Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
Authors: Ankit Ghosh
Gargee Kashyap
Sarthak Mittal
Nupur Jain
Raghavan B Sunoj
Abir De
Venue: ICML 2025 poster (2025)
Keywords: Yield Prediction
GNN
Atom Mapping
Primary area: applications->chemistry_physics_and_earth_sciences
TL;DR: Learns yield by approximating atom mapping and then a condensed graph of reaction (CGR), where CGR serves as a surrogate of the transition state.

Abstract: Yield of chemical reactions generally depends on the activation barrier, i.e., the energy difference between the reactant and the transition state. Computing the transition state from the reactant and product graphs requires prior knowledge of the correct node alignment (i.e., atom mapping), which is not available in yield prediction datasets.  In this work, we propose YieldNet, a neural yield prediction model, which tackles these challenges.  Here, we first  approximate the atom mapping between the reactants and products using a differentiable node alignment network. We then use this approximate atom mapping to obtain a noisy realization of the condensed graph of reaction (CGR),  which is a supergraph encompassing both the reactants and products. This CGR  serves as a surrogate for the transition state graph structure. The CGR embeddings of different steps in a multi-step reaction are then passed into a transformer-guided reaction path encoder.
Our experiments  show that YieldNet can predict the yield more accurately than the baselines. Furthermore, the model is trained only under the distant supervision of yield values, without requiring fine-grained supervision of atom mapping.

Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
Ankit Ghosh"
Gargee Kashyap Sarthak Mittal
Nupur Jain'
Raghavan B Sunoj
Abir De
Abstract
& Sunoj, 2022; Schleinitz al., 2022; Lu & Zhang, 2022) have primarily relied on quantum chemically computed molecular descriptors, molecular fingerprints, or SMILES based representation rather than a more holistic representa- tion built on molecular structure itself. In contrast, recent approaches (Saebi et al., 2021: Gong et al., 2021; Kwon et al., 2022; Li al., 2023) leverage graph neural networks (GNNs) to compute the embedding using molecular graphs.
Yield of chemical reactions generally depends the activation barrier, i.c., the energy differ- ence between the reactant and the transition state. Computing the transition state from the reactant and product graphs requires prior knowledge of the correct node alignment (i.c., atom mapping), which is not available yield prediction datasets. In this work, we propose YEELDNET, neural yield prediction model, which tackles these chal- lenges. Here, we first approximate the atom map- ping between the reactants and products using a differentiable node alignment network. We then use this approximate atom mapping to obtain a noisy realization of the condensed graph of reac- tion (CGR), which a supergraph encompassing both the reactants and products. This CGR serves as a surrogate for the transition state graph struc- ture. The CGR embeddings of different steps in multi-step reaction are then passed into a cransformer-guidee reaction path encoder. Our experiments show that YIELDNET can predict the yield more accurately than the baselines. Further- more, the model is trained only under the distant supervision of yield values, without requiring fine- grained supervision of atom mapping.
Yield of a reaction depends strongly on the activation energy. through Arrhenius equation (Arrhenius, 1889), which is as- sociated with the potential energy of a transient chemical entity called the 'transition state'. is possible to infer the transition state graph from the reactant and product graphs if we know the atom mapping. i.e., the correspondence be- tween the atoms of the reactants and the products (Kim et al., 2024). But most existing yield prediction datasets only include the reactants and products, lacking both tran- sition states and atom mappings. Furthermore, the compu- tation of the true atom mapping from the graph structures of reactants and products is a challenging task. Atom map- ping computation involves solving various pairwise graph matching tasks, which are NP hard problems (Astero & Rousu, 2024). In practice, this is usually mitigated by using expert-curated rules. However, such inputs require manual intervention which could, in turn, limit their widespread de- ployment (Schwaller et al., 2021a). As a result, the existing works on yield prediction do not consider atom mapping or transition states into their model, leaving significant room for improvement.
1. Introduction
The yield of chemical reaction is expressed as the per- centage of conversion of the reactant(s) to product(s). Early prediction of yield before wet-lab validation of reactions can have an immense impact on ML-driven reaction discov- ery. It allows the identification and removal of low-yielding reactions, thereby helping in the design and optimization of chemical synthesis. Previous works on yield prediction (Ah- neman et al., 2018; Nielsen et al., 2018; Zahrt et al., 2019; 9 Sandfort et al., 2020; Schwaller et al., 2020; 2021e; Singh
1.1. Our contributions
We address these challenges by designing novel yield prediction network (YIELDNET). Specifically, make the following contributions.
Differentiable approximation of atom mapping In the context of chemical reaction, atom mapping refers to an alignment map between the reactant and product nodes, which ensures that the atom composition is preserved. Such an alignment can be obtained by solving graph match- ing task between the reactants and the products (Körner & Apostolakis, 2008; Astero & Rousu, 2024). However, this necessitates exploration of vast permutation space,
Equal contribution 'Department of Chemistry, IIT Bombay *Department of Computer Science and Engincering, IIT Bombay. Correspondence to: Ankit Ghosh <ankitghosh@iing in...........
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
which is daunting. To address this challenge, we introduce fully differentiable node alignment network built the Gumbel-Sinkhomn iterative procedure. This network takes GNN embeddings of the reactants and products as input and outputs an alignment (doubly stochastic or soft permutation) matrix which approximates the atom mapping.
designing ML models for atom mapping (Schwaller et al., 2021a; Nugmanov al., 2022; Astero & Rousu, 2024) and transition state (Pattanaik et al., 2020; Jackson et al., 2021; Makos et al., 2021; Choi, 2023; Duan et al., 2023; Kim et al., 2024; Duan et al., 2025). However, the problem setup of these works is very different from ours. For in- stance, the aforementioned works on transition states focus only on predicting the transition states but not yield. To do so, they use the fine-grained supervision of ground truth transition states for training. Similarly, the goal of Nug- manov et al. (2022); Astero & Rousu (2024) is to predict the atom mapping, after training under the supervision of fine- fine-graing -neeggainedd ground truth atom mapping. In another work on atom mapping, Schwaller et al. (2021a) employed a large transformer over SMILES representations. In contrast the above works, our key goal is to predict the yield. Hence, we train our network on datasets containing only yield values, not any ground truth atom mapping or transition states.
Approximate condensed graph of reaction We use the atom mapping to compute a condensed graph of reaction (CGR) (Varnek al., 2005; Nugmanov et al., 2019), as supergraph containing both reactants and products. Owing to such construction, the CGR serves as a surrogate of the transition state. Subsequently, we feed this CGR into a second GNN to compute its embeddings.
Since the atom mapping is approximated by a neural net- work, the CGR becomes a trainable weighted graph. There- fore, this second GNN must allow differentiation with re- spect to the input CGR, which is not supported in standard off-the-shelf GNN models. To address this, we design a GNN that relies exclusively on differentiable operations on the input adjacency matrix, enabling end-to-end training using the CGR.
Our work is also related to graph neural networks (GNNs), other applications of ML in chemistry, and attention mecha- nisms in different domains. We briefly review each of them as follows.
Neural reaction encoder reaction may involve multiple sequential steps, often referred to as reaction path. Using the computed representations of the CGR for each step in hand, we compute the representation for the entire reaction path. For each elementary step in a multi-step reaction, we concatenate the embeddings of the CGR with reaction- step encodings, akin to positional encodings. Finally, these embeddings are fed into transformer, and the output from the transformer is then used to compute the reaction yield.
Representation learning for graphs Graphs are structured objects that are different from images and texts. They need specialized representation learning methods to compute the graph embeddings. These graph representation learning methods can be broadly divided into two categorics. The first set of works consists of transductive models, where the node embeddings are computed independently of each other (Grover & Leskovec, 2016; Perozzi et al., 2014). This requires us to train O(IV)) embeddings separately for each node in a graph. The second set of works consists of induc- tive models, referred to as graph neural networks or GNNs (Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Veličković et al., 2018; 8; Zhang & Chen, 2018), that employ message passing neural networks with shared pa- rameters. At the outset, their goal is collect information from the neighborhoods at different distances from node and cast it into a low dimensional representation vector. Such methods are widely used in link prediction (Zhang & Chen, 2018), node classification (Kipf & Welling, 2017), graph matching (Li et al., 2019b; Bai et al., 2019) etc.
Learning under distant supervision The key goal of YIELDNET is to predict reaction yield. While doing so, it learns an approximate atom mapping using our node align- ment network. Notably, YIELDNET is trained solely under the distant supervision of the yield values, without any fine grained supervision of ground truth atom mapping or CGR. Since transition state plays an important role in reaction yield, our approach of using the continuous approximation of the CGR as surrogate for transition state from the dif- ferentiable atom mapping, can enhance the inductive bias of the model.
Our experimental evaluation across multiple datasets show that YIELDNET is able to outperform several baselines by a significant margin. Furthermore, we observe that YIELD- NET can effectively approximate atom mapping under the supervision of only reaction yields.
ML applications in chemistry In recent years, there has been surge of work that uses machine learning in wide variety of chemical applications (Hirohara et al., 2018; 8; Bjerrum, 2017: Liu et al., 2018; Huang et al., 2020; Honda et al., 2019; Wang et al., 2019c; Chithrananda et al., 2020). Several works have represented molecules as a cus- tomized string called SMILES (Anderson et al., 1987) and then applied CNNs (Hirohara et al., 2018) or sequence en- coders (Bjerrum, 2017; Liu et al., 2018; Huang et al., 2020; Honda et al., 2019; Wang et al., 2019c; Chithrananda et al.,
2. Related work
Apart from the aforementioned work on the yield predic- tion, in recent years, there has been an increasing interest in
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
2020). However, the same molecule can result in multiple SMILES strings, which might affect the expressivity of the trained model. Therefore, wide body of work represents a molecule as a molecular graph and designs graph based machine learning models (Kearnes et al., 2016; Schütt et al., 2018; Qiao et al., 2020; Klicpera et al., 2020b;a; Fuchs et al.,
r = {A B1 + Y1 + Z1,
AA Bn + Yn kn}
(I)
At step i S n, we denote R. = (A,, B.} and I; = (Y; 7 Chemical yield for a reaction can be defined as the observed amount of the desired final product (Y, in Eq. (1)), which is generally less than the theoretical maximum amount of the product given by stoichiometric calculations.
2020; Samanta et al., 2020; Jin et al., 2018; Cao & Kipf,
2018). Such graph based models for molecular graphs can be used for generative modeling (Samanta et al., 2020; Jin et al., 2018; Cao & Kipf, 2018), property predictions (Yang 2022; Liu et al.,
We do incorporate both catalyst and solvent, when present, into the reactant set R. Temperature was excluded as is invariant in the case of wightthroughput experimentation datasets (e.g., DF and NS datasets (Section 5)), which main- tain consistent reaction conditions throughout, or varies mildly for others (e.g., SC). Integrating temperature into the node/edge features might enhance CGR or yield prediction quality.
et al., 2019; Axelrod &
2021; Chithrananda et al., 2020) etc.
Learning aligmment in the context of graphs Graph matching naturally leads to quadratic assignment prob- lem (Anstreicher, 2003; Gold & Rangarajan, 1996; Caetano et al., 2009). Fey et al.: Zanfir & Sminchisescu (2018); Wang et al. (2019b) provide a learning model for training graph matching, under distant supervision. In last few years, there are some works on learning node to node alignment for subgraph matching (Roy et al., 2022b; Ramachandran et al., 2024; Raj et al., 2025), maximum common subgraph computation (Kriege et al., 2019; Roy et al., 2022a), graph edit distance (Jain al., 2024; Li et al., 2019b), graph rerp- resentation learning (Roy et al., 2021).
Activation energy and its relationship with yield In an elementary step reaction R + I, reactants R form transient and unstable chemical entity (shown as TS in Figure 1), before producing I. This chemical en- tity, called as transition state, has the highest poten- tial energy in the reaction, and the energy required to overcome this barrier is the activation energy (AEt).
A lower activation en-
TS
ergy leads to a faster reaction rate and higher yield (Kozuch & Shaik, 2011).
Potential energy
3. Preliminaries
Art
Notations We represent a single-step reaction as A + B + Y + 7, where R = {A, B} are the reactants and / = -Y, are the products. Here, I typically consists of intermediates in multi-step reactions. Each step involves at most two reactants and two products, consistent with our datasets. For any molecular graph G = (VA, EA). we use VA E RR/V xdy and A E R to denote the node and edge feature matrices, respectively. In practice, we build the node feature for u, i.e., : using the atomic number, degree of connectivity, hybridization, etc., and the edge feature for (u, v), i.e., [A[(u, :] using the type of bonds respectively. We denote the reactant and product graphs as Gn (Vn, Ex) and G₁ (Vi, E,), and their adjacency matrices as AdiR and Adiy. respectively. We denote the set of N x N permutation matrices as PN. While P represents both the hard permutation matrix and the doubly stochastic matrix. For a hard permutation matrix P. the alignment map æ satisfies Plu, u'] = π Tuu] [u]=u'. n'.
Atom mapping In a chemical reaction, only the bonds in the reactants R are serranged-broker or newly formed- to produce the prod- ucts I, while the atoms themselves remain conserved. This conservation establishes a correspondence, represented as a permutation ा. V/- - VR mapping the atoms in the prod- uets to those in the reactants. Here, n is the atom mapping.
Reaction path
Figure 1: Reaction path vs po- tential energy
Our goal We are given set of training dataset contain- ing multi-step reactions along with the ground truth yield values, i.e., D = {(ri, yield(r))) (rip), (r))d Any reaction has elementary steps as described in Eq. (I), where each elementary step contains a set of two-dimensional representations of the molecular graphs { RR = (A,, B,), I = (Y,, Z,) during both training and test. Our goal is to develop a yield prediction model that can approximate atom mapping and utilize this map- ping compute surrogate for the transition state, enabling accurate yield prediction for new reaction.
Reaction and their components A reaction path r com- prises n elementary steps, with each step represented as A; + B, + Y; + Z;, where Y; and Z; are intermediates. Inputs A; and B; may be initial reactants, catalysts, or in- termediates from prior steps (Y or Zj. < i). The final product Yn from the n-th step is the reaction's s desired out- put, whose yield is the focus. Thus, r follows a sequential structure:
4. Proposed approach
For brevity of exposition, we first outline a yield prediction model for an ideal setup, where the atom mapping is known.
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
Ga
Gena == GfUGf
(I) We integrate graph neural network (GNN) that com- putes both node embeddings and edge embeddings. (II) We feed these node embeddings into a differentiable node alignment network, Align, to obtain an alignment ma- trix P N P
(III) To ensure the differentiability of the graph structures of CGR, we use the edge embeddings to obtain a smooth approximation of the adjacency matrix of CGR.
AdbooR
P'A
Addy
maxx
Figure 2: Top row: GeGR is computed as the supergraph containing GR and G7 = GG. where G7 is obtained by applying atom- mommiig π on V1. Bottom row: Adjacency matrix of CGR is computed as max (Adjæ: P Adjy P'T), where P is 0/1 permutation matrix corresponding to I, Given GR and G,, π can be obtained by minimizing the number of edges in the supergraph GR U GT: this renders in the maximum overlap between GR and GT.
(IV) We compute the embeddings of CGR using an input- differentiable GNN.
(V) Finally, we combine the embeddings of CGRs of each elementary step using sequence encoder and obtain the yield.
In the following we describe the above steps in details.
Embedding computation using GNN Given a multistep reaction r, each elementary step consists of a reactant graph Gn a product graph G2 and their node and edge features VR, ER and V,, Er, each with N nodes, obtained after padding. We apply a graph neural network GNN,, with pa- rameter & on these graphs to compute their node embeddings H. and edge embeddings M.. as follows:
Then, we introduce YIELDNET, a neural model that first learns approximate atom mapping when its ground truth is unavailable, and then computes a condensed graph of reaction (CGR) as surrogate for the transition state.
4.1. Yield prediction when the atom mapping is known
Activation energy directly impacts yield, making it impera- tive to incorporate transition states in yield prediction mod- els. If the atom mapping I is known, we can design a yield prediction model in the following steps, which will enable us to express the yield as a trainable function of the CGR that approximates the corresponding transition state.
Ha: MR GNNo Vn, En), H, GNNo(G,, Ei).
(3)
Here H. E Rxd and M. E RNNNND For brevity, we present our analysis with D = 1 and defer the general discussion using tensor form in Appendix C. To elaborate, for the reactants R. HR[u, :] = hR(u) E IR is the em- bedding of node u, while MR [u, v] is the message value mR(u, [u, v) for an edge (u, v) E ER and zero for non-edges (u, v) & En- Similarly, compute H, and M,.
(1) Given the reactants R and the products I in an elemen- tary step reaction, permute Vj using the atom mapping to construct GT = (Vi), (x
which is isomor- phic to Gj. Then, we compute the graph of transition state as the condensed graph GOGR == Gn U G7. i.e., a super- graph which subsumes the structures of both GR and G₁ (Figure 2). Suppose P E Pv is the gold permutation ma- trix corresponding to n, where N [Rrl [Vr] obtained after padding. This allows to approximate the adjacency matrix of the CGR using the adjacency matrices Adjæ and Adj, as follows.
Differentiable approximation of atom mapping To ad- dress the challenge of learning P", i.e., the permutation matrix corresponding to the unknown atom mapping, we adopt a data-driven approach. Ideally, Pa should be a bi- nary 0/1 matrix, but such discrete values attenuate gradient signals and prevent backpropagation. To overcome this, we approximate P using node alignment network Align. This network takes the node embeddings of the reactant and product graphs, H and H, as input, and outputs an alignment matrix P which approximates Pt. The matrix P is a continuous doubly stochastic matrix, which enables smooth end-to-end optimization.
(2) (2) We compute the node embeddings of GeGR. for each elementary step i < n of an ststep reaction (1). (3) We feed these node embeddings into sequence-to- sequence encoder to compute the yield.
AdiCGR max (Adin. P P'Ad)
p == Align H H,)
(4)
4.2. YIELDNET model
Sinkhorn mmax H O. Hy (
(5)
Overview For most real-life chemical datasets, the atom mapping π or the permutation matrix P* is typically not readily available. Hence, we design neural networks to ap- proximate atom mapping (item (1) above), which leads to an approximate CGR for each reaction step. We achieve this in the following steps, as illustrated in Figure 3.
Here, Sinkhorn(-) performs iterative Sinkhorn normaliza- tions on the input matrix (Cuturi, 2013; Mena et al., 2018). Given a temperature > O, an input matrix C and the number of iterations T, Sinkhorn = (inkhorn(")
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
V Rt. En
Adia
H
M R
V1, Bi VR. BR
H CGRR
GR
GNN
mAX AdjeGR
HOGR
HOOR.
yield
Aliga
InportiiforaataabacNN,
G
GNN
M
Step R = R.,
H HOGR
Add
Vr, E,
Reaction- Path- Encoder
Figure 3: Illustration of YIELDNET. For each reaction step i, we process the reactants Rt and products / using a GNN, GNN,, to obtain node embeddings HR, H, and edge embeddings MR. M,, We feed them into a fully differentiable node alignment network, Align, to obtain an alignment matrix P which approximates the permutation matrix corresponding to the atom mapping. Then, we use P to obtain a continuous approximation of the adjacency matrix of CGR, AdjCGR Next, we feed this approximate AdicoR with the node and features into new GNN (InputDiferenttaalleNNN HCGR, the node embedddings of CGR. Finally, reaction path encoder use (HCGR,) } to predict the yield.
obtain
the "set" of node embeddings and relax the optimiza- tion (8) into the problem of minimizing the approximate size of the superset containing Hn and H, Specifically, we seek to solve: minpePN Σ E max H PH)) This is same as solving the following optimization task, due to the fact that in each row of P, exactly one element equals to one.
H E NNdd
Gn
Sinkhorn
max
P [0, N
munn ( 40
G1
min PEP
ΣΣ max (Hn(u, E]. Hy(u', Hrlu,,))P(uuui 26,30
P(u,u) w').
(9)
Figure 4: Differentiable approximation of atom mapping using node alignment network Align. It takes H n and H the node embeddings of Gn and Gy input; and, outputs an alignment matrix P. by computing iterative Sinkhorn iter- ations on the matrix |- Ee max(HRl f), H,(,,, (
The above problem is a linear optimal transport (OT) prob- lem (Kuhn, 1955; Villani, 2008) and is solvable in poly- nomial time. However, the optimal solution of the above optimization (9) is still a 0/1 hard permutation matrix, which diminishes gradient signals. To address this challenge, we perform further relaxation of the optimization (9) and solve the following entropy regularized linear OT problem.
is computed follows:
Sinkhorn (0) (C) eCp(C/A)
(6)
Sinkhorn (++1) (C)= = Row (2) (C)))
min ΣΣ max
(HR[u, 8. Hr ( Plu, n']
(7)
Here, RovDiv (ColDiv) indicates division by sum of rows (columns). Note that the limit limy-+oc Sinkhorn (T) (C)
ē=1 a.u
A Entropy (P),
(10)
(C)
such that: P P20,P1=PT1-1
(11)
converges to a doubly stochastic matrix and the limit lim ooo Sinkhorn (T) (C) converges to a permu- tation matrix. We run the iterations in Eq. (7) a total of T= 10 times.
Here, À >> 0 is the regularizer coefficient. As shown by Cu- turi (2013); Mena et al. (2018), our alignment matrix P obtained using Eq. (5) is the solution of the above OT prob- lem (10) - (11). Therefore, our node alignment network Align (4) can approximate the true atom mapping, enhanc- ing the overall inductive bias of our model.
Rationale behind our design choice: Here, we justify the choice of node alignment network in Eq. (5) by connecting this design decision to combinatorial heuristic for deter- mining atom mapping. From the combinatorial viewpoint, we can compute the permutation matrix P corresponding to atom mapping, by minimizing the size of the supergraph containing the structures of R and (Figure (2)). terms of the adjacency matrices, this may be written as:
Continuous approximation of CGR Having computed our alignment matrix P (5), we make a continuous ap- proximation of AdjOGR (2), which serves as a surrogate of the transition state of the underlying reaction. This is achieved by performing elementwise multiplication of Adj and Adj, with the messages collected from edge embed- dings MR and M,, generated by GNN in Eq. (3), i.e.,
P arrgmin PEPN 21.2
max (Adi PAdi, PT) [u, vj. (8)
AdjeGR ss max(Adjjn M PAdj, O M,PT) (12) Even without this continuous approximation, the discrete adjacency matrix max(Adj PAdi, PT) is differentiable, since the matrix P is now the output of the node alignment
The above optimization problem a quadratic assignment problem (QAP) due to the quadratic involvement of P, and hard as well. To tackle this challenge, we view a graph
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
network. However, there are two key advantages of using the continuous approximation (12).
Node
H HCGR,, ,tep,]
S
(I) M. can model transient bonds: CGR represents the transient state, where the edges represent transient fractional bonds. This can be captured better using continuous weights M rather than binary 0/1 values.
H HCRR, step.)
S:
2
yield
Step do
MLP du
H HCGRg step3]
S
$3
Transformer
(2) Continuous weights enhance backprapagation: Bi- nary adjacency matrices suppress gradient signals, lim- itiig model training. whereas the continuous weights help in backpropagation through the GNN. For instance, if instead of Adj, @ M,, we use only the binary ad- jacency matrix Adjı, then the gradient term will in- volve PAdj, dP
Figure 5: Reaction path encoder. For an n-step reaction r with n = 3, we encode each step i I n by concatenating HCGR, the node embeddings of CGR,, with step encod- ing. A transformer then processes this to obtain S, Next, S, is aggregated into s; via NodeAggr dz and {s, i n} is passed through StepAggT d3 to compute the reaction em- bedding 2,. We use an MLP on Z, to predict the yield.
do + dP do Ad Since P involves it- crative normalization starting with exponentials, AP in- de cludes products like Plu, Ple,, v'), due to the fact that do E²s Ex E" = -e< (ee"))(e (E,²t) when K + j. Thus, PAdi, ap Plu, w¹) PIe, P(w, w'), with each P(...] E [0, 1]. which weakens the gradient signals.
features VCGR and ECGR. However, standard GNN imple- mentations assume fixed, discrete graph structures and do not allow differentiation through the input graph. To over- come this limitation, we design a custom GNN that performs message passing and aggregation using tensorized, differen- tiable operations on the entries of AdjCGR. VeGR. and ECGR. We denote this GNN as InputtDifccrentiallGGN vi param- eterized by v. which computes the embeddings HOGR R xdæ as shown below. Appendix C contains more details. HeOR = InputDifferentiabllee (.
de contains terms of the form
Instead, in our approximation (12), the gradient will involve: PAdj, @ dM: PT + PAdj, C M2 dP dP do
do + Adjr O M,PT. The first term will include entries like
do
& (AdicaR: VCGR, ECOR) (15)
Plu, Ple, e''[ - w']. While the last two terms re- sult in the entries like P(u, u'| v/] P/] (w, M']M[ w'), instead of Plu, x')PP Plu, P(W, w/) in binary adjacency ma- trix representations. These terms with M will enhance the gradient signals.
Reaction path encoder For a multistep reaction r, the elementary steps occur in unique sequence. Therefore, encoding the reaction path requires that CGR of each step should be associated with signal that captures its position in the reaction path. However, the node embeddings HCGR for the different steps are independent of the ordering of the steps. Therefore, to embed condensed graphs in a sequence- aware manner, we concatenate HCGR with step-encodings similar to positional embeddings and pass them through a layer of transformer. Given a n-step reaction r, first compute the sequence-aware node embeddings S;
Embedding computation for CGR Next, we compute the node embeddings of the continuous approximation of the CGR, derived in Eq. (12). We first compute the initial node and edge features VeOR and ECOR, which are required to ini- tialize the GNN embeddings and the message propagation process. These features are composed of two key compo- nents. The first component is identical to VR and En, i.e., the initial features of the reactants, reflecting the reactants' role in transitioning to the product via the condensed graph. These features are critical as they capture the reactants' structure prior to transformation. The second component is the difference between the features of the reactant set R and the product set I, which captures the effect of the reacting atoms and edges. To compute this difference, the nodes and edges of the reactants are aligned with those of the prod- ucts using the approximate atom mapping P derived from Eq. (5). Therefore, we have:
H. = |HORR, St,..., Sn = Transformer,, (H....,H.)) Here, CGR, is the CGR for the i-th elementary step. Next, we aggregate S, to their graph level embedding 8; us- ing a NodeAggr, which is a permutation invariant set en- coder (Vinyals et al., 2016). We compute 8; as:
i [I....n]
(16)
(17)
s; = Nooeeggr o, (S, :].., S,[, i (18) We could alternatively apply NodeAggr to HCUR, first, fol- lowed by the Transformer. However, this approach would aggregate the node embeddings of CGR into its graph em- bedding before introducing interactions. Hence, this would only allow interactions between CGRs at a more coarse level. In contrast, applying the Transformer before NodeA Agrr al- lows the model to capture finer interactions between atoms within the CGRs across different steps.
VCGR = [RR; VR PV1),
(13)
ECGR ERi ER P..
(14)
Next, we input the features VEGR: ECGR and the weighted adjacency matrix AdjcaR (12), into a specialized GNN model to compute the node embeddings for the CGR. This process requires the GNN to support backpropagation through both the input adjacency matrix AdjeGR and the
Next, we aggregate all the graph embeddings (s; : m}
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
to obtain the reaction embedding 2, using StepAgeres ds which is another set encoder of the same architecture as proposed by (Vinyals et al., 2016). Finally, we feed the 2, into a multilayer perceptron (MLP) to predict reaction yield value for the underlying multistep reaction r.
formation reaction (Zahrt et al., 2019), viz. (2) NSI, (3) NS2, (4) NS3, (5) NS4, (6) NS5; one dataset on (7) Suzuki coupling reaction (SC); and, another dataset on (8) De- oxyflurorination (Nielsen et al., 2018) (DF). GP is the sim- plest dataset, where each reaction is single-step reaction and the ground truth atom mapping is available. The remain- ing seven datasets involve multi-step reactions and do not contain any ground truth atom mapping. Availability of atom mapping in GP helps to better evaluate our atom mapping approximator and understand the effect of CGR on yield prediction, which is not possible for the other more commonplace datasets. Appendix D contains more details about the datasets.
2, = StepAggr 05 ($1...., sn))
(19)
yield = MTP &L (z,)
(20)
Here, the set ø = {01, o2, Ø3, Φ1 is the collection of all trainable parameters.
4.3. Training
We conduct end-to-end training of the entire network to minimize the mean squared error (MSE) between the pre- dicted and gold yield values, which serves as the loss func- tion. Note that our node alignment network does not explic- itly ensure that atoms of the reactants only match with the same atoms in the products, e.g., it does not ensure carbon matches with carbon and not oxygen. We ensure this goal by explicit regularization described as follows.
Baselines We evaluate our model against several base- lines including (1) YieldBERT (Schwaller al., 2021c), (2) DeepReac+ (Gong et al., 2021) and several variants of graph neural networks: (3) graph convolutional network (GCN) (Kipf & Welling, 2017), (4) heterogenous graph transformer (HGT) (Hu et al., 2020), (5) topology adaptive
graph convolutional networks (TAG) (Du et al., 2017) and (6) graph isomorphism network (GIN) (Xu et al.).
Regularizer to constrain atom mapping We introduce a regularizer that penalizes the Plu, u| values for those (u, v) pairs where u and v have different atomic identities (or, atomic numbers) since an atom doesn't 't convert to another atom throughout the reaction. Specifically, we introduce matrix A E N where A(u, u] I(atomæ(u) * (ttmm(()) and I(-) is the indicator function. Then, we com- pute the regularizer Reg( R, /) = P@ AlE for each CGR to limit the exploration space of P. where P is computed in Eq. (5).
Evaluation metries We partitioned the datasets into 70% training, 10% validation, and 20% test folds. We generated ten random splits using different random seeds. For each split, we measure the performance in terms of mean absolute error (MAF) between the predicted and ground truth yield values on the test set, and then average it across all ten splits to report the overall performance.
5.2. Results
Loss
(ri,yield(rr) Given set of reactions D (nip , yield(r the sum of the squared error between predicted yield, "eeld....of7 computed using Eq. (20) and the ground truth yield yicld(r), in the presence of the atom mapping regularizer as described above. Given a regularization parameter p. the training optimization becomes:
Comparison with yield predictor baselines First, we com- pare YIELDNET against the baselines across eight datasets. We also report on a skyline variant (YIELDNET (sky)), which predicts yield using the true CGR, instead of an approximation. Since true CGR is only available in GP, skyline performance is limited to only GP dataset. Table I shows the results in terms of MAE. We make the following observations.
we minimize
min Σ [(yielda,,.o()) - yield(r))² +P Σ Reg( R2 7)] 8.0.0
(1) YIELDNET (sky) outperforms YIELD- NET, highlighting CGR's 's importance in yield prediction. (2) YIELDNET outperforms all baselines in seven out of eight datasets; and, in five datasets, the performance gains are statistically significant. The baselines do not explicitly approximate CGR and instead use the embeddings of the reactants and products as-they-are, leading weaker perfor- mance. (3) There is no consistent second- best model; GCN, TAG, DeepReac+, and YieldBERT share the second-best across datasets.
FE
(R.))e
(21)
5. Experiments
In this section, we provide comprehensive evaluation of our method across eight datasets. Appendix E con- tains additional experiments and results on more datasets including USPTO dataset (Lowe, 2017). Our code is available in https: https:////tthbb com/ ankitthreo/ YieldNet. . git.
Evaluation of our atom mapping approximation Next, we compare our approximate atom mapping computed us- ing Eq. (5), with four existing atom mapping methods. They include (1) RDKit (Landrum, 2013): the alignment strategy built into the RDKit library, (2) RXNMapper: a pre-trained
5.1. Experimental setup
We carry out our experiments using eight datasets. They include- - (1) GP dataset which is derived from Gas- phase Isomerization reactions (Grambow et al., 2020b); five datasets derived from catalytic asymmetric N, S-acetal
Datasets
7

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
Model
GP
NS1
NS2
NS3
NS4
NS5
SC
DF
GUN
38.057 0.337
11,162 0.735
9,416 + 1.045
8.813 0.857
8.777 0.221
4,453
0.293'
10.682
0.396
12.516 0.329
HGT
39.405 +: 0.231
13.385 = 0.844
9.573 = 1.016
9.421 = 0.835
8.629 = 0.249
4.561 =: 0.277
13.665 :: 0.334
19,500 + 0.257
TAG
36.887 0.412
13.246 + 0.760
9.560 0.994
9.232 0.860
8.603 + 0.230
4.547 0.286
12,603 0.450
18.150 + 0.447
GIN
38.307 +: 0.450
13.318 = 0.825
9.588
1.017
9.152 0.860
8.617 0.217
4.568 0,285
12.899 0.389
17.126 : 0,479
DeepReac+
27.837 = 0.346
12.985 = 1.452
10.729 + 0.690
9.968 = 0.981
9.439 = 0.775
5.125 +: 0.301
16.953 +: 1.800
14.022 : 1.694
YieldBHR
40,910 0.300
12.446 + 0.820
9.593
1.002
8.780 0.918'
9.236 0.290
4.532 0.255
10.437 + 0.328
12,470 + 0,412
YIELDNET
23.152 + 0.393
9.245
0.518
8.387 + 0.907
7,914 + 0.931
7.015 + 0.495
4.382 + 0.249
8,751 + 0.438
6.941 0.192
YIELDNET (sky)
17.396 0.357
NA
NA
NA
NA
NA
NA
NA
Table I: Comparison of yield prediction performance for YIELDNET against all the competitive baselines, viz., GCN (Kipf
& Welling, 2017), HGT (Hu et al., 2020), TAG (Du et al., 2017), GIN (Xu et al.), DeepReac+ (Gong et al., 2021),
YieldBERT (Schwaller et al., 2021c), on the 20% test examples, across all datasets. Performance is measured in terms of
Mean Absolute Error (MAE). YIELDNET(SKy) represents a skyline of our model, where we use the true CGR. Only GP dataset contains the true CGR and therefore, such skyline performance is not available for other datasets. Numbers in green (yellow) indicate the best (second best) performer. Our improvement in performance over the next best baseline, where YIELDNET is the best performer, is statistically significant with p-value < 0.05, except in the cases marked with *
Model
NSI
NS2
NS3
SC
alignment matrix. We compare our method with attention- based method (Astero & Rousu, 2024), which was the best alternative to our model, as shown in Table 2. Our method gives the average error using PP - P- P = 3.708, whereas attenion based method gives P P' || = 5.096. Notably, our method achieves such high accuracy in atom mapping prediction solely through end-to-end learning from yield data, without any access to the ground- atom mapping.
RDKit
9.910
8,845
8,567
RXNMapper
9.610
8,871
8,470
10.046
Random
11.195
8,984
8.695
10.223
Attention
9.548
9.024
8.432
8.826
YIELDNET
9.245
8.387
7.914
8.751
Table 2: Comparison between different atom mapping strate- gies in terms of MAE of yield prediction
node alignment model (Schwaller et al., 2021a) on a large set of reactions, (3) Random: Uniformly sampled alignment, and (4) Attention: an attention network between reactants and products based on Astero & Rousu (2024), that induces non-injective alignment, unlike our proposed alignment network which outputs an approximate injective alignment. Here, RDKit requires a well-defined reaction template ob- tained through expert-curated rules and subsequent manual inspection of atom mapping for each reaction in the dataset. Consequently, we cannot generate such templates for SC datasets due to the high diversity of reactions. In Table 2, report the results in terms of the MAE of the predicted yield. We make the following observations: outperforms the alternatives. (2) The attention network out- performs RXNMapper, RDKit and Random in most cases. However, since it induces a non-injective mapping, it as- signs a product atom to multiple reactant atoms with high probability. As a result, our method outperforms it, since our alignment network provides the atom mapping through a doubly stochastic matrix, which in turn induces an injec- tive mapping (one atom in R is mapped to exactly one atom and vice-versa). (3) The random permutation generation strategy performs the worst. This underscores the necessity of good approximation of atom mapping.
