tuned it using CCSD(T) data?\n\n3.  Can you provide any results of applications for which DFT-based NNP fails, as Smith et al., 2019 did?\n\n\n[1] Sina Stocker, Johannes Gasteiger, Florian Becker, Stephan Günnemann, and Johannes Margraf. How robust are modern graph neural network potentials in long and hot molecular dynamics simulations?  2022.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "pRb_uPE66E", "reviewer": "Reviewer_K6wT", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors of this paper present a training strategy to improve performance in ML MD. In this strategy, first bias of inaccurate data is calculated using the model trained on accurate, then the bias-aware inaccurate data is used to train an ML MD model from scratch, and finally, the model is fine-tuned using accurate data. Pretraining using unlabeled data is also explored in this work.  ", "summary_of_the_review": "Overall, this paper shows a carefully design pretraining-finetuning training strategy for ML MD, and their results support the claim that inaccurate data can help boost the performance of ML MD models as accurate data are very expensive to generate. \n\nUpdate: I have read through all the reviewers' comments and the authors' response. I think the authors addressed most of the concerns. Even though I still have concerns about the novelty of this work, it may be an informative paper for researchers in the MLFF field. Thus, I increased my rating to 6: marginally above the acceptance threshold. \n", "strength_and_weaknesses": "Strengths:\n1. The training strategy proposed in this paper makes sense for ML MD. The authors also investigated data weighting based on quality and explored the use of unlabelled data for pretraining.\n2. The experiments conducted in this research are thorough and their results support the main claim.\n\nWeaknesses:\n1. The authors should provide more details about the derivation of Eq. 3. \n2. ML FF methods mentioned in the related work were not compared with the proposed method. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "j3sYNljm8im", "reviewer": "Reviewer_2odL", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "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": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "In the paper, the authors aim to improve methods to train machine learning force fields for MD simulations. These methods rely on accurate data, which is expensive to generate. They augment the training procedure with less accurate data being cheaper to generate. This is achieved by first training a model with a limited amount of accurate data, which is then used to estimate the bias of the less accurate data. The data is reweighted according to this bias, followed by training a model on those samples. In a third step, the model is fine-tuned on the accurate data.\n\nThe authors evaluate their method on five molecules. The learned force field seems more accurate than when using standard training, especially when using a small amount of accurate training data and performs similar or better than a related approach.", "summary_of_the_review": "In conclusion, I favor to reject this paper. However, I am open for a discussion with the authors and the other reviewers.\n\n\nEdit after rebuttal:\n\nI appreciate the author's effort to improve their paper. The toy example they added is interesting. However, the setup seems counterintuitive to me. First, there should be noise being added to the data and its standard deviation could depend on the accuracy of the simulated data generation process. Second, I like the idea of adding a sampled bias, but why is it sampled uniformly from the values {0, 2, 4, 8, 16}? Thereby, 20% of the \"biased\" data is actually unbiased. I suggest drawing the bias from an interval such as [2, 6] or perhaps [0, 8] instead, limiting the number of accurate samples, or even using a constant bias with additional noise centered around 0.\nGiven that improved runtime is one of the major selling points of the paper, I think a runtime analysis is vital for the paper.\n\nTherefore, I will not change my score, but encourage the authors to improve their draft and submit it to another venue.", "strength_and_weaknesses": "### Strengths\n\nThe method is simple to implement. It allows using various kinds of additional data to improve the model's performance. The authors test their procedure thoroughly on five molecules and use different types of auxiliary data.\n\n### Weaknesses\n\nThe procedure is basically limited to adding additional data to the training procedure, which is in itself not new. The novel idea is training a model to estimate the bias and reweight the data accordingly. However, this bias-aware training does not improve the model significantly, see Table 3. The weights based on the estimated bias are apparently very noisy, but they could be improved by updating them during training in a bootstrapping fashion.\n\nThe method has been tested empirically on several molecules, but the paper lacks theoretical validation that the method is sound. Moreover, it should be tested in a toy setting, such as doing regression with a few variables on generated data with varying noise levels.\n\nMost importantly, the authors did only compare the performance of their models with respect to the number of accurate training samples. However, they should do so with respect to the overall runtime used to generate both the accurate and the inaccurate training data. Since the relative improvement decreases rapidly when increasing the number of accurate samples, it is not obvious whether it would not be better to just generate more accurate data.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject