Gradual Optimization Learning for Conformational Energy Minimization
Authors: Artem Tsypin
Leonid Anatolievich Ugadiarov
Kuzma Khrabrov
Alexander Telepov
Egor Rumiantsev
Alexey Skrynnik
Aleksandr Panov
Dmitry P. Vetrov
Elena Tutubalina
Artur Kadurin
Venue: ICLR 2024 poster (2024)
Keywords: energy minimization
conformational optimization
geometry optimization
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We propose a data-efficient framework for conformational energy minimization with neural networks

Abstract: Molecular conformation optimization is crucial to computer-aided drug discovery and materials design.
Traditional energy minimization techniques rely on iterative optimization methods that use molecular forces calculated by a physical simulator (oracle) as anti-gradients.
However, this is a computationally expensive approach that requires many interactions with a physical simulator.
One way to accelerate this procedure is to replace the physical simulator with a neural network.
Despite recent progress in neural networks for molecular conformation energy prediction, such models are prone to errors due to distribution shift, leading to inaccurate energy minimization.
We find that the quality of energy minimization with neural networks can be improved by providing optimization trajectories as additional training data.
Still, obtaining complete optimization trajectories demands a lot of additional computations.
To reduce the required additional data, we present the Gradual Optimization Learning Framework (GOLF) for energy minimization with neural networks.
The framework consists of an efficient data-collecting scheme and an external optimizer.
The external optimizer utilizes gradients from the energy prediction model to generate optimization trajectories, and the data-collecting scheme selects additional training data to be processed by the physical simulator. 
Our results demonstrate that the neural network trained with GOLF performs \textit{on par} with the oracle on a benchmark of diverse drug-like molecules using significantly less additional data.

Published as a conference paper at ICLR 2024
GRADUAL OPTIMIZATION LEARNING FOR CONFORMATIONAL ENERGY MINIMIZATION
Artem Tsypin | Leonid Ugadiarov Kuzma Khrabrov', Alexander Telepov',
Egor Rumiantsev Alexey Skrynnik 1.2 Aleksandr Pa.4
Dmitry Vetrov Elena Tutubalina 1.3,6 Artur Kadurin 1,72
AIRI, Moscow ²FRC CSC RAS, Moscow SSer AI, Moscow
*MIPT, Dolgoprudny 'Constructor University, Bremen
IIPP RAS Research Center for Trusted Artificial Intelligence, Moscow
Kuban State University, Krasnodar
(Tsypin, Kadurin]@airi, net
ABSTRACT
Molecular conformation optimization is crucial to computer-aided drug discovery and materials design. Traditional energy minimization techniques rely on iterative optimization methods that use molecular forces calculated by physical simulator (oracle) as anti- gradirndsents. However, this a computationally expensive approach that requires many interactions with physical simulator. One way to accelerate this procedure is to replace the physical simulator with a neural network. Despite recent progress in neural networks for molecular conformation energy prediction, such models are prone to errors due to distribution shift, leading to inaccurate energy minimization. We find that the quality of energy minimization with neu- ral networks can be improved by providing optimization trajectories as additional training data. Still, obtaining complete optimization trajectories demands lot of additional computations. To reduce the required additional data, we present the Gradual Optimization Learning Framework (GOLF) for energy minimiza- tion with neural networks. The framework consists of an efficient data-collecting scheme and external optimizer. The external optimizer utilizes gradients from the energy prediction model to generate optimization trajectories, and the data- collecting scheme selects additional training data to be processed by the physical simulator. Our results demonstrate that the neural network trained with GOLF performs on par with the oracle on a benchmark diverse drug-like molecules using significantly less additional data.
INTRODUCTION
Numerical quantum chemistry methods are essential for modern computer-aided drug discovery and materials design pipelines. They are used to predict the physical and chemical properties of candidate structures (Matta & Boyd, 2007 Oglic et al. 2017 Tielker et al. al.] (021)) Ab initio property prediction framework for specific molecule or material could be divided into three main steps as follows: (1) find a low-energy conformation of given atom system, (2) compute its electron structure with quantum chemistry methods, and (3) calculate properties of interest based on the latest. The computational cost of steps (I) and (2) is defined by the specific physical simulator (oracle O) varying from linear to exponential complexity w.r.t the number of atoms or electrons in the system (Sousa et al. 2007). Overall, the more accurate the oracle is, the more computationally expensive its operations become.
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The traditional approach to the problem of obtaining low-energy molecular conformations is to run an iterative optimization process using physical approximations, such as those provided by the Density-functional theory (DFT) methods (Kohn & Sham (165), as they are reasonably accurate. However, for large molecules, even a single iteration may take up several hours of CPU-compute (Gilmer et al.; 2017). Therefore, it is crucial develop alternative approaches (such as Neural Network-based) that reduce the computational complexity of iterative optimization.
The recent growth in computational power led to the emergence of molecular databases with com-
puted quantum properties (Ruddigkeit et al. 2012 Ramakrishnan et al., 20144) Isert et al. 2022]
Khrabrov et al. 2022 Jain et al. 2013). For example, nablaDFT (Khrabrov al. [022] consists
of more than 5 x 10"h conformations for around 10 drug-like molecules. This data enabled deep
learning research for many molecule-related problems, such as conformational potential energy and quantum properties prediction with Neural Network Potentials (NNP) (Chmiela et al., 2017 Schütt
al.
2017 Chmiela et al., 2008:: 2020; |Shhitt et al.] 2021 Shuaibi et al.] 2021] Gasteiger et a
2020
2021 Chmiela et al. (023). and conformational distribution estimation (Simm & Hernández-
Lobato 2019; Xu et al.] 2021 Ganea et al., 2021; Xu et al. 2022 Jing et al.] 2022] Shi et al,
2021 Luo et 20211. p. Naturally, there have been several works that utilize deep learning to tackle
the problem of obtaining low-energy conformations. One approach is to reformulate this task as a conditional generation task (Guan al. 2021 Lu et al.] 2023); see Section X
for further details. Another solution is to train an NNP to predict the potential energy of molecular conformation and use as force field for relaxation (Unke et al. [222]) Assuming the NNP accurately predicts the energy, its gradients can be used as interatomic forces (Schütt et al. (0177) Such technique allows for gradient-based optimization without a physical simulator, significantly reducing computational complexity.
In this work, we aim to improve the training of NNPs for obtaining low-energy conformations. We trained NNPs on the subset of nablaDFT dataset (Khrabrov et al., 2022) and observed that such models suffer from the distribution shift when used in the optimization task (see Figure [T]. To alle- viate the distribution shift and improve the quality of energy minimization, we enriched the training dataset with optimization trajectories (see Section 4 generated by the oracle. Our experiments demonstrate that it requires more than 5 x 10t additional oracle interactions to match the quality of a physical simulator (see Table T). These models trained on enriched datasets are used as baselines for our proposed approach.
In this paper, we propose the GOLF- of NNPs to generate low-energy conformations. GOLF consists of three components: (i) genuine oracle Oa (ii) an optimizer, and (iii) a surrogate oracle Os that is computationally inexpensive. The Og is an accurate but computationally expensive method used to calculate ground truth energies and forces, and we consider a setting with a limited budget on Og interactions. The optimizer (c.g., Adam (Kingma & Ba 2014) or L-BFGS (Liu & Nocedal, 1989)) utilizes NNP gradients to produce optimization trajectories. The Os determines which conformations are added to the training set. We use Psi4 (Smith et al. (020). a popular software for DFT-based computations, as the OG and RDKit's (Landrum et al.) 2022) MMFF (Halgren) 1996) as the Os The NNP training cycle consists of three steps. First, we generate a batch of optimization trajectories and evaluate all conformations with Os Then we select the first conformation from each trajectory for which the NNP poorly predicts interatomic forces w.r.t. Os (see Section 5)., calculate its ground truth energy and forces with the OG and add it to the training set. Lastly, we update the NNP by training on batches sampled from initial and collected data. We train the model until we exceed the computational budget for additional Og interactions. We show (see Section 6.2) that NNPs trained with GOLF on the nablaDFT (Khrabrov et al.] (2022) perform on par with Og while using 50x less additional data compared the straightforward approach described in the previous paragraph. We also show similar results on another diverse dataset of drug like molecules called SPICE (Eastman et al.] 2023)) We publish"" the source code for GOLF along with optimization trajectories datasets, training, and evaluation scripts.
- Gradual Optimization Learning Framework for the training
Our contributions can be summarized as follows:
We study the task of conformational optimization and find that NNPs trained on existing datasets are prone to the distribution shift, leading to inaccurate energy minimization.
https://githubs com AARIARRII-- nnt nstiiute//Gl..
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We propose a straightforward approach to deal with the distribution shift by enriching the training dataset with optimization trajectories (see Figure D.. Our experiments show that additional 5 x 10y conformations make the NNP perform comparably with the DFT-based oracle Og on the task of conformational optimization.
We propose novel framework (GOLF) for data-efficient training of NNPs, which includes a data-collecting scheme along with an external optimizer. We show that models trained with GOLF perform on par with the physical simulator on the task of conformational opti- mization using 50x less additional data than the straightforward approach.
RELATED WORK
Conformation generation Several recent papers have proposed different approaches for predict- ing molecule's s 3D conformers. Xuu et al.] (2021) utilize normalizing flows to predict pairwise dis-
tances between atoms for given molecular structure with subsequent relaxation of the generated conformation. Ganea et al. (202T) construct the molecular conformation by iteratively assembling it from smaller substructures. Xu et al. (2022): Wu et al. (2022); Jing et al. (2022): Huang et al.
(2023); Fan et al. (2023) address the conformational generation task with diffusion models (Sohl- Dickstein et al.] 2015). Other works employ variational approximations (Zhu et al. 2022 Swanson et al. 2023). and Markov Random Fields (Wang et al.] 2022). We evaluate these approaches in woot-mean-square deviation of atomic positions (RMSD), on the tasks reported in the various papers, these models perform poorly in terms of geometry and potential energy on the optimization task. In most cases, additional optimization with a physical simulator is necessary to get valid conformation.
Section 6.1 Despite showing promising geometrical metries, such
Geometry optimization Guan al. 2021 Lu et al. 2023) frame the conformation optimization problem as a conditional generation task and train the model to generate low-energy conformations conditioned on RDDiit-geeerated (or the randomly sampled from the pseudo optimization trajectory) conformations by minimizing the RMSD between the corresponding atom coordinates. As RMSD may not be an ideal objective for the conformation optimization task (see Section we focus on accurately predicting the interatomic forces along the optimization trajectories in our work.
Additional oracle interactions Zhang al. (0188) show that additional data from the oracle may increase the energy prediction precision of NNP models. Following this idea, Kulichenko et al. (2023) propose an active learning approach based on the uncertainty of the energy prediction to reduce the number of additional oracle interactions. The main limitation of this approach is that it requires training a separate NNP ensemble for every single molecule. Chan the molecule as set of rotatable bonds and utilize the Bayesian Optimization with Gaussian Process prior to efficiently search for low-energy conformations. However, this method requires using the oracle during the inference, which limits its applications. The OC2022 (Tran* et al. (022) provides relaxation trajectories for catalyst-adsorbate pairs. However, no in-depth analysis of the effects of such additional data on the quality of optimization with NNPs is provided.
(019) parametrize
To sum up, we believe it necessary to explore further the ability of NNPs to optimize molecular conformations according their energy. Our experiments (see Section 66 show that additional oracle information significantly increases the optimization quality. Since this information may be expensive, we aim reduce the number of additional interactions while maintaining the quality on par with the oracle.
NOTATION AND PRELIMINARIES
[x of the molecule as pair of atomic numbers z = (21,..., 2n, }, 2, E N and atomic coordinates X {x1,.... } x; E R23, where n. is the number of atoms the molecule. We define the oracle O function that takes conformation s as input and outputs its potential energy Egracle E R and interatomic forces Feracle Rnx3, ; E Ferack (s). To denote the ground truth interatomic force acting on the i-th atom, we use Exrude, We use different superscripts to denote energies and forces calculated by different physical simulators. For
We define the conformation
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example, denote the RDKit's MMFF-calculated energy E and the Psi4-calculated energy as -.
We denote the NNP for the prediction of the potential energy of the conformation parametrized by
weights 0 (s; 0) {z, IR. Following (Schitt al. 2017 Schütt et al. 2021). we derive forces from the predicted energies:
(s; 0f(;;) dx;
where F; E IR is the force acting on the i-th atom predicted by the NNP. We follow the standard procedure (Schütt et al., 2017; Schüit et al., 2021 Gasteiger et 2020 Musaclian et al. (0022) and train the NNP minimize the MSE between predicted and ground truth energies and forces:
C(s, Egracee poracke 0) = pl - f(s; 0]|
Foracle
F,(s
(2)
where [(s, Es Eqracle F.orele 0) is the loss function for a single conformation s, and is the hyperpa- rameter accounting for different scales of energy and forces.
To collect the ground truth optimization trajectories (see Section 4. we use the OPTIMIZE method from Psi-4 and run optimization until convergence. Optimizer Opt (L-BFGS, Adam, SGD-
momentum) utilizes the forces (s; E 12a to get NNP-optimization trajectories .o.....
where so is the initial conformation:
8 = 8, + (ppt(F(ss:)))
Here, a is the optimization rate hyperparameter, and T is the total number of NNP optimization steps.
In this work, we use NNPs trained on different data. To train the baseline model pbasetine (.; (:; use the fixed subset of nablaDFT Dn (see Appendix [ for more details). It consists of approxi- j. The Do be extended with the ground truth optimization trajectories obtained with Psi-4 to get datasets denoted according to the total num- ber of additional conformations: Punj-10kk Panj-100k, and so on. The resulting NNPs are dubbed [trij-10k (:; (:;0), O), and so on respectively. We call the models trained with GOLF (see Sec- GGLLF (-;0), pGOLF-10K (-; etc.
mately 10000 triplets of the form {s, EPPT FDFT).
prajj 11k
tion
To evaluate the quality of optimization with NNPs, we use a fixed subset of the nablaDFT dataset Drest that shares no molecules with Do. For each conformation s E Diest we perform the opti- mization with the Og to get the ground truth optimal conformation Sopt and its energy EDPT The quality of the NNP-optimization for 8t 80...., is evaluated with the percentage of minimized energy:
EDIT DFFT
pct(s:) = 100%
80
(4)
E
DFT
Sopt
By aggregating pct(s;) over Diea. we get the average percentage of minimized energy step
pct.
pet(s:);
Dresa
sED
Another metric is the residual energy in state s;: (sts It calculated as the delta between EDFT and the optimal energy:
5,
(s)) = D.
EDFT
(6)
Similar to pct,, this metric can also be aggregated over the evaluation dataset:
Fres
Σ
E (8t).
(7)
Deest
DE
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10
10 1
10
Energy MSE, Hartree'
Forces MSE. Hartree!
10
10
10
10
( - 10c
M -1k
1000
500k
10
10"
10 Evaluation step
10a
10a
10 Evaluation step
104
Figure 1: Mean squared error (MSE) of energy and forces prediction for NNPs trained on Do, Dusj-1uk. D Duaj- 100k Sooj--S0kk- To compute the MSE, we collect NNP-optimization trajec- tories of length T 100 and caleulate the ground truth energies and forces steps t 1,2, 3, 5, 8, 13, 21, 30, 50, 75, 100. Solid lines indicate the median MSE, and the shaded regions indicate the 10th and the 90th percentiles. Both the x-axis and y-axis are log scaled
Generally accepted chemical precision kcal/mol (Helgaker al. 2004). Thus, another impor- tant metric is the percentage of conformations for which the residual energy is less than chemical precision. We consider optimizations with such residual energies successful:
suteuss
Σ
([ET < 1].
DDest
D
CONFORMATION OPTIMIZATION WITH NEURAL NETWORKS
Energy prediction models such as SchNet, DimeNet, and PaiNN can achieve near-perfect quality on tasks of energy and interatomic forces prediction when trained on the datasets of molecular confor-
mations (Schütt et al. 2017 Gasteiger et al.] 2020 Schütt et al. 1 Ying et al. 202 1 Shuaibi
et al., 2021: Gasteiger et al. 2021 Batzner et al. 2022 Musaelian et al.) 2022). In theory, the gradients of these models mization, replacing the computationally expensive physical simulator. However, in our experiments (see Section(6). 6). this scheme often leads suboptimal performance in terms of the potential energy of the resulting conformations. We attribute this effect to the distribution shift that naturally occurs during the optimization: As most existing datasets (Isert al. 2022 Khrabrov al., 2022 East- man et al. 2023) Nakata & Maeda 2023) do not contain conformations sampled from optimization trajectories, the accuracy prediction deteriorates the conformation changes along the optimiza- tion process. The lack of such conformations in the training can result in either divergence (initial potential energy is lower than the final potential energy) the optimization or convergence to conformation with higher final potential energy than the optimization with the oracle.
be utilized by external optimizer perform conformational opti-
To alleviate the distribution shift's effect, we propose enriching the training dataset for NNPs with the ground truth optimization trajectories obtained from the Og To illustrate the effectiveness of our approach, we conduct series of experiments. First, we train a baseline model pbaseline (.: O) on fixed subset D, of small molecules from the nablaDFT dataset. The Do ((2pl 10000) contains conformations for 4 000 molecules, with sizes ranging from 17 to 35 atoms, and the average size of 32.6. Then we train NNPs fuaj (): on enriched datasets Duuj-10k. Deraj Dunj-Sook, containing approximately 10 10a 5 103 additional conformations respectively. The additional data consists of ground truth optimization trajectories obtained from the Og Then, we evaluate the NNPs by performing the NNP-optimization on all conformations in D ([Dreal R3 20000, contains Rs 10 000 molecules) and calculating the MSE between ground truth and predicted energies and forces. We use the L-BFGS as Opt due to its superior performance compared to other optimizers (see Appendix B). We run the optimization with an NNP for a fixed number of steps observe that this number is sufficient for the optimization to converge (see Figure
(+;
100 as we
3 Figure
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Table Optimization metries for NNPs trained on enriched datasets
NNP
phaseline
1okj-1Ok
firi 100k
firij-500k
pct f
77.9 + 21.3
95.1 + 7.6
96.2
8.6
98.8 + 7.6
(kss(kcal/mo)) +
8.6
2.0
1.5
0.5
success (%)
8.2
37.0
52.7
73.4
illustrates the effect of the distribution shift fraseline (+; 0) (the prediction enror increases as the optimization progresses) and its gradual alleviation with the addition of new training data.
Table presents optimization metrics pCtt, F potsueesss for T 100. Note that the potential energy surfaces of molecules often contain a large number of local minimas (T'ai & Jordan 1993). Due to this fact and the noise in the predicted forces, the ANP-optimization can converge to bet- ter local minimum than the Og resulting in the optimization percentage greater than a hundred: pCt(sT) > 100% (see Appendix and the violin plots in Figure 2 We say that the NNP matches the optimization quality of Og its average residual energy E E is less than the chemical precision. Table 1 shows that it takes approximately 5 10 additional oracle interactions to match the optimization quality of the Og However, takes on average 590 CPU-seconds to perform a single DFT calculation for conforma- tion from Do with the LuB97X-D/def2-3 D/d level of theory (R) Gold 2.60Hz CPU-cores (assuming there are 240 parallel workers each using four threads). This amounts to approximately 9.36 CPU-years of compute for x 10 additional conformations.
for examples). This explains the range of values in Table
our cluster with total of 960 Intel(R)
GOLF
Motivated by the desire to reduce the amount of additional data (and compute) required to match the optimization quality of the OG we propose the GOLF. Following the idea of Active Learn- ing, we want to enrich the training dataset with conformations where the NNP's prediction quality deteriorates. We propose select such conformations by identifying pairs of consecutive confor- mations 8t, 8141 in NNP-optimization trajectories, for which the potential energy does not decrease: E S This type of eтror indicates that the NNP poorly predicts forces in 8t, so we add this conformation to the training dataset.
Algorithm GOLF
Require: training dataset Do, genuine oracle OG surrogate oracle Os optimizer Opt, optimiza-
tion rate a, NNP [(; 0), number of additional Og interactions K, timelimit T, update-to-data
ratio U
1: Initialize the NNP (-: 0) with the weights of the baseline NNP model
2: Set D + Copy (Po). set
3: Sample D, and calculate its energy with Os Epree MMFF
4: repeat
5: s' ++ (Fpt(F(s;
Get next conformation using NNP
6: Calculate new energy with the Os: Eeur
MMFF
7: if Fear > Eprreo or f. 2.2 T then
> Incorrect forces predicted in or T reached
8:
Calculate EDFT .DDFT Og (s)
D add {s, D.FT _FPFT]
9:
> Add new data to D
10:
Train (::00 )) on D using Eq. U times
11:
Set l +-
12:
Sample 8 ~ D, and calculate its energy with Os: Eprev f- MMFF
else
14:
sts'
Epreo t- Eeur t. 1
16:
end if
18:
until
[D] 1 < K

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However, this scheme requires estimating the energy for all conformations in generated NNP- optimization trajectories, which makes it computationally intractable. To cope with that, we employ computationally inexpensive surrogate oracle Og to determine which conformations evaluate with the Og and add to the training set. Although the energy estimation provided by the Os is less accurate, such simplification allows to efficiently collect the additional training data and success- fully train the NNPs. We chose the RDKit's ' (Landrum et al. (2022) MMFF (Halgren) 1996) as the Og due its efficiency. In our experiments, takes 120 microseconds on average on a single CPU core to evaluate a single conformation with MMFF, which is about 5 x 10 times faster than the average DFT calculation time.
Algorithm D describes the GOLF training procedure. We start with an NNP f(:: pretrained on the Doo We calculate a new optimization trajectory on every iteration using forces from the current NNP and choose conformation from this trajectory to extend the training set. Then, we update the NNP on batches sampled from the extended training set D. This approach helps the NNP learn the conformational space by gradually descending towards minimal conformations.
6 EXPERIMENTS
We evaluate NNPs and bascline models on a subset of nablaDFT Diea: (Desl = 19477, containing conformations for 10273 molecules. The evaluation dataset D Die shares molecules with either Do or additional training data. We use PaiNN (Schiitt et al.] 2021) for all NNP experiments. First, we train a baseline NNP graseline (+: on Du for x 10 training steps. To train (. 0) we first initialize the weights of the network with pbaseline (-: 0) and then train it on the corresponding dataset (D 10k Punji 100k D my sook) concatenated with Do for additional 5 10a training steps. The only exception is the ftrjj-500k (-; 0), which is trained for 10 training steps due a larger dataset.
To train the pGOLF (-: 0) models, adjust the update-to-data ratio U' to keep the total number of updates equal to 5 10S. For example, if K is set to 10a, we perform U 50 updates for each additional conformation collected (see line 10 of Algorithm [ The Algorithm | describes a non-parallel version of GOLF with a single OG To parallelize the OG calculations (line 8
select the total number of additional Og interactions K and
we use batched version of the Algorithm [ where a batch of NNN optimiiiinn trajectories is generated and then processed by large number of parallel DFT
oracles.
To evaluate NNPs, we use them to generate optimization trajectories S0...., ST. T = 100 for all s E Diea- We then calculate EDFT steps f {1, 2, 5, 8, 13, 21, 30, 50, 75, 100), as calculating in every step computationally expensive. Having calculated EDFT for all s E Diea. we can compute pct(sT), (s)), s E Dies along with pet,, Frre,, sacesss In all our experiments, we use the I.-BFGS as Opt. except for Appendix B, where we test the effect of different external optimizers on the model's performance. We run the optimization with an NNP for fixed number of steps T= 100 as we observe that this number is sufficient for the optimization converge (see Figure 3). We report the optimization quality of RDKit's MMFF non-neural baseline. If EDFT > EDIT we say that the optimization has diverged and do not take such conformations into account when computing pct,, E , pct We denote the percentage diverged optimizations as pcl.v- We also report well-known metrics COV and MAT (Xu et al. (021). More information on these metrics can be found in Appendix F We present all metrics in Table 2
6.1 GENERATIVE BASELINES
To compare approach with other NN-based methods, we adapt ConfOpt (Guan et al. 2021), Torsional diffusion (TD) (Jing et al. 2022), and Uni-Mol+ (Lu al. 2023) for the task of conformational optimization. The training dataset is composed a single conformation for each 4000 molecules Dp. first optimize geometry for each conformation with Og and then train the generative models to map initial conformations to final conformations from corresponding optimization trajectories. Table 2 reports the best metries for each model type. Refer to Appendix G for an in-depth discussion of results. The training details and metrics for all the variants of the models are also reported in Appendix G
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Table 2: Optimization and recall-based metrics. We set 5 = 0.5A bold for the best value in each column.
when computing the COV. We
Methods
(C)t
(%
piccmma)))
pct success (%)
COV(%))
MAT (Å)
RDKit
85.5 + 8.8
0.6
5.5
4.1
54.9
0.61
TD
23.8 + 19.8
61.4
33.8
0.0
10.0
1.42
ConfOpt
39.1 + 22.8
71.1
27.9
0.2
25.0
1.13
Uni-Mol+
54.6 + 20.4
8.1
18.6
0.2
56.3
0.53
fhoseline
77.9 + 21.3
7.5
8.6
8.2
58.8
0.55
frikit
93.0 + 11.6
4.4
2.8
35.4
63.8
0.51
fing
95.1 7.6
4.5
2.0
37.0
63.3
0.52
flny-10ck
96.2 + 8.6
2.8
1.5
52.7
65.6
0,49
finy SO0k
98.8 + 7.6
2.0
0.5
73.4
67.0
0.48
GOLLF-IK
97.3 + 5.1
3.9
1.1
62.9
71.0
0.42
GGLFF.
98.8 + 5.0
3.0
0.5
77.3
71.2
0.42
105
105
100
100
% of optimized energy
% of optimized energy
95
95
90
pal-11t mt
mal Apol
proi 2008
Annior
marl keot
pose
Rau
(a) Distribution (Ct( NNPs on nablaDFT
(b) Distribution pct(sT) ) NNPs on SPICE
Figure 2: Violin plots of the percentage of optimized energy pct(sT) calculated for various NNPs on Dres and DSPICE Blue marks denote the mean percentage of optimized energy pctz, the 10th, and the 90th quantile.
test
6.2 NNPS TRAINED ON NABLADFT DATASET
To iluuttraee performance various NNPs trained on molecules from the nablaDFT dataset (Khrabrov et al. 2022). we plot the distribution of pct(#r) using violin plot (see Figure 2ap. To highlight the data efficiency the proposed GOLF framework, we report (GOLF-IK(;; (:;)) 0),, as well as our primary model /GOLF-10k (.: 0). To demonstrate the significance of our proposed data-collecting scheme, we compare the NNPs trained with GOLF against an NNP trained on Dana {sopt D which is composed of optimal conformations obtained by the Os
As shown in Figure 24 and Table 2 the NNPs benefit from additional training data and outperform the baseline in terms of all optimization metrics. The pct., and puc.useess gradually increase with the amount of (:; additional training data both for (.; 0) and ,GOLF. (.; 0) models. However, the NNPs trained with GOLF require significantly less additional training data: (GoLF-ik (:;0) outperforms pumj-110k (.; 0), while using 100 times less data; our main model, GGLF--10 9) outperforms junj (:; in terms of pet while using 50 times less data. NNPs trained with GOL.F also outperform pruiit 0), which shows the importance of enriching the dataset with conformations based the proposed Active L.aaming-insppred data collecting scheme.
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6.3 NNPs TRAINED ON SPICE DATASET
To demonstrate the generalization ability of our approach. we perform a similar set of experiments on another diverse dataset of small molecules called SPICE (Eastman et al. (2023). Namely, we select a subset Do (sec Appendix [ for detailed description) from the SPICE dataset to be roughly the same size Do and trained a baseline model SSPCEE (:; 0). We then use the same DFT-based oracle Og to get ground truth optimization trajectories and obtain enriched training datasets D D.PPCT D Finally. we train JSICE( (::0) 0) mod- els SSPCCE 10 (:: 0) model. All models are evaluated on D dataset ((D
17724) that shares Figure 2b and Table 3 It should be noted that the hyperparame-
molecules with D The results used these experiments were not specifically optimized the SPICE dataset, suggesting potential for further improvements the metrics with tailored adjustments.
Table 3: Optimization metries for NNPs trained on DSPICE
NNP
ftaa--1Ok
fura--100k
GGOLF-IK
p (%)
90.4 + 12.0
93.4
10.0
94.3
9.4
93.9
9.6
94.2 + 8.9
dit (%) 1
4.7
6.8
2.4
2.4
3.2
Tes (kcal/mol) /
3.6
2.4
2.1
2.3
2.1
pct (%)
19.7
37.4
44.2
41.6
40.9
6.4 LARGE MOLECULES
Finally, test the ability of our models generalize to unseen molecules of bigger size. To do that, collect a dataset DLM (LM for Large Molecules) of 2000 molecules from the nablaDFT dataset. Sizes of molecules in DIM range from atoms atoms with an average size of atoms.
Table 4: Optimization metries for NNPs trained on Do
NNP
chaseline
(traj 500k
GOLF 10k
pCt, (%) f
77.7 + 19.7
97.4 + 6.7
97.7 + 4.1
dic (%)
5.1
1.9
2.7
E (Kcal//mll) +
9.6
1.1
1.0
pct (%)
4.8
58.2
61.4
(-; 0) matches the quality of ground truth optimization 7 < 1), the only downside being a lower pct,,ueesss compared to results in Table [ We hypothesize that this percentage can be increased by adding small amount of larger molecules to Do but leave this for future work.
As can be seen in Table 4 the GOLF-100
CONCLUSION
In this work, have presented new framework called GOLF for molecular conformation optimization learn- ing. We show that additional information from physical simulator can help NNPs overcome the distribution shift and incrcase their quality on energy prediction and optimization tasks. We thoroughly compare our ap- proach with several baselines, including recent conformation generation models and an inexpensive physical simulator. Using GOLF, we achieve stale-of-the-art performance the optimization task while reducing the number of additional interactions with the physical simulator by factor of 50 compared the naive approach. methods" optimization quality on diverse set drug- like molecules.
The resulting model matches the In addition, we find that our models generalize to bigger molecules unseen during training. We consider the following two directions for future work. First, we plan to adopt the proposed approach for molecular dynam- ies simulations. Second, we plan account for molecular environments such a solvent protein binding pocket.

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ACKNOWLEDGMENTS
The work was supported by grant for research centers in the field of artificial intelligence, provided by the Analytical Center in accordance with the subsidy agreement (agreement identifier 000000D7303311S50000 and the agreement with Ivannikov Institute System Programming dated November 2021 No. 70- 2021-00142.
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A EXPERIMENTAL SETUP
Our implementation of GOLF is based Schnetpack2.0 (Schitt et al. 2023 Namely, Schnetpack2 0's implementation carried out on a cluster with Nvidia Tesla V100 and 960 Intel(R) Xeon(R) Gold 2.60Hz CPU-cores, and the total computational cost is = 80 CPU-ycars and x 1900 GPU-hours.
PaiNN
data processing pipeline. All the experiments were
To train fGOLF (+; 0). we use a batched version Algorithm | that simultaneoslly generates several NNP- optimization trajectories with the same NNP and calculates energies and forces using "number of parallel Og DFT-workers running parallel. We use a smaller value "number parallel Og "= 48 for fGoLF-I 0) reduce the number correlated samples in the replay buffer. To prevent the biasing newly collected conformations, we sample 10% of cach mini-hatch from the initial training dataset Do during training.
model towards
We list all the hyperparameters in Table 5
molecules, do not employ When evaluating the NNPs the terminate the optimization trajectory and instead use fixed timelimit Tova
Table 5: Hyperparameter values for GOLF-10k.
GOLF-10k
NNP hyperparameters
Backbone
PaiNN
Number of interaction layers
3
Cutoff radius
5.0 Á
Number radial basis functions
50
Hidden size basis)
128
Training hyperparameters
Number of parallel OG
120
Batch size
64
Optimizer
Adam ConcanAn
Learning rate scheduler
Initial learning rate
1 x 10
Final learning rate
x 10-7
Gradient clipping value
1.0
Weight coefficient p
1 10
Total number training steps
5 x 10
Number of additional GO interactions K
10000
Update-to-data ratio U
.50
Timelimit T Train
100
Timelimit Tema
100
Conformation optimizer hyperparameters
Conformation optimizer
L-BFGS
Optimization rate
1.0
Max number of iterations in the inner cycle
5
B EXTERNAL OPTIMIZERS
The external optimizer Opt is a crucial component of GOLF, as it generates the NNP-optimization trajectories from which sample the additional training data. To test the effect of the external optimizer on the training and the evaluation of NNPs, we conduct series of experiments with SGD (Robbins & Monro 1951) with mo- mentum, Adam (Kingma & Ba 2014), and I.-BFGS (Liu & Nocedal 1989). We the optimizer for the
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100
divergee conformtiions
% of aptimized energy
80
(COLT-L0rGS
(GOL/-APAT
60
FOOL LSPGS
2
yoo AAam
10OL5 550 momerom
40
102
Evaluation stop
Evaluation step
Figure 3: pct, and pet 1, 1,2,3,5,8, 13, 21, 30, 75, 100. Shaded regions indicate the 10th and the 90th percentiles of the pct(s+), 8 E Drea distribution. The x-axis is log-scaled.
Table Hyperparameter values GOLF with different external optimizers.
GOLF-IBFGGS
GOLF-Adam
GOL.F-GGD-
Training hyperparameters
Total number of training steps
2 x 10
2 10
2 10
Update-to-data ratio U
20
20
Timelimit 7 (training)
100
200
200
Timelimit 7 (evaluation)
100
500
500
Conformation optimizer hyperparameters
Conformation optimizer
L.BFGS
Adam
SGD 5 10
Optimization rate
1.0
5 10-3
Max number of iterations in the inner cycle
Momentum
0.9
training and the evaluation of NNPs. We dub the resulting models as GGOLF- 10k SOL----OO-GD/// (-: O). JOLF- 10k Adam (:: 0) and
jooLF-io-LBBrS (-; 0) respectively. the pytorch implementation L-BFGS includes inner cycle with up 5 (empirically chosen hyperparameter) NNP evaluations,
f GGOLF-I0k-Alam 0) and / (GOLF-10-SGD
for 500 steps instead of 100 for GGOLF-IO 10k LBRGS (-; 0). We train such models for 2 10 training steps instead
of 5 x 10 to save computational resources. We provide training hyperparameters for & (.; 0) with different external optimizers in Table 6 and omit hyperparameters identical those in Table 5 Such
ber of training steps is enough show the superiority of the L-BFGS external optimizer compared other optimizers.
As it can be seen in Figure GooLF 10k LBEGS (-: 0) outperforms other optimizers in terms of pct.y. However, COLFF-Il-Aaan 10k-Admm (-: (=:0) performs better in terms of pct..- We hypothesize that GGOLF- (-; 0) can be tuned to match the optimization quality yGOLF-10kLBBGS (-; 0), while retaining close-to-zero petages but leave this for future work.
C MSE FOR GOLF
To further show that GGOLF-10K (.; 0) and 10 (+: 0) perform similarly, we evaluate the prediction quality of GOLF-10L
0) along the NNP-generated trajectories and plot the MSE for predicted energies and forces in Figure
D NABLADFT DATASET
Throughout this work, several subsets of nablaDFT (Khrabrov et al. 2022) dataset. The nablaDFT datasct is based the Molecular Sets (MOSES) dataset, which is diverse subset of the ZINC dataset, con-
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10
10 1
10
Energy MSE, Hartree'
Forces MSE. Hartreel
10
10-5
10
10
GOLF
GOLF-11K
10
10"
10 Evaluation step
10a
10a
10
104
Evaluation step
Figure 4: Mean squared error (MSE) of energy and forces prediction for NNPs trained on Do and s0nj- and NNP trained with GOLF. To compute the MSE, collect NNP-optimization trajectories of length T = 100 and calculate the ground truth energies and forces in steps I == 1,2, 3, 8, 13, 21, 30, 50, 75, 100. Solid lines indicate the median MSE, and the shaded gions indicate the 10th and the 90th percentiles. Both the x-axis and y-axis are log scaled
taining approximately one million drug-like molecules with atoms N. S. O. CL Br. and For each molecule from the dataset, the authors ran the conformation generation method Wang et al. (200)) from the RDKit software] SaraareeLaadde Landrum et al. (2022). Next, they clustered the resulting conformations with the Butina clus- tering method [Barnard & Downs (1992.. Lastly, they selected the smallest number of clusters that cover least 95% of conformations and used their centroids as set conformations for given molecule. This pro- cedure has resulted in to unique conformations for each molecule, with 340 152 total conformations in the full dataset. Finally, these conformations were evaluated with DFT-based oracle. The baselines and GOLF models are trained on the train set Do: subset of nablaDFT, which contains 4 000 molecules and zs 1=00000 conformations 2.5 conformations per molecule). The test set Dise contains = 10 000 different molecules and 19 447 conformations. Optimization trajectories for f (:; were obtained with a DFT-based oracle by opti- mizing conformations from the train The average length of trajectory is z 100 steps. Finally, generative baselines trained to map conformations from Do to their optimal counterparts.
E SPICE DATASET
Another dataset that is used in work SPICE (Fastman al. 2023). It is subset of the PuhChem dataset (Kim et al., 2023) and contains a diverse set of drug-like molecules. The total number molecules in SPICE is 14644. The dataset contains 25 high-energy conformations and 25 low-energy near-optimal con- formations per molecule. Molecules contain the following atoms: C. N. S. O, F, Cl. Br. I. P. and H. To cross-validate models trained SPICE and nablaDFT, filtered molecules containing and P atoms. This procedure resulted 13231 filtered molecules. To make training setup consistent with nablaDFT, we selected zs 3500 molecules and ze 9500 conformations for the SPICE training set D& The training set only contains the high-energy conformations, as we observed that training on near-optimal conformations leads instabilities. The test set D includes z 7000 molecules and z 18000 conformations. The test set contains both high-energy and low-energy conformations in equal parts. Note that initially, DSPICE supposed to match the size of Dien but the DFT-based optimization for some molecules did not converge, so we excluded them from the test set. Similar to Section optimization trajectories were obtained with a DFT-based oracle by optimizing conformations from PPPIEE The only difference is that we used optimization in spher- ical coordinates instead of Cartesian. The change of coordinates resulted in shorter optimization trajectories (around steps on average). The biggest trajectories dataset for SPICE DSPICE thus only contains 200 conformations.
F DISTRIBUTION MATCHING METRICS
Consider the evaluation of the NNP on the dataset Dies. Let S = (ST).ED... denote the set of all final conformations in the NNP-optimization trajectories. and S, { J Dia denote the set of all ground truth optimal conformations obtained by the GO. To measure the difference between S and S,, we the Get Best RMSD in the RDKit package and denote the root-mean-squaee deviation RMSD(s, 8). The
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recall-based coverage and matching scores defined follows:
M E 8r (MSD(s, 8) < < E s,)l:
[+p
MAT (.,,
min RMSD(s, 3).
[
COV is number conformations in 3, that are "reasonably" close (RMSD 6) to some conformation from S,. MAT is the average over all s E S, RMSD to the closest conformation from Sy- Note that both COV and MAT are not ideal metrics for the optimization task because they do not consider the energy of the final conformation.
G GENERATIVE BASELINES
Table 7: Energy and Recall-based scores. We set = 0.5A when computing the COV.
pctai (%)
COV(%))
MAT (A)
Methods
Mean
Mean
TD
24.04 + 21.3
54.1
12.53
1.284
ConfOpt
33.36 + 22.0
92.5
24.08
1.004
Uni-Mol+
13.49
1.25
TDpr
25.63 + 21.4
46.9
11.25
1.33
ConfOptpr
36.48 + 23.0
84.5
19.88
1.05
Uni-Mol+pr
69.9 + 23.1
23.2
15.29
1.23
Uni-Mol+wntt
54.92 : 20.5
8.0
63.41
0.44
Uni-Mollpr+inn
62.20 + 17.2
2.8
68.79
0.407
The energy-based metrics for the Uni-Mol+ model are not reported due to the problems with energy compu- tation.
In this section, we provide additional details considering the training of generative baselines and the corre- sponding metrics (see Table We consider three architectures designed for conformation generation (Energy- inspired molecular conformational optimization (ConfOpt) (Guan et al.) 2021). Torsional diffusion (TD) (Jing et al. 2022), and Uni-Mol+ (Lu et al. ) and adapt them to the task of geometry optimization. For the first two models, we follow the same setup proposed in the corresponding papers and train models to generate optimal conformations from the ones generated by RDK In the case of Uni-Mol+, we compare two setups: i) the model trained to generate optimal conformations conditioned on geometries from RDKit; ii) the model trained to generate optimal confornations conditioned on non-optimal conformations from DFb add a subscript init in the latter case. Moreover, we also experiment with starting the training with randomly ini- tialized weights and pretrained checkpoints. We use checkpoint obtained on the PCQM4MV2 dataset (Nakata) & Maeda 2023) the case of Uni-Mola and the GEOM-DRUGGE dataset (xelrod & nomez-Bombarel 2022 otherwise. We add a subscript pr for pre-trained models.
To save computational resources, all the models from Table 7 were evaluated on a subset of Dis that we call Diss" ([Dise") = 2044). Our findings are as follows: ConlOpt and TD models perform much worse on the energy optimization task in our setup than the tasks reported in the corresponding papers. Uni-Mol+ performs on par with NNP baselines but worse than the models trained on additional data. We suspect that the reason such behavior TD is the small amount of data and the necessity to model a discrete distribution over the optimal geometries instead of the whole conformational space. The TD authors also report that the resulting conformations differ by a large margin in terms of energies and other quantum chemical properties from the reference conformations and require additional optimization with the simulator. We hypothesize that in the of ConfOpt, the main problems the choice of architecture and the fact that the model generates optimal conformations from SMILES and does not use initial geometries.
In Table we observe that i) all generative baselines benefit in terms of pct., from using pre- pre-trained weights. Even though the pre- pre-training was done using data generated by DFT DFT-based methods with different levels of the- ory than in nablaDFT; ii) starting from non-optimal conformations from nablaDFT greatly benefits all metrics for Uni-Mol+, indicating that reasonable initial conformation is crucial for generative baselines.
17

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Published as a conference paper at ICLR 2024
we-sin
Figure 5: Visualization of final conformations obtained by various models, the 2D view of the molecule, and the reference optimal conformation obtained with the Og.
H FINAL CONFORMATIONS COMPARISON
To highlight the difference in the quality of conformation optimization, visualize final conformations for ConfOpt, Torsional Diffusion, Uni-Mol+, and our best-performing model (GOLF-10K) with py3Dmol (Rego &
Koes 2013). Figure we provide visualizations molecules from test D also provide 2D visualization of the molecule obtained with RDKit and a visualization of the reference optimal conformation obtained with the Oc.
Molecules I and 3 are an example of the case where the conformation optimization with GOLF-10k converges the same local minima Og the RMSD the reference conformation is close zero, while the pct close to 100%. is also hard to spot any visual differences. the other hand, molecules and illustrate the case where the conformation optimization with GOLF-10k converges to the different local minima: RMSD is larger than zero, but the pct 1o0 is 100% or even greater than 100% in case of the molecule 4. The visual difference between the resulting conformations is prominent.
Negative values for the pct often caused hy distorted distances between atoms in cycles (ConfOpt optimization for molecules 3 and 4). Low positive values of pct,20 generally indicate conformations with correct distances between atoms but incorrect dihedral angles between different parts of the molecule (ConfOpt optimization for molecule 2, Torsional diffusion for molecule 1, Uni-Mot ni-Mol+ optimization for molecule 2).
18


Reviews:
[{"review_id": "C3U2D8w3Hf", "reviewer": "Reviewer_DBvP", "summary": "This paper introduces GOLF, a framework for improving molecular conformation optimization with neural networks. GOLF addresses distribution shift issues, enhancing energy prediction and optimization. It outperforms traditional methods and reduces the need for physical simulator interactions by 50 times.", "questions": "Same with the 'weaknesses' part:\n1. Why not use more datasets besides nablaDFT?\n2. What's the complexity of GOLF? It seems that you only show experiment results on subsets of nablaDFT. Is it because the computational complexity of the method is very high?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The development of the proposed GOLF is clear.\n2. Demonstrated outstanding performance in conformation optimization tasks.\n3. Generalization to larger molecules.", "weaknesses": "1. Dataset Limitation: The paper may be limited by the availability and diversity of datasets used for testing, potentially impacting the generalizability of the results.\n2. Complexity: It seems that the complexity of GOLF is not clearly discussed in the paper.\n3. Practical Implementation: Though the algorithm is not very complicated, this paper does not release code, which leaves me cautious about the practical implementation and complexity of the algorithm.\n\nOverall, while the paper presents valuable contributions, addressing these weaknesses could enhance its overall impact and relevance in the field of molecular conformation optimization.", "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": "", "extra_scores": {}}, {"review_id": "7fDXHfPUSM", "reviewer": "Reviewer_bemc", "summary": "This paper proposes a gradual optimization learning framework (GOLF) for molecular conformation minimization. The framework is designed to improve the training of Neural Network Potentials (NNP). The authors first claim that NNPs trained on existing datasets suboptimal in energy minimization due to the distribution shift and perform experiments to show (a large amount of) additional data from the optimization trajectories can help improve the NNP's performance.  The GOLF framework uses a surrogate oracle (MMFF) to evaluate the conformation energy and expand the training data by selecting the incorrect prediction and re-evaluating with the genuine oracle (DFT), which reduces the required additional data. The experiments on the nablaDFT dataset demonstrate the effectiveness of the proposed method.", "questions": "- Are the conformations in nablaDFT dataset equilibrium ones or the intermediate state sampled from the optimization process? How large is the training set D0? More description about this dataset is needed. \n- ConfOpt and TorsionDiff are designed to generate equilibrium low-energy conformers, and not guaranteed to achieve a lower energy by repeatedly applied. Thus, I think it's unfair to compare these models with GOLF in terms of pct. \n- The statement of \"We hypothesize that in the case of ConfOpt, the main problems are the choice of the architecture and the fact that the model generates optimal conformations from SMILES and does not use initial geometries. \" doesn't make sense to me. ConfOpt takes the 2D molecular graph as input and also utilizes initial 3D conformations. \n- Does $f^{traj-10k /100k }$ keep the total number of updates equal to $5 \\times 10^5$? If so, please provide more training details, otherwise, the comparison between them and $f^{GOLF}$ is unfair.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The proposed method sounds reasonable and the experiments shows its effectiveness.  The method also looks easy to implement, which can improve the conformation energy minimization performance at a small cost.", "weaknesses": "- The writing is not very clear, especially in the introduction section. It takes me a while to understand simply enriching the training dataset is actually a preliminary baseline method the authors want to compare with. Also, lots of experiment details are mixed with the method, which makes the paper not very easy to read. \n- The calculation of COV and MAT looks problematic. It seems the authors optimize **one conformation per molecule** and take them of the entire test set as the generation set. However, in the conformation generation setting, models generate **multiple conformations per molecule** to construct the generation set, and then COV and MAT are calculated per molecule,  and finally the average / median of them on the entire test set is reported.", "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": "", "extra_scores": {}}, {"review_id": "Be7tgWcJ1A", "reviewer": "Reviewer_HQch", "summary": "In the paper, the authors present Gradual Optimization Learning Framework (GOLF), a framework for improving the efficiency of generating low-energy molecular conformation prediction models, a crucial technology used in computer-aided drug discovery and materials design.\n\nOverall, the paper is well written and presents some valuable insights into applying active learning efficiently for the discovery of energy minimized conformations. Traditional approaches, such as Density-functional theory (DFT) models, use high-fidelity physics-based numerical quantum chemistry simulators whose computational costs are exponential with respect to the complexity of the molecule under study. Unfortunately, this limits their applicability to simple molecules with few atoms or electrons. To address computational complexity and to scale conformal optimization to more complex molecules, researchers have explored various alternatives based on lower fidelity linear models and, more recently, neural network models that leverage the availability of computed quantum property molecular databases. Sadly, these alternate approaches lead to inaccurate predictions and suffer from distribution shift. To scale conformal energy prediction to larger molecules while addressing computational cost, the authors propose GOLF, an automated data augmentation scheme and a hybrid computational approach that combines the use of both high and low-fidelity simulators as well as neural networks.\n\nIn Section 1, the authors present the concept of low-energy molecular conformations and their uses. This section is well written and provides an adequate background of the approach and the insights that motivate the solution. The authors indicate that the fundamental problem with traditional approaches, such as DFT, to obtain optimal conformations is their high computational cost. These approaches, which are based on numerical quantum chemistry simulations that calculate anti-gradients representing molecular forces, are iterative by nature and, when given a sufficiently complex molecule or physical system, may fail to complete even a single iteration. For this reason, the authors claim that “reducing the number of interactions with the physical simulator” is crucial for efficiency. The authors then go on to describe current methods that apply Neural Network Potentials (NNP), a form of deep neural networks, to the problem. NNP-based techniques significantly reduce computational complexity by using the gradients inherent to neural networks to model the molecular forces, thereby obviating the need for expensive simulations. Unfortunately, the NNP approach suffers from distribution shift resulting in inaccurate predictions. The authors then introduce GOLF, which employs a data augmentation active learning scheme to improve the diversity of the training dataset, thereby alleviating the distribution shift. By doing so, GOLF achieves energy minimized conformation prediction accuracy comparable with that of high-fidelity simulations while retaining the intrinsic efficiencies gained by using neural networks. A novel framework for data-efficient training of NNPs, GOLF comprises three components: 1) a computationally expensive high-fidelity simulation that is a genuine oracle (GO) used to calculate the ground truth energies and forces; 2) an optimizer that uses the NNP gradients to produce optimization trajectories that constitute the additional training dataset; and 3) a computationally inexpensive low-fidelity simulation that is a surrogate oracle (SO) used to augment the training dataset. Finally, the authors then conclude this section by summarizing their contributions.\n\nHowever, there are various weaknesses in this section as well that can be addressed to improve the quality of the paper. 1) The statement that “reducing the number of interactions with the physical simulator” is unclear and the reviewer assumes the efficiency objective is attained by reducing the number of iterations required to produce optimal low-energy conformations. 2) The authors state that they augment the dataset with “optimization trajectories” without explaining what such a trajectory is and how the trajectories address distribution shift. At a minimum, providing a reference to the discussion of augmented data in Section 5 would be useful. 3) Moreover, as GOLF requires running the high-fidelity simulation, GO, to produce the anti-gradients, it is unclear whether GOLF can be successful when GO fails to complete a single iteration. This is a serious flaw in the paper and needs to be addressed. 4) In addition, the “Ab initio property” phrase is used without a definition or description and seems rather superfluous to the narrative. 5) The adequacy of the requirement for 5 X 105  “additional oracle interactions”, which presumably means optimization trajectories that augment the training dataset, is likely anecdotal based on the molecules selected for the experiments. If that is not the case, an explanation of why it is generally applicable should be articulated.\n\nIn the “Related Works” section (Section 2), the authors describe a variety of contemporary approaches to conformation generation. The benefits and drawbacks of these methods are discussed. However, it is not clear to the reviewer how GOLF addresses the drawbacks of these approaches. Successfully addressing the drawbacks could demonstrate GOLF’s superiority. Also, form the exposition in this section, it is not clear how significantly different the GOLF approach is to the active learning technique presented by Kulichecnko et al. (2023). Finally, the phrase “we believe it is necessary to explore further the ability …” is confusing. Are the authors proposing future work or teeing up the discussion in the remainder of the paper?\n\nIn the “Notations and Preliminaries” section (Section 3), the authors summarize the theoretical foundation of their approach. Although informative, the notation is somewhat cryptic and can benefit from slightly greater verbosity or additional graphics. Also, mentioning GOLF models in this section, without any discussion as to what they are or how they differ from ftraj, seems premature and confusing. At the very least, there should be a forward reference to Section 5 that articulates how GOLF intelligently identifies the datasets that promote diversity, which enhances prediction performance. Moreover, a small discussion of the NNP architecture used in the experimentation would be useful for the sake of completeness.\n\nSection 4 presents “Conformation Optimization”. This section is well written and the both the graphic, Figure 1, and the table, Table 1, provide valuable insight. The Figure 1 graphic clearly depicts the distribution shift, in terns of Mean Square Error (MSE), increasing as the optimization progresses. The graphic also depicts that the prediction accuracy improves – MSE decreases – when augmenting the training dataset with GO produced optimization trajectories. This is an important result but without highlighting it, the reader can easily miss that it is one of the contributions of the paper. Table 1 seems to be highlighting precision of the approach, but the word is not used in the discussion. It is unclear to the reviewer as to the innovativeness of the approach, which may be construed as a weakness. Moreover, using the GO to provide the baseline training dataset may limit the scalability of the approach.\n\nThe authors present a sound argument in Section 5 where they present the GOLF algorithm and discuss using a high-performance low-cost surrogate oracle to make the data generation computationally tractable. The “Experiments” section, Section 6 is reasonably complete though much of the discussion seems anecdotal. The reviewer is unable to determine how many of the experimental results were achieved through a fortuitous selection of the molecules under study. Also, the authors report that the GOLF technique can produce “a high percentage of diverged conformations”. This is not surprising as the training dataset is likely to be much noisier as a result of the choice to use a low-cost simulation. It would be nice to get better characterization of the noise and its effects, including the loss in efficiency resulting from these unusable conformations.\n\nSections 7, 8, and Appendices conclude the paper. An explicit tie back to the goals and contributions identified in the Introduction would be beneficial.", "questions": "Suggestions:\n1) Clean up the use of \"interactions with the physical simulator\" and the \"number of iterations\" in several locations in the paper. They seem to imply the same concept. If they are, just use a single phrase for both.\n2) Additional references early on in the paper to the results later on in the paper will help the reader question many of the seemingly unsupported statements.\n3) A better tie-in to how GOLF addresses the drawbacks of the \"Related Works\" section will improve the quality of  the paper.\n4) Reduce the amount of mathematical notation in the “Notations and Preliminaries” section (Section 3) to simplify the narrative and improve understandability.\n5) The main result in Section 4, the decrease in MSE by augmenting the dataset, should be highlighted and tied-in to the wording of the contributions listed in the introduction section of the paper.\n\nQuestions:\n1) GOLF requires running the high-fidelity simulation, GO, to produce the anti-gradients initial training data. In a different section of the paper, the claim is that for a sufficiently complex system, the physical simulation may not succeed in completing even a single iteration in a reasonable amount of time. Taken together, these two statements seem to suggest that GOLF is atomic complexity scale limited. How do the authors claim to address this apparent limitation?\n2) In the Experiments section (Section 6), how much of the results are related to the selection of the molecules under study? Stated differently, how do the authors plan to address the generalizability of the approach?\n3) The additional data generated for Active Learning are selected based only on errors. Without some sort of approach to balance the introduction of new data, does the approach bias the dataset distribution causing the learned distribution to fail to generalize to other molecules?\n4) How different is the GOLF approach from the active learning technique presented by Kulichecnko et al. (2023).", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, the paper is well written and informative. It seems relevant to improving the computational tractability of conformational energy minimization. The insight to use active learning to address the distribution shift and improve accuracy is valuable. Also, the approach to active learning by using low-cost simulation to augment the training data set without impacting the quality of the subsequent model is somewhat innovative.", "weaknesses": "The various weaknesses are already detailed above. Here I summarize the most important ones. Using GO to generate the baseline dataset may limit the scalability of the approach. It is unclear why the authors do not use the molecular databases they mention in the “Introduction” section to extract the baseline dataset. Some of the discussion is somewhat cryptic and can benefit from some additional discussion or graphics. The results seem anecdotal, tied to the selected dataset and molecules, and thus may not generalize particularly well.", "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": "", "extra_scores": {}}, {"review_id": "23OgQ8yqRK", "reviewer": "Reviewer_RbRE", "summary": "The authors propose an active learning approach to train neural network potentials by employing a cheap surrogate oracle before querying the much more expensive genuine oracle.", "questions": "I would consider raising my score if the authors can address my concerns around baselines and metrics as mentioned in the Weaknesses section.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "The paper is very well written with clear insight and motivation. The method is very neat and the results are promising. I am excited to see more work that follows these mixed genuine and surrogate oracle approach.", "weaknesses": "Additional baselines and metrics could make the result stronger.\n\nIn terms of the baseline, the true innovation of the framework is the \"active learning\" component, and not \"conformer generation\" itself. Therefore, the baseline conversation probably should focus on other active learning approaches such as Kulichenko et al (2023) or Chem et al. (2019). While these methods require OG, the author can still compare to these methods by contrasting the OG query budget to the same amount, but use random selection instead of SG estimates as proposed by GOLF. The comparison to TD/ConfOpt, while interesting, the problem setup and training data requirement are very different from what the authors are trying to demonstrate here.\n\nIn terms of the metrics, it would be very helpful if the authors can provide more context about why \"percentage of minimized energy\" is a meaningful metric, and why >98% is considered solving the optimization. If >98% is broadly considered as solving the optimization, can the authors report what percentage of targets in the test set is \"solved\" under different experiment setup?", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)