Interplay of atom mapping with predicted yield Here, we examine the extent to which the performance of YIELDNET can be attributed to its ability to learn the atom mapping. We investigate the relationship between the mismatch in prediction er- ror AAE,, and the difference between the true and learned atom mapping: AP, for each test reaction as follows. We consider GP datasets since it is the only dataset contain- ing atom mapping. Our ap- proach begins by training a different variant of our model where we replace P. for each reaction r with P i.e., the permutation matrix cor- responding to the ground truth atom mapping. We call this model as reference model. Then, for the test dataset, we gather all the learned alignments P, from our trained model yield, a and transform them into hard permuta- tion matrices Punndrr via the Hungarian algorithm. Next, we apply both P; and Paadd to the reference model to ob- tain the predictions yield (r) and yield (r) for each reaction r in the test set. Note that yieldp: gives high ac- curacy as we leak the true atom mapping into the model. Next, we calculate the difference in the absolute errors (AE) for each test reaction as AAE, = =|AE', - AA where AE* , = =Iyield (r) - yield(r) and A = lyield (r) - yield(r)) for the gold yield values yicld(r). Finally, we quantify the dissimilarity between two permutation matrices using AP, = =||P; - Pl
40.3
0.1
AP i Figure 6: AAE, vs A..
(1) Our method
Next, we evaluate the accuracy of our learned atom mapping by measuring its proximity to the ground-trute -tuth atom map- ping available in the GP dataset- the only dataset that pro- vides such ground-trume mappings. Specifically, we compute the average Frobenius norm of ||P PP PI tions, where P is the node permutation matrix representing the ground-trum tround-truth atom mapping and P the learned node
across all reac-
8

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
and plot the correlation between I E[AAE,, (P,)) and A A., by averaging AAF, at particular A.,. Figure 6 shows strong correlation between AP, and AAE,, indicating strong association between the learned alignments' quality and the model's predictive performance.
6. Conclusion
In this work, we present YIELDNET, a novel model for yield prediction of the final product in multi-step reactions. The energy of the transition state plays a critical role in determin- ing reaction yield. The graph structure of the transition state can be inferred using atom mapping between the reactants and products. As most datasets lack atom mapping, we employ a differentiable node alignment network, which can be trained end-to-end using supervision from yield rather than fine-grained atom mapping labels.
A Ablation study on CGR representations Here, instead of incorporating the embeddings of the CGR into the reaction path encoder, we directly feed the embeddings of the reac- tants Rd and the products I; into the encoder. Specifically, we omit both our node alignment network Align and CGR representation network InputDifierentiableGC..
and in- stead input the embeddings from the GNN into the trans- former, alongside the step encoder. Table summarizes the results. We observe that incorporating CGR embeddings into the reaction path encoder is more effective than directly utilizing the embeddings of the reactants and products.
Our work opens several promising directions for future research. YIELDNET can be used to predict free energy, reaction rates, and more. Yield depends on factors beyond activation energy, like reaction time, purification procedure, etc., which current datasets lack. While the datasets also lack the underlying atom mapping, the graph structures of the reactants and products allowed us to approximate it. A key direction for future work is to incorporate the other factors into our model when such datasets become available.
Method
NSI
NS2
NS3
SC
w/o CGR
11.356
9.459
9.342
10.686
YIELDNET
9.245
8.387
7.914
8.751
Table 3: Effect of ablation of CGR (MAE).
Ablation study on reaction encoder components Here, we investigate the benefits offered by different components of our reaction path encoder (Eq. (16)-Ea) (20)). Specifi- cally, we focus on ablations of its two key components, viz., the transformer (17) and StepAggr (19). Table 4 summa- rizes the results for four multi-step reaction datasets mea- sured in terms of MAE of the predicted yield. We make the following observations: (1) the transformer encoder con- tributes to the improved accuracy across most cases; (2) utilizing Seq2Seq for sets (Vinyals et al., 2016) yields better performance than DeepSet (Zaheer et al., 2017) and simple sum aggregation methods; (3) ablation on the set encoder has a stronger impact on MAE than the transformer because the set encoder is applied in the final layer.
Impact Statement
This paper offers a valuable tool optimizing reaction yields. It can be used in prediction of yield when the datasets con- tain molecular graph structures. However, it is imperative to acknowledge the possibility of incorrect predictions by any yield prediction model, which demands human inter- ventions. Additionally, misuse of the model may lead to predictions for undesirable compounds, potentially causing adverse societal consequences. Careful attention to ethical guidelines is paramount to ensure a responsible and positive impact on the model.
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction (Appendix)
A. Limitations
1. Currently our work is limited reactions having reasonable number of atoms (~ 200- 250). However, implementing our model on larger molecular reactions catalysis (Sheldon et al., 2020) has some drawbacks. Computation of P using minkhorn-iteration
(5) has complexity of ((N2), where N is the number of total atoms of reactant(s) or product(s) in reaction step. Thus, computing P in such cases would be challenging. However, this issue can be mitigated by low-rank factorization (Scetbon et al., 2021) for Sinkhorn routine, which reduces the complexity to (NN. 2. The standard yield prediction datasets currently have only a single reaction path r for reactions leads to one single product. Thus, we consider only single reaction path-based reactions to train our model. However, our method can be adapted to take multiple reactions into account, where we can add more stochasticity into Sinkhorn layers to enable generation of multiple permutations from the same set of reactants and products. They will lead to multiple atom mapping and CGRs, which in turn can generate multiple reaction paths.
3. The standard yield prediction datasets contain only molecular graphs of any molecular component and these datasets are commonly used in previous baselines. Hence, our model, as of now, does not incorporate information about geometry of the molecules. However, our model can be easily adapted to incorporate geometry by extending our GNN into the spatial graphs and taking into account three-dimensional coordinates for our atom mapping approximators.
B. Additional discussion on related work
Apart from the general applications in chemistry mentioned in Section 2, there has been a recent interest in modeling chemical reactions (Bradshaw et al., 2019; Benayad et al., 2024; Joung et al., 2024; Hoque et al., 2024a) and solving reaction- related problems (Schwaller et al., 2021b; Burés & Larrosa, 2023). These also include retrosynthetic planning (Segler & Waller, 2017; Liu et al., 2017; Coley et al., 2017b; Dai et al., 2019; Shi et al., 2020; Chen et al., 2020; Yan et al., 2020; Fortunato et al., 2020; Somnath et al., 2021; Tu & Coley 2022: Chen et al., 2023), reaction outcome prediction (Wei et al.,
2016; Coley et al., 2017a; Jin et al., 2017: Schwaller et al., 2018: 2019; Coley et al., 2019; Struble et al., 2020: Guan et al.,
2021; Bi et al., 2021: Chen & Jung, 2022; Nippa al., 2023; 2024; King-Smith al., 2024; Pereira al., 2024), reaction
optimization (Raccuglia et al., 2016; Zhou et al., 2017; Gao al., 2018; Shields al., 2021; Wang et al., 2024; Li et al.,
2024; Hoque al., 2024b), atom mapping (Schwaller et al., 2021a; Nugmanov et al., 2022: Chen et al., 2024; Astero &
Rousu, 2024), transition state geometry prediction (Pattanaik et al., 2020; Lemm et al., 2021; Makoš et al., 2021; Choi, 2023; Duan et al., 2023; Kim et al., 2024; Duan et al., 2025), etc (Qian et al., 2023; M. Bran et al., 2024). In the case of atom mapping, RXNMapper (Schwaller et al., 2021a), and GraphormerMapper (Nugmanov et al., 2022) are the two pioneers of deep learning-bai through an unsupervised task of reaction SMILES reconstruction. Due this unsupervised nature, RXNMapper needs a substantial amount of data and large transformer mncoer--bsse model (Lan et al.). On the other hand, GraphormerrMpppr is pre-trained in multiple levels on different tasks, making the model very computationally expensive. As both models use ceenntion-guide paths for atom mapping, these mappings aren't 't injective i..,, there are multiple atom mapping possible for one single reaction instance and that should not happen in real case scenarios. Recently, Chen et al. (2024) use active learning for the atom mapping task, which also attention breention-basedd.. In another recent work, AMNet (Astero Rousu, 2024) makes the atom mapping injective by imposing an extra layer of WL. test algorithm (Weisfeiler & Leman, 1968) on top of a non-injective graph attention. However, YIELDNET model (1) tries to approximate atom mapping as doubly-stochastic matrix, which by architecture yields injective mapping, (2) is relatively simple. Our key goal is predict yield which is also different from the above described atom mapping models. We achieve this by designing architecture, that approximates atom- hom-mapping on the fly.
atom-mapping tools. RXNMapper uses post-processing on the attention heads, learned
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
C. Neural architectures different components of YIELDNET
C.1. Architecture GNN,
Our GNN, is built upon a communicative message passing network (CMPNN) (Song et al., 2020). In contrast to a simple message- passsegg neural network (MPNN) (Gilmer et al., 2017) or directed caassage-passingg neural network (DMPNN) (Yang al., 2019), where either only node states or edge states are updated independently in each iteration, our model updates both node and edge states simultaneously. Moreover, the update states for nodes and edges are treated interdependent instead of treated independently. We elaborate each step, starting from initial features input to the final embedding as output.
In the following, we describe the embedding computation only for the reactant set R. which can subsequently be extended for the product set J. GNN, takes input graph Gn as Adja E {0. 1} x node features VR E TRNxdy; and edge features ER E I IxNxxx We use two different MLPs for each of the features to transform into initial embeddings, nodeEmb, IR P and edgeEmb, 8) R accordingly,
nodeEmbo (u) = ML.P,, (Vn) edgeEmb, v) MLP by (AdiR En)(u,
(22)
(23)
Here, MLPp, are just single-layer networks followed by ReLU activation function. In Eq. (23), Adi (0, 10 and ER E NNN xdg, Here, in @ operation. we first broadcast Adj, into the third dimension to have an adjacency tensor of dimension N x N x dg and then perform Hadamard product with ER. Next, we perform message passing for ls number of propagation layers. For a E fl,.., K - 1), the message passing steps are given by,
(v) = o (max, (nbr(()) edgeEmb;-1 u)) @ Euenbe(e) edgeEmba-1 nodeEmb,,()) (v) nodeEmbx-1(v) + pax (v); nodeEmbx(o) E R ua(u, v) Adja(u, (nodeEmbbx(u) edgeEmb,.-
(24)
(25)
(26)
edgeEmb,, (u, v) = (,, ReLU edgeEmb, v) + 03. 2e (2, v)
(27)
exp Adjn|u, [u, w) nodeEmb,, (u) T "nodeEmb,
where (, Ev' exp Adinlu', u] nodeEmb,, nodeEmbx
(28)
The above computations were performed for & E {L,..K 1}. For k K, we update the embeddings as follows:
(k( o (maauuenr(()) edgeEmb (u, Emenbo(o)) edgeEmbx- - v))
(29)
nodeEmb x (v) = MLPe, (no(v), nodeEmbx-(()))) Mx (v)): nodeEmbx (v) E Rd
(30)
edgeEmb x (u, v) edgeEmbR-1(a4
edgeEmb (v) E ER"
(31)
Finally, we compute node embedding vector hu(ww) per node u, node embedding matrix Hn E Rxd, edge embedding vector m (1, v) for edge (u, v), edge embedding matrix MR N follows:
hR(u) = nodeEmb (u)
(32)
m (u, u) edgeEmb x (u,
(33)
H :] == teR(u)
(34)
MR|u, v == mR(u,
(35)
C.2. Architecture of InpurDifferrntiiaaleGNN
This architecture is similar to the GNN,, except that we realize this in tensor space and use adjacency tensor AdjegR NxxxD This is because during the continuous approximation of AdiCAR in Eq. (12), we need Mg, M, E N (3). Thus, to obtain AdjEGR- we need to do @ operation i.e., broadcasting Adj. E N*N into a third dimension to have an adjacency tensor of N x N x followed by Hadamard multiplication with corresponding M.. Finally, we obtain an AdjCGR E NxND through Eq. (12). Now, similar to the GNN,, to perform the Eq. (23), we feed AdJOGR into an MT.P
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
to bring its final dimension from D to dg same E, that the Hadamard product in Eq. (23) compatible.
AdjEGR - MLP vi (AdieGR)
(36)
nodeEmbTo = MLP (VR)
(37)
edgeEmbT, MLP (AdjCOR @ ECGR)
(38)
Then the tensorized version for Eqs. (24)-(28) for E {l,..., K -K-1 1] is written as (aT indicates tensorized version of a vector z described GNN)): nT [T, = a mmax seaxeegeEmbb v,
edgeEmbT x-1 v.
(39)
nodeEmbTa, [u,:] = nodeEmbb: |v. + TT (u,
(40)
u, = AdicGR (4, v,:] @ nodeHEmbTx [v. edge (u, v.:]
(41)
edgeEmbT, [u, v,:] Bxlu, :
ReLU (dggeEmb,lu, u, + t 1 vTglu,,.))
(42)
exp (nodeEmbTT, nodeEmbTT AdiCGR [u, v,
where (T&( w,
(43)
Ew exp (nodeEmbT,, nodeEmbT,) nodeemmbbbdddodmoyy... @ AdjCGR Ju', v.
For k = K, we update the embeddings as:
uTx [u o(max edgeEmbT k-1 v.:) @
xdgeEmbbTx-ltt.t v.
(44)
1
nodeEmbTT, x [v.:] = MTP (nodeEmbTa (lv, [v.]
(((;:))
(45)
Finally, HCGR is computed nodeEmbT Rxdd
C.3. Details about neural path encoder
Neural path encoder composed of four components Transformer NodeAggT StepAggras and MLP
1. Transformers, Before going Transformer encoder, HEAR, supposed to have concatenation with step,, which is similar to the concept of positional encoding (Vaswani et al., 2017). Here, i S where is the number of total steps in reaction path r. However, we used one-hot positional encoding, which leads us to step; E 10,1) Thus, according to Eq. (16), H. E IR (dn+ As our Transformer,, is a single transformer encoder layer having the same encoder layer architecture as Vaswani et al. (2017), our final S E N +n) (17).
2.
NodeAggr: The main architecture the same as Seq2Seq for sets or known as Set2Set (Vinyals et al., 2016). This type of aggregation inflates the final output dimension of embeddings to twice the input dimension i.e. 2(d + n). Thus, use Lincar layer to reduce back the output dimension to the same the input dimension (d n). This breaks down the Eq. (18) into the following
8; Linear de (Se22Set (S, for i S
(46)
3. StepAggre, This architecture also similar NodeAggray, Thus, Eq. (19) can be rewritten as,
2, Linear Sanh (Set2Sel ($1, "t)......))) 2, E (d)
(47)
4. MLP de The final MLP de layer is Linear ReLU Linear (LRL) layer, the input z, has final dimension of (d + n).
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
D. Additional details about experimental setup
D.I. Dataset preparation
To assess the efficacy of our model, we focus on four reactions that have garnered recent attention. These reactions- Gas- phase isomerization (GP) (Grambow et al., 2020b), deoxyflurorination (DF) (Nielsen et al., 2018), asymmetric a,S-acetal formation (NS) (Zahrt et al., 2019), and Suzuki coupling reaction (SC) exhibit diverse data sizes and varying number of steps.
Each reaction in these datasets is annotated with sets of reactants, intermediates, and products based on mechanistic considerations. The measured yields ranging from 0% to 100% are compiled from prior wet-lab experimental data, except in the case of the GP dataset, which we will discuss shortly. Several of these datasets, including (DF) (Nielsen et al., 2018) and asymmetric N, S-acetal formation (NS) (Zahrt et al., 2019), are obtained through High-Throughput Experimentation (HTE). The experimental process for HTE typically involves selecting a limited set of reactants and conducting reactions on all possible combinations within those chosen reactants. In contrast, real-life datasets contain a significantly larger pool of diverse reactants, with only small fraction of the potential reactant combinations explored. This discrepancy results in substantial reduction in dataset size for rial-life scenarios (Saebi et al., 2021; Schleinitz et al., 2022). Additionally, real-life datasets exhibit higher sparsity. making it less likely to encounter a reactant from the test sample in the training set.
Due to the above issues, we create subsets of these HTE datasets, which create multiple low throughput (LTE) datasets, making them closer to reality. Specifically, we derive GP, GP1, GP2 from the gas-phase isomerization dataset; NS1, NS2, NS3, NS4, NS5 from the NS dataset; DFI, DF2 and DF3 from DF dataset. Moreover, we collect a new dataset consisting of samples of the reaction type Suzuki Coupling (SC) using the reaction data from 50 peer reviewed publications. In contrast to the existing HTE datasets, this dataset is low throughput, i.e., they consist of reactions of only limited number of combinations between the reactants. We also use USPTO from Lowe (2017). These finally lead to total fifteen datasets.
D.1.1. LIST OF DATASETS
The fifteen datasets used are: GP, GP1, GP2, USPTO, NS1, NS2, NS3, NS4, NS5, NS, DFI, DF2, DF3, DF, and SC. Among them, we present GP, NSI, NS2, NS3, NS4, NS5, SC and DF in the main part and present the rest in this Appendix. In the following part, we describe the datasets in full detail.
D.1.2. DATASETS DERIVED FROM GAS PILASE ISOMERIZATION
The gas-phase isomerization reaction dataset (Grambow et al., 2020b) contains around 12000 elementary step organic reactions involving small molecules composed solely of C, H, N and O atoms. The dataset has the activation energy for the reaction as the label. Previous attempts to predict activation energies involved using fingerprints and graph-based models (Choi et al., 2018; Grambow et al., 2020a; Heid & Green, 2021). We transformed the activation energy (AE1) into yield values, by using the Eyring equation (Eyring, 1935). This approximates a rate constant (7) value from activation energy. We then assume every elementary step reaction following first-order kinetics to get the yield value. The steps are shown below. Notations used in Eqs. (48) (51) are only limited this section and may share overlap with symbols used in other parts of the paper. Here, AGt represents Gibbs free energy of activation (or assumed to be activation energy) in Jmol-! 1_ kB h. R are the Boltzmann constant (1.38 x 10 23 JK-1), Planck constant (6.626 x 10 (3)),,), and universal gas constant (8.314 15.314 JKmol-1 respectively. Here, here is the reaction time, we take a fix t of 1 hour i.c., 3600s.
KB Temp
-AG!
7 =
exp Temp
(48)
h
yield = (1 - exp( (t)) x 100
(49)
Temp is the reaction temperature. In our case, we consider fixed Temp 1105.26266 for our GP dataset. We have activation energy in this dataset in terms of kcal mol-1. To convert it to proper unit, we multiply our activation energy value by factor of 4183 and feed the values in place of AGF, After all these, our final equation becomes.
1.38 x 10 23 x 1105.26
- x 4183
exp 8.314 x 1105.26
(50)
6.626 x 10-34
yield = (1 - exp( 7 x 3600)) x 100
(51)
We derive GP and its other two variants GP1 and GP2 based on clustering of the gas- gas-phase reaction dataset. We observe that the reactants in GP1 mostly contain very small, 3 or 4-membered rings (e.g., cyclopropanes, oxiranes, aziridines,
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
cyclobutanes, etc.). Similarly, reactants in GP2 are mostly substituted five-membered aromatic heterocycles (e.g.. pyrrole, pyrazole, imidazole, furan, etc.). In total, we have used three datasets GP (main), GP1 (Appendix), and GP2 (Appendix).
D.1.3. DATASETS DERIVED FROM DEOXYFLUORINATION
The deoxy feoxyfluorination (DF) reaction is an important method for converting inexpensive alcohol to the corresponding fluorinated compounds by using sulfonyl fluorides, in the presence of a suitable base (Nielsen et al., 2018). Here, the whole dataset represents a collection of combination reactions involving 37 unique alcohols, 5 sulfonyl fluoride, and 4 bases, totaling 740 reactions and their measured yields. Previous studies on yield prediction for this reaction employed the random forest (Nielsen et al., 2018) and NLP-based transfer learning (Singh & Sunoj, 2022). Apart from the full dataset (DF), we have extracted three subsets, each containing 200 instances, from the full dataset. We create these subsets based on the yield labels. Yields have a skewed distribution, where yield has mean around 40. We create three subsets having an average yield of low (10), medium (75) and high (90). We find that there are different predominant compounds. For instance:
(1) The dataset DF1 has he--hy/oooyyycooeeyyssiioddiiii 1,3dione, 2,3,4 2,3,4,,,7,,,9,1 611..//8.0..0..taaayyyymmmm1111. 1,2-aaazzeiiinn 4-nitro benzenesulfonyl fluoride as predominant reactants.
(2) The dataset DF2 has [[[l::///rrrizooooll_aaaaalppp// http:iidiiioollppp----e-ee 1-oo,, 2-(tert-buty))- tetrametthylguanidnne,
1, 11122.3...444444oafffrrrroo.....
-1 1-sulfonyl fluoride predominant reactants.
(3) The dataset DF3 has 1:445-/dphhnnyxxxx222-211/ppppp 4442333344444-ooaafuuuruo
2-(lert-butyy--1111.33 tetramethyylguanddnne
- -I-sulfonyl fluoride as predominant reactants.
In total, we have used four datasets DF (main), DF1 (Appendix), DF2 (Appendix), and DF3 (Appendix).
1.1.4. DATASETS DERIVED FROM ASYMMETRIC N, S-ACETAL FORMATION
The asymmetric N, S-acetal formation reaction (NS) is catalytic transformation involving the addition of a thiol to aldimines. Zahrt et al. (2019) released dataset comprising 1075 such examples, where the labels are in terms of activation barrier difference. Later, Singh & Sunoj (2022) released another version of the prior dataset, which comprises 1027 samples having labels as percentage values. This dataset consists of 5 thiols, 15 different imines as substrates, and 43 different chiral phosphoric acids as catalysts (Cat). The label for this dataset is expressed in terms of enantioselectivity, which is effectively the yield of the major enantiomer produced in the reaction. Similarly to the previous dataset, we have sampled the complete dataset into five subsets of size 300. Like DF, we create datasets with average yield of 25, 40, 50, 60, 75.
(1) The dataset NS1 has 2.w.sh/mmmhhhooyyyygl022222333ad (2. (2.1-d:1',22 dixxaplosphepinee44odddd (ww.e//eAAA.aaoooooaaaaaaaaaaaad as predominant reactants.
(2) The dataset NS2 has htwpa//wwhhhi.iooo....../.y/ninn
(E) N- benzyylidenebeezzamiee oww.://rr..miie..ooooooooonooaaa
cyclohexaaethiiol as prwdominanttraaciiiii.. reactants.
(3) The dataset NS3 has hatpah/hhhhhiiioooooooooyyyyyhha hixp::/::iiooapppppppppp--oooee
(E) oow.lllhiee.iiiiieeeeeeeeeeeeeed predominant reactants.
(4) The dataset NS4 has
http:///haaaaaaaa...///./eeeaaaa cychoeeaeethooo as predominant
[2,1-
d:1' http://l.a.aaitooooooeeeeee//e/a htt.://a......ooooooooooooooeeoe
reactants.
(5) The dataset NS5 has 4hyylooy-22--iiss44eeeooxyyyyyo htwprh/oooopppocoooiimiaattttt.. dinaphtth/22.1-:::1222-1111 diopppooppppppppioooooiiii. -eettyylbenzeenthholl]] as predominant reactants.
We also utilized the full dataset (abbreviated as NS) to check the performance. In total, we have six datasets NS1 (main), NS2 (main), NS3 (main), NS4 (main), and NS5 (main), and NS (Appendix).
1.1.5 DATASETS DERIVED FROM SUZUKI COUPLING REACTION
The calladiuu-caatllyzer Suzuki Cross-coupling (SC) is versatile method for generating biaryl uodducts-ubiquitour in natural compounds, pharmaceuticals, and chiral ligands (Kantchev et al., 2007). This dataset comprising 481 1 reactions is manually curated from 25 publications filtered out from numerous reports within this reaction class (Çakir et al., 2021; O' O'Brien et al., 2006; Navarro et al., 2006; Li et al., 2a; Peh et al., 2010; Chen & Kao, 2017; Lu et al., 2017; Micksch
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
al., 2014; Diebolt al., 2010; Tu et al., 2012; Ouyang et al., 2018; Han al., 2018; Zhang et al., 2020; Karataş, 2019;
Yang, 2017; Lv et al., 2014; Organ et al., 2009; Xia et al., 2021; Navarro al., 2004; Wu et al., 2011; Sahin al., 2017; Kuriyama et al., 2013; Hartmann et al., 2009; Dastgir et al., 2010; Arci et al., 2021; Azpiroz et al., 2017; Kumar et al., 2009; Mercan et al., 2011; Izquierdo et al., 2015). The selection criteria were specifically tailored to type of catalyst where palladium is intricately bound to an N-heterocyclic carbene (NHC) ligand, with additional selection criteria involving heteroatoms (N, O, and P) within the catalyst structure. The dataset is labeled based on the yield of the resulting biaryl product. We used the SC dataset in our main paper.
1.1.6. DATASETS DERIVED FROM USPTO
This is US patent chemical reaction dataset collected by Lowe (2017). Later the dataset is filtered to classify the reaction classes (Schwaller et al., 2021b). The reaction dataset has around 44k reactions with yield labels and reaction classes. We use these reaction classes to filter out the dataset. With the further exclusion of reactions with missing reagents, we selected 1 1700 reaction samples from the original dataset. Next, we select only those reactions which occur through a two-step reaction pathway. For instance, the protection of functional groups (c.g., alcohol, amines), functional group interconversion, including nucleophilic substitution, etc. Finally, we curated 1 1150 two-step reaction samples for our study. We used the USPTO dataset in the Appendix.
D.2. Details about our implementations of the baselines
While implementing the baselines, we performed few modifications to ensure that the comparison between our method and the baselines is fair- (1) number of parameters is approximately same; and (2) no component of a baseline is exposed/pre- trained in external dataset, since that would give additional signals to them, which is not provided our method and rest of the baselines.
GCN Here in GNN, the message function employs the graph convolution layer (GCN) (Gilmer et al., 2017), which collects adjacency node embeddings through edges. We have developed other versions of it by replacing this GCN with alternative node-based convolution message functions. Finally, following Kwon et al. (2022), we predict the yield by taking molecular graphs of reactants and products as input. It aggregates reactant vectors, concatenates them with the product vector, and channels them through an FNN to predict the yield. We set the dimension of node embeddings 18.
HGT Here in GNN, we utilize Heterogeneous Graph Transformer- treat the molecular graph as a heterogeneous graph (Hu et al., 2020). Then, we used the same method proposed by Kwon et al. (2022), which is described in the context of GCN. We set the dimension of node embeddings as 17.
Convolution (HGT) as our message function to
Here in GNN, we utilize TAG, which is based on the topology adaptive graph convolution network (Du et al., 2017). Then, we used the same method proposed by Kwon et al. (2022), which described in the context of GCN. We set the dimension of node embeddings as 20.
TAG
GIN Here in GNN, we employ the GIN message convolution function, inspired by graph isomorphism networks, showcasing Graph Neural Network (GNN) that exhibits comparable power to the Weisfeiler-Lehman (WL. test for isomorphism (Xu et al.). Then, we used the same method proposed by Kwon et al. (2022), which is described in the context of GCN. We set the dimension of node embeddings as 20.
The overall architecture of the (Gong et al., 2021) model is founded on a two-level graph attention convolution framework (Veličković et al., 2018). In the initial stage, the graphs for each reaction component are encoded using graph attention network (GsT)- GNN. Later, these individual component graphs are treated as nodes within fully connected reaction graph, using the graph embeddings serving node features. Another layer of GAT is then applied to this complete graph. Finally, the embeddings of all components are fed into a FFN network. We set the dimension of node embeddings as 48.
DeepReac+
YieldBERT YieldBERT uses a previously trained BERT encoder (Devlin et al., 2019) to forecast chemical reaction yield as a function of reaction SMILES (Weininger, 1988). But that would expose the model to external datasets possibly containing examples in test data too. Hence, we train BERT encoder along with a regressor, end-to-end for yield prediction. We set the dimension of hidden size as 12.
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
D.3. Implementation details
Training We maintain a 70: 10:20 split for training, validation, and testing across all models, employing 10 different splits for robust evaluation. We kept the batch size same across all models. We set b 50 for GP datasets and b = 8 for the rest. We train each model with 100 epochs and evaluate their performance the test dataset, selecting the epoch with the lowest validation MAE. We use Adam optimizer for each model. We use Noam learning rate with 2 warmup epochs and an initial and final learning rate of 10-4 and a maximum learning rate of 10-3. Additionally, we keep the regularizer parameter fixed at a value of 0.1 in our model.
Model
1. GNNg. We set the dimension of node embeddings, d 20 (30) and dimension of edge embeddings, D 20 (31).
2. Align. We set temperature À 0.1 in Eq. (6), Gumbel noise factor 1.0, and Sinkhorn iterations T = 10.
3. InputDifferentiablGGNN v. It shares parameters with GNN,, except tho and 8/4 (Appendix C). We set dæ = 20 where is the number of steps.
4. Transformer As d = 20 2 n = +n) = 20, which is the input and output dimension of the Transformer We set the number of heads to 5, and the feedforward dimension to 2048.
5. NodeAggrós and StepAggr As described in the Appendix C, the set encoders of Eqs. (46) and (47) of both the aggregator modules
borrowed off-the-shelf from Vinyals et al. (2016), where have input dimension (dæ + n) 20, number of layers 1, number of iterations 3. The final output of the Aggr modules will be of the dimension of (dı + n) 20.
6. MLP,,, d1. We have input dimension of (dæ + n) = 20 and output dimension of as per Eq. (20).
Baseline For a fair comparison, we maintain a similar number of parameters across all the models. We used the same learning rate for all GNN models and YieldBERT of 10-4 For DeepReac+, the learning rate is 10-3
Number of parameters The number of parameters for multi-step reactions is around 1 112k, and almost 26k for a single step one. The number of parameters is almost maintained throughout all the models (Table 6).
#Parameters
GP
rest
GCN
26825
115753
HGT
28405
151895
TAG
27965
108725
GIN
27165
105525
DeepReac+
27074
119522
YieldBERT
28861
110997
YIELDNET
26478
112583
Table 6: Number of parameters for all models
D.4. Hardware details
All the models are trained on NVIDIA A100 80GB GPU, All the models are fully based on PyTorch (Paszke et al., 2019).
We run in Ubuntu 20.04.6 LTS machine having 2TB RAM with 64 bit CPU and AMD EPYC 7742 64-Core Processor.
D.5. License
We utilize CMPNN (Song et al., 2020), which comes under MIT License. For baseline comparisons, we use dgl-based (Wang et al., 2019a) GCN (Kwon al., 2022), HGT (Hu al., 2020), TAG (Du al., 2017), GIN (Xu et al.), and DeepReac+ (Gong al., 2021)- all of which come under the Apache 2.0 License. YieldBERT (Schwaller et al., 2021b) comes under MIT License. The Gas-Phass reaction datasets (Grambow al., 2020b) are licensed under CC-BY 4.0. The USPTO dataset used here comes under the MIT License. The original USPTO dataset (Schwaller et al., 2021 by Lowe (2017) comes under CCO 1.0 License. RXNMapper (Schwaller et al., 2021a) tool comes under MIT License. RDKit tool comes under BSD-3-Clause License. NS (Zahrt et al., 2019) and DF (Nielsen et al., 2018) datasets used here are available in (Singh & Sunoj, 2022). However, we couldn't 't find licenses for those datasets. PyTorch (Paszke et al., 2019) and NetworkX (Hagberg al., 2008) both come under BSD-3-Clause License.
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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E. Additional results
E.1. Comparison in yield prediction terms of RMSE
From Table 7 we observe a similar trend as Table in RMSE for seven out of eight datasets. different trend RMSE for GP is due to using MAE on the validation set during early stopping. The metric used for early stopping was MAE in the validation set, which is why may show alternate trend for RMSE in GP. Like Table I, YIELDNET (sky) (skyline variants of our model, which uses true CGR for yield prediction) outperforms other models, reflecting CGR's importance in
yield prediction.
Model
GP
NST
NS2
NSA
NS4
NSS
SC
DF
GCN
42,778 + 0.276
15.080 1.151
12,090 + 1.011
10,438 + 0.864
11.667 + 0.286
6.631 0.532
16.314
0.762
16.245 + 0.392
HGT
43.442 + 0.181
17,550 + 1.038
12.241 + 1.12
11.222 1 0.758
11.639 1 0.334
6,805 + 0.552
19.136 + 0.612
23.358 = 0.195
TAG
41.698 +: 0.336 0.336
17.470 0.97
12.212 0.993
10,953 0.798
11.575 0.318
0.318
6.766 0.563
18.792 0.661
22.177 0.473
GIN
42,712 +: 0.351
17.449 + 1.052
12.285 + 1.011
10.760 + 0.799
11.424 + 0.290
6.782 +: 0.552
18.467 0.635
21.320 + 0.428
DeepReac+
35,748 + 0.375
16,349 + 1.474
13.048
0.699
12,041 + 0,960
12.526 1 0.921
6,992 = 0.391
19.852 1.604
17.188 =: 1.680
YieldBERT
44.733 0.260
16.438
1.153
12.090 0.992
10,955 0.894
11.911 0.342
6.803 +: 0.557
16.104 = 0.620
15.993
0.384
YIELDNET
38.195 +: 0.436
12.415 + 0.866
10.701 +: 0.966
9.702 + 0.969
9.383 1: 0.491
6.507 = 0.535
14.029 = 0.795
9.231 + 0.258
NA Table 7: Comparison of yield prediction performance for YIELDNET against all the competitive baselines, viz., GCN (Kipf
YIELDNETGKy)
31.598 1: 0.448
NA
NA
NA
NA
NA
& Welling, 2017), HGT (Hu et al., 2020), TAG (Du et al., 2017), GIN (Xu et al.), DeepReact (Gong et al., 2021), YieldBERT (Schwaller et al.. 2021c), on the 20% test examples, across all datasets. Performance is measured in terms of Root Mean Squared Error (RMSE). Numbers in green (yellow) indicate the best (second best) performer. Our improvement performance over the next best baseline, where YIELDNET is the best performer, is statistically significant with p-value 0.05, except the cases marked with".
E.2. Statistical significance test
To check the statistical significance of our results, we perform the paired t-test between YIELDNET and each baseline. We report the value of MAE and RMSEs for Table and Table 7 in Table 8. If the p-value between the performance metrics is lower than the 5e-2 margin, we consider the corresponding performance significant.
p-value for MAE table
Model GCN
GP
NS1
NS2
NS3
NS4
NS5
SC
DF
1.5e-10
3.6e-03
7.7e-03
2.7e-02
1.6e-02
3.7c-01*
1.6e-05
1.1e-07
HGT
5.1e-11
3.1e-05
1.7e-03
2.3e-03
2.5e-02
3.7e-02
1.0e-07
5.5e-12
TAG
2.1e-09
1.7e-05
1.6e-03
4.6e-03
2.7e-02
3.9e-02
3.3e-06
4.2e-10
GIN
1.0e-09
2.8e-05
1.8e-03
5.8e-03
2.0e-02
1.1e-02
7.8e-07
4.8e-09
ReepReac+
3.3e-06
3.7e-03
2.4e-03
6.5e-02*
1.8e-02
9.3e-02*
1.0e-03
6.5e-03
YieldBERT
9.6e-11
1.8e-04
4.3e-04
9.8e-02*
1.5e-02
4.9e-02
1.0e-03
3.9e-07
p-value for RMSE table
Model
GP
NSI
NS2
NS3
NS4
NS5
SC
DF
GCN
3.7e-06
4.1e-03
3.5e-03
9.2e-002
4.8e-03
3.1e-02
3.0e-04
8.2e-08
HGT
1.6e-06
9.8e-06
1.9e-03
6.0e-03
5.5e-03
8.3e-03
4.1e-06
4.6e-12
TAG
1.5e-04
8.7e-06
1.9e-03
1.5e-02
5.8e-03
2.7e-02
1.le-05
2.8e-10
GIN
6.7c-06
1.3e-05
1.7e-03
2.5e-02
7.9c-03
2.5e-02
6.9e-06
1.5e-10
DeepReac+ 3.2e-02
4.9e-03
3.1e-03
2.6e-02
7.9e-03
1.7e-01"
2.1e-02
4.6e-03
YieldBERT 3.0e-07
1.9e-04
1.0e-03
5.2e-02*
6.5e-03
3.1e-02
2.0e-03
7.7e-08
Table p-value for MAE and RMSEs for all bascline models compared our model. Our improvement in performance over the next best baseline, where YIELDNET is the best performer, is statistically significant with p-value < 0.05, except the cases marked with".
25

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E.3. Additional results comparison across various atom-to-atom alignment methods
Here, we report the MAE and RMSE with standard error for all atom-to-atom alignment methods introduced in Section 5.2. Table 9 and 10 report the performances, which reveal that our alignment method is the best performer in the majority of the cases. Due diversity in the reaction for GP and SC, they don't 't follow common reaction template, thus atom mapping using RDKit aren't It feasible for them.
Mean Absolute Error (MAE)
Model
GP
NSI
NS2
NS3
RDKit
9,910 0.630
8,845 + 0.873
8.567 + 0.931
RXNMapper
20.604 0.433
9.610 0.602
8.871 0.871
+ 0 0.970
Random
25.187 + 0,407
11.195 0.890
8.984 + 0.993
8.695 + 0.796
Attention
25.730 1 0.758
9.548 + 0.414
9.024 + 0.894
8.432 1 0.934
YIELDNET
23.152 0.393
9.245 0.518
8.387 0.907
7,914 + 0.931
Mean Absolute Error (MAE)
Model
NS4
NS5
SC
DF
RDKit
7.237 + 0.375
4.533 0.246
7.187 = 0.208
RXNMapper
6,871 1 0.406
4.677 +: 0.315
10.046 1: 0.434
6.879 1 0.173
Random
7.991 0.256
4.504 0.243
10.223 0.474
10.595 + 0.244
Attention
6.955 + 0.259
4.317 + 0.220
8.826 + 0.399
7.593 + 0.220
YIELDNET
7.015 + 0.495
4.382 0.249
8.751 + 0.438
6.941 + 0.192
Table 9: Comparison between different alignment strategies in terms of MAE of yield prediction. Brief results were presented in Table 2. Numbers in green (yellow) indicate the best (second best) performer.
Root Mean Squared Error (RMSE)
Model
GP
NS1
NS2
NS3
RDKit
13.133 + 0.802
11.381 = 0.910
10.451 + 0,912
RXNMapper
35.429 1 0.455
12.769 1 0.854
11.366 1 0.852
10.365 1 1.002
Random
37.428 1 0.482
14.827 1 1.185
11.467 1 0.998
10.852 1 0.772
Attention
39.513 + 0.659
12.703 + 0.713
11.484 0.933
10.500 + 0.933
YIELDNET
38.195 + 0.436
12.415 + 0.866
10.701 + 0.966
9.702 + 0.969
Root Mean Squared Error (RMSE)
Model
NS4
NS5
SC
DF
RDKit
9.972 0.599
6.732 + 0.557
9.770 + 0.334
RXNMapper
9.483 + 0,740
6,984 + 0.684
15,486 + 0.773
9.202 + 0.221
Random
10.908 0.421
6.644 1 0.562
15.419 0.839
14.098 0.324
Attention
9.531 + 0.421
6.423 + 0.565
13.719 + 0.710
10.533 + 0.382
YIELDNET
9.383 1 0.491
6.507 1 0.535
14.029 1 0.795
9.231 1 0.258
Table 10: Comparison between different alignment strategies terms of RMSE of yield prediction. Brief results were presented in Table 2. Numbers in green (yellow) indicate the best (second best) performer.
E.4. Additional results on ablation study on CGR representations
Here, we present the MAE and RMSE values for the ablation study on CGR computation introduced in Section 5.2, which extends the results of Table 3. From Table 11 we observe that the influence of CGR computation is benefitting on the yield prediction task for almost all of the datasets.
E.5. Additional results on ablation study of reaction path encoder components
Next, we present the MAE and RMSE values for the ablation study on components of the reaction encoder for all the datasets (Section 5.2), which extends the results of Table From Table 12 we can observe that the influence the SetEncoders way more significant than that the Transformer. Since the GP is a single-step reaction dataset, these ablation studies are not applicable it.
26

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
Mean Absolute Error (MAE)
Method
GP
NS1
NS2
NS3
NS4
NS5
SC
DF
w/ CGR
24.921 = 0.427
11.356 - 0.702
9.459 = 0.953
9.342 + 0.822
7.726 - 0.428
4.495 1 0.270
10.686 + 0.453
9.064 - 0.242
YIELDNET
23.152 = 0.393
9.245 = 0.518
8.387 = 0.907
7.914 = 0.931
7.015 = 0.495
4.382 + 0.249
8.751 = 0.438
6.941 = 0.192
Root Mcan Squared Error (RMSE)
Method
GP
NSI
NS2
NS3
NS4
NSS
SC
DF
CGR
36.68614
0.353
15.249
0.881
12.075
0.949
11.316
0.775
10,506
0.54
6.679
0.585
16. = 0.748
11.827 0.276
YIRLDNET
38.195
0.436
12.415
0.866
10.701
0.966
9.702 0.969
9.383
0.491
6.507
0.535
14.029 + 0.795
9.231
0.258
Table 11: Ablation study CGR computation. Brief results were introduced in Table 3. Numbers in green indicate the best
performer.
Mean Absolute Firor (MAF)
Model
NSI
NS2
NS3
NS4
NSS
SC
DF
StepAggr = DecpSet
9.643 = 0.703
9.008 = 0.915
8.283 + 0.948
6.124 + 0.202
4.395 0.253
8,551 = 0.292
6.712 + 0.264
StepAggr = SumAggr
10.192 + 0.743
8.858 + 0.806
8.349 + 0.909
6.937 1 0.490
4,433 + 0.262
8,853 : 0.309
7.165 1 0.183
Without transformer
9.467 0.516
8.680 0.786
8.164 + 0.849
6.407 + 0.224
4.461 + 0.252
8.628 = 0.504
6,448
0.140
YIELDNET
9.245 0.518
8.387 0.907
7.914 0.931 7.015 + 0.495
4.382 + 0.249
8.751 + 0.438
6.941
0.192
Root Mean Squared Error (RMSE)
Model
NS1
NS2
NS3
NS4
NS5
SC
DF
StepAggr = DecpSet
12,493 + 0.930
11.594 0.912
10.102 + 0.928
8.297 + 0.250
6.451 + 0.543
13.117 + 0.584
8.924 + 0.309
StepAggr = SumAggr
13.341 0.998
11.413 0.916
10.078
0.910
9.222 + 0.596
6.62 0.615
13.442 + 0.551
9.531 0.223
Without transformer
12.407 0.727
10.871 0.845
10.055 0.878 8.900 + 0.424
6.577 0.577
13.485
0.777
8.761 0.182
YIELDNET
12.415 + 0.866
10.701 0.966
9.702 + 0.969 9.383 = 0.491
6.507 + 0.535
14.029 + 0.795
9.231 + 0.258
Table 12: Ablation study on different components of the reaction encoder. Brief results were introduced in Table 4. Numbers
in green (yellow) indicate the best (second best) performer.
E.6. Additional results on ablation study on regularizer
In this section, we present the MAE and RMSE values for the ablation study on regularizer Reg( R, 7) by taking the hyperparameter = (Section 5.2), which extends the results of Table From Table 13 summarizes that the addition of the regularizer improves the performance in almost all of the cases.
Mean Absolute Er (MAE)
Method
GP
NSI
NS2
NS3
NS4
NS5
SC
DF
P=0
23,650 0.465
9.929
0.627
8,590 0.885
8.035
0.903
7.162
4.397
0.244
8.810
0.308
8.959
0.704
YIELDNET 23.152
0.393
9.245 0.518
8.387 0.907
7.914 0.931
7.015
0.495
4.382
0.249
8.751
0.438
6.941 + 0.192
Root Mean Squared Firor (RMSF)
Method
GP
NS1
NS2
NS3
NS4
NS5
SC
DF
0
39.666 + 0.494
13.228 + 0.820
10.892
0.940
9.919 * 0.943
9.815 + 0.313
6.524 + 0.547
13.351 + 0.540
11.942 + 0.822
YIELDNET 385 1 0.436
12.415 1 0.866
10.701 1 0.966
9.702 1 0.969
9.383
0.491
6.507 1 0.535
14.029 1 0.795
9.231 1: 0.258
Table 13: Ablation study on regularizer Reg( (R, 1) in Eq. (21). Brief results were introduced in Table 5. Numbers in green indicate the best performer.
E.7. Comparison between different ways to compute alignment matrix
Here, we compare an alternative way to construct alignment matrix P mentioned in Eq. (5) by choosing Clu, o] hR(u) hr (t) (referred Coor) to chosen one, Clu, v] - max (kR(u), (y())) [$] (referred as Cmax).. From Table 14 we observe almost similar performances for both the C matrices.
Mcan Absolute Error (MAE)
Method
GP
NS1
NS2
NS3
NS4
NS5
SC
DF
Can
22.735 1 0.543
8.709 1 0.416
8.654
0.885
8.121
0.896
6.606 1 0.247
4.365 1 0.267
8,481
0.330
6.964 1 0.132
Cmax
23.152 + 0.393
9.245 + 0.518
8.387 0.907
7.914 + 0.931
7.015 + 0.495
4.382 + 0.249
8.751 + 0.438
6.941 = 0.192
Root Mean Squared Frror (RMSF)
Method
GP
NSI
NS2
NS3
NS4
NSS
SC
DF
Can
37.639 + 0.639
11.459 + 0.599
11.010 = 0.938
10.002 + 0.916
967 = 0.463
6.514 - 0.618
12,984 = 0.601
9.322 - 0.165
Cmax
38.195 1 0.436
12.415 1 0.866
10.701 0.966
9.702 1 0.969
9,383 1 0.491
6.507 1 0.535
14,029 0.795
9.231 1: 0.258
Table 14:
Performance of YIELDNET with different ways to calculate P mentioned in Eq. (5)
27

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E.8. Comparison against competitive baselines with additional datasets
Meun A Absolute Error (MAB)
Model
GPI
GP2
USPTO
DF1
DF2
DF3
NS
GCN
34.771 + 0.399
37.316
0.309
34.33 + 0.345
9.610 + 0.692"
10.098
0.775
10.451
0.823
6.542
0.139
HGT
38.345 0.241
41.798
0.276
34.767 + 0.320
11.246 = 0.843
10.721
0.889
10.989
0.916
8.608
0.218
TAG
36.504 + 0.292
39.757
0.239
34,693 + 0.321
11.200 = 0.774
10.553 0.919
10.877 = 0.919
9.083 + 0.132
GIN
36.184 + 0.184
39.008
0.280
34.896 +: 0.304
11,169 = 0.839
10.373 0.871
10.610 = 0.805
8.883 + 0.155
DeepRcact
29.700 = 0.210
32,577
0.402
36.890 + 0.994
12.302 + 1.693'
10.665 0.928
10.091 = 0.689
9.450 + 0.905
YieldBERT
39.587 + 0.278
44.072
0.280
35.352 + 0.356
10.126 + 0.822*
9.912 0.925
10.475 0.656
7.837 0.154
YIELDNET
24.269 = 0.610
27,563 0.618
33.460 + 0.474
8.832 = 0.747
8,552 0.894
8.815 0.384
6.278
0.194
Root Mean Squared Error (RMSE)
Model
GP1
GP2
USPTO
DF1
DF2
DF3
NS
GCN
40.945 0.347
42.132
0.300"
38.257 0.383
13.165 + 0.963*
12,065
0.688
13.523 0.842
9.055 0.214
HGT
42.667 : 0.209
44.633 0.239
37,913 = 0.374
14.38 = 1.072
12.575
0.819
14.172 0.836
0.2251
+: 0.251
TAG
41.551 + 0.221
43.536 + 0.317
37.970 = 0.397'
14.274 = 1.011
12,602
0.831
14.054 + 0.835
11.974 + 0.188
GIN
41,83 = 0.226
0.257 42.983
38.006 = 0,385
14.399 = 1.071
0.784 12.296
13.741 = 0.748
11.764 + 0.225
DeepReact
37.576 + 0.241
40,961 0.371
42.876 1.157'
15.618 + 1.724"
13.136
0.922
12,902 + 0.839*
12.028 0.917
YieldBERT
44.099 0.165
46.380 0.220
38.446 + 0.396
13,669
1.146*
12.110
0.822
13,529 0.726
10.678 0.217
YIELDNET
38.862 =: 0.622
42.852 0.659
41.191 + 1.122
12.181 = 0.903
10.790 0.865
11.813 0.478
8,777 0.310
Table 15: Comparison of yield prediction performance for YIELDNET against all the competitive baselines, viz., GCN (Kipf
& Welling, 2017), HGT (Hu et al., 2020), TAG (Du et al., 2017), GIN (Xu et al.), DeepReac+ (Gong et al., 2021), YieldBERT (Schwaller et al., 2021c), on the 20% test examples, across the additional datasets. Performance is measured in terms of MAE (top half) and RMSE (bottom half). Numbers in green (yellow) indicate the best (second best) performer. Numbers with indicate that either the model outperforms YIELDNET in that case or the difference not statistically significant (i.e. p-value 0.05). The results reveal the observations in Table
In this section, we present the MAE and RMSE for all the methods the additional datasets viz., NS4, NS5, DFI, DF2, DF3, and NS. Table 15 summarizes the results, which shows that our method outperforms the baselines, similar to Table 1. For the rest of the datasets YIELDNET outperforms the baselines by significant margin in most cases. We observe the results are more prominent in the case of MAE as compared to RMSE.
28

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E.9. Results on approximating the gold permutation for additional datasets
We compute |P P' (mentioned in Section 5.2) for attention and YIELDNET for additional GPI and GP2 datasets
Model
GP
GP1
GP2
Attention
5.096
4.611
4.496
YIELDNET
3.708
3.627
3.165
Table 16: Comparison | -P
as true labels of permutations for Gas-phase reaction datasets are available. Table summarizes that the atom mapping approximation in YIELDNET is superior that of attention.
E.10. Results on the interplay between the quality of learned alignment and predicted yield for additional datasets
We plot (A,,,, AP, vs AP, same Figure 6 for additional GP1 and GP2 datasets in Figure This shows a similar trend i.e. high correlation between error in yield prediction and atom-to-atom alignment for GP1 and GP2 datasets.
0.10
1
0.125
0.08
E BAAEE, E AP.
0.100
0.075
0.04
0.050
0.025
0.02
0.000
0.00
2
P.P
P.
Figure E(AAE, AP,) AP, plot for GP1 (left) and GP2 (right) dataset
29

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E.11. Visualization of condensed graphs where they show the exact transition states
Condensed graphs serve as surrogates of the transition states. We are visualizing some cases for the GP dataset, where they reflect the exact transition state. We select the reactions with low MAE values and collect the alignment matrix, P corresponding to those reactions. Then, apply the Hungarian algorithm to get hard permutation matrices, Pard- We obtain the AdjeGR as follows, AdjoGR max max(AdjjR: P h. Finally from the connections obtained from the AdiCGR. we draw the molecular structures of condensed graphs as shown in Figure 8.
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HI
H
H
Figure 8: Representative examples of the condensed graphs where they show the exact transition states, (shown on the top of the arrow in each case) as generated by YIELDNET during yield prediction.
We use NetworkX (Hagberg al., 2008) get the connections between atoms from AdICOR After obtaining those NetworkX raw graphs manually draw figures of Figure 8 in ChemDraw for better visualization, maintaining the connectivity obtained through the NetworkX graphs.
30

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Learning Condensed Graph via Differentiable Atom Mapping for Reaction Yield Prediction
E. Time complexity
We also report the inference time for the NS3 dataset with the test size of 60 and batch size of 8. From Table 17, is clear that inference time for our model is comparable with the some of the GNN based models (HGT and TAG) and not high YieldBERT.
Model
Inference time in sec
GCN
1.727 + 0.010
HGT
4.684 + 0.087
TAG
3.109 + 0.023
GIN
1.693 = 0.020
DeepReac+
0.900 + 0.026
YieldBERT
84.983 + 0.186
YIELDNET
3.440 + 0.298
Table 17: Inference time for different methods
31


Reviews:
[{"review_id": "frYrQJUQcl", "reviewer": "Reviewer_brLH", "summary": "The paper introduces YIELDNET, a neural yield prediction model designed to predict the yield of multi-step chemical reactions without explicit atom mapping supervision. The key contributions include a differentiable node alignment network to approximate atom mapping, the construction of a CGR as a surrogate for transition states, and a transformer-guided reaction path encoder to model multi-step reactions. The approach enables end-to-end learning with supervision only from yield values, outperforming all baselines across eight datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their suggestions, which we address as follows.\n\n> *theoretical analysis*\n\nWe provide the following theoretical underpinning. We start with the QAP in Eq 8. If $\\mathcal{P}$ is the set of permutations,  then QAP is\n$$\\min  _{P\\in \\mathcal{P}}\n\\underbrace{\\sum _{u,v}  \\max (Adj  _R,P Adj _I P^{\\top}) [u,v]} _{:=c(R,I;P)} --- (QAP)\n$$\nThe standard way to minimize it is to use Gromov Wasserstein (GW) based projection [A1], where P is updated as\n\n$$    P  \\leftarrow  \\text{argmin}  _{P} \\textrm{Trace}\\left(P^T\\nabla  _{P}\\ c(R,I;P) \\right)  -- (grad) $$\nIn our work, we provide a neural approximation of the gradient $\\nabla  _{P}\\ c(R,I;P) \\approx [\\sum  _{i=1} ^d  \\max (H  _R[u,i], H  \n _I [u',i])] _{u,u'}$. We use max to keep  aligned with max() in Eq (QAP). This approximates Eq. (grad) as the following linear OT (Eq 9): \n\n$$ \\min  _{P\\in \\mathcal{P}}  \\underbrace{ \\sum  _{u,u'}  \\sum _{i=1} ^d  \\max(H  _R[u,i], H  _I [u',i]) P _{uu'}} _{F(P)}  -- (a) $$\n\nEq (a) is optimization over permutations, which may appear to be computationally hard. However, we can write it as an instance of the following optimization over the space of doubly stochastic matrices $B = (P: P   \\ge 0,  P\\mathbf{1} = P^T\\mathbf{1} = 1)$.\n\n$$ \\min _{P}F(P),  \\text{  s.t. }  P\\in B --- (b)  $$\n\nEq (b) is equivalent to Eq (a) since it is an linear program.The optimal solution of a linear program always coincides with some vertex of the space induced by the set of linear constraints. Thus, the optimal solution of Eq (b) is a permutation matrix $P^*$, as the set of permutation matrices are the vertices of B, the space   of doubly stochastic matrices.  \n\nHowever, $\\arg\\min _{P}F(P)$ in Eq (b) is non-differentiable. To enable differentiation, we approximate it using Sinkhorn iterations, analogous to how argmax is approximated using softmax. Given n numbers $G_1,..G_n, \\ \\ \\mathbb{1}[i=\\arg \\min _{j} G_j] \\approx p_i=\\frac{e^{-G_i/\\lambda}}{\\sum _{j}e^{-G _{j}/\\lambda}}$, where $\\mathbb{1}$ is indicator function. One can show that $p_i: i\\in [n]$  minimizes the following entropy regularized optimization problem:\n\n$$\\min _{p} \\sum _{j} p _j G _j +\\lambda. \\sum _{j} p _j \\log(p _j),\\text{ such that } 0 \\le p _j \\le 1, \\sum _{j} p _j =1$$\n\nUsing the same technique, we can show that Sinkhorn mechanism minimizes the entropy regularized linear OT problem $\\min _P F(P) - \\lambda \\text{Entropy}(P)$ [A2].\n\n[A1] Xu et al. Gromov-wasserstein learning for graph matching and node embedding, ICML 2019\n\n[A2] Mena et al. Learning Latent Permutations with Gumbel-Sinkhorn Networks. ICLR 2018.\n\n> *Computational overhead* \n\nFollowing table shows the computational overhead in terms of inference time. It shows: our model is comparable with the some of the GNN based models (HGT, TAG) and not as high as YieldBERT.\n\n|Model|time (sec)|\n|-|-|\n|GCN|1.73|\n|HGT|4.68|\n|TAG|3.11|\n|GIN|1.69|\n|DeepReac+|0.9|\n|YieldBERT|84.98|\n|YIELDNET|3.44|\n\nMoreover, the training time per epoch for our model is 8 seconds (s), which is comparable with baselines: 6s for HGT, 10s for YieldBERT (with same batch size)\n\n*Efficiency enhancement:* Low rank OT provides highly efficient Sinkhorn iterations. See Scetborn et al. Low-Rank Sinkhorn Factorization, ICML 2021. One first apporximates $M = \\sum _{i=1} ^d  \\max (H  _R[u,i], H  _I [u',i])\\approx AB^T$ where $A$ and $B$ are $N \\times d$ and $N \\times d$ low rank matrices. Similarly, we approximate the transport matrix $P$ as $P \\approx Q D(1/g) R^T$ where $Q, R$ are $N \\times r$ matrices, $g$ is $r \\times 1$, and $D(1/g)$ is a diagonal matrix with entries $1/g_i$. To ensure $P$ is doubly stochastic, we enforce:\n\n$$Q\\mathbf{1} = \\mathbf{1}, \\quad \\mathbf{1}^T Q = g^T$$\n\n$$R\\mathbf{1} = \\mathbf{1}, \\quad \\mathbf{1}^T R = g^T$$\n\n$$g^T \\mathbf{1} = 1, \\quad Q, R, g > 0$$\n\nInstead of minimizing $Tr(P^T M)$, we minimize:  $Tr((Q D(1/g) R^T)^T AB^T)$ using alternating minimization wrt $Q, R, g$. To optimize $Q$, we rewrite:\n$$\nTr((Q D(1/g) R^T)^T AB^T) = \\langle Q, AB^T R D(1/g) \\rangle\n$$\nWe apply Sinkhorn iterations to: $X = AB^T R D(1/g)$, computed in 3 steps. Note that complexity of multiplying $m \\times n$ and $n \\times p$ matrices is $O(mnp)$ \n\n1. Compute $R D(1/g)$, which is $O(Nr)$ since $R$ is $N \\times r$, $g$ is $r \\times  1$.  \n2. Compute $B^T R D(1/g)$ which is $O(dNr)$ since $B^T: d \\times N$ and  $R D(1/g): N\\times r$\n3. Compute $X = AB^T R D(1/g)$ which is $O(Ndr)$  since $A: N \\times d$ and  $B^T R D(1/g): d\\times r$\n\nThus, complexity reduces from $O(N^2)$ to $O(Ndr)$. The same efficiency gain applies when optimizing $R$ and $g$.  \n\n> *lighter model*\n\nWe replaced some complex components with simpler components in our model. MAE numbers are as follows.\n\n||NS1|NS2|NS3|\n|-|-|-|-|\n|step-Aggr=SumAggr|10.19|8.86|8.35|\n|No-Transformer|9.47|8.68|8.16|\n|Our|9.25|8.39|7.91|\n\nWe observe that there is some drop in performance. Depending on available resources, one can decrease the complexity to build a lighter model.", "claims_and_evidence": "The paper claims that YIELDNET can predict reaction yields more accurately than existing methods by leveraging an approximate transition state representation. This claim is supported by experimental comparisons showing significant improvements over baselines. The ablation studies on different components of the model, such as CGR representations, reaction path encoder components, and the regularizer, provide evidence for the importance of these elements in the model.", "methods_and_evaluation_criteria": "The proposed methods make sense for the problem. The differentiable node alignment network is a novel approach to approximate atom mapping, which is a crucial step in the reaction yield prediction problem. The evaluation criteria, including using MAE and RMSE to measure the performance on test sets, are standard and reasonable for this type of prediction task. The datasets cover a variety of reaction types for evaluating the model's performance.", "theoretical_claims": "The paper does not present complex theoretical proofs. The design of the node alignment network is based on relaxation of a quadratic assignment problem (QAP) to a linear optimal transport (OT) problem. The authors claim that the alignment matrix obtained is the solution of the entropy regularized linear OT problem. While the theoretical background is well presented, it would be beneficial to have a more detailed proof or reference to a more in-depth theoretical analysis to ensure the correctness of this claim.", "experimental_designs_or_analyses": "The experimental setup is sound, with multiple datasets and comparisons against reasonable baselines. The paper provides quantitative evidence of YIELDNET’s superiority. The ablation studies are well designed to analyze the impact of different components of the model", "relation_to_broader_scientific_literature": "The key contributions of the paper are closely related to the broader scientific literature. In the area of reaction yield prediction, previous works mainly rely on quantum chemically computed molecular descriptors, molecular fingerprints, or SMILES based representation. YIELDNET differentiates itself by leveraging graph neural networks and approximating atom mapping and transition states.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths: 1. The combination of differentiable atom mapping, CGR approximation, and reaction path encoding in a single model for yield prediction is novel.\n2. Strong empirical performance across multiple datasets.\n3. The work has practical significance as accurate yield prediction can help in chemical synthesis design and optimization.\nWeaknesses: The method may face challenges when applied to larger molecular reactions due to the O(N^{2}) complexity of computing the alignment matrix P. Although the authors mention a possible mitigation strategy, it needs further verification.", "other_comments_or_suggestions": "No", "questions_for_authors": "1.What is the computational overhead of this method compared to baselines? The Sinkhorn operator involves iterative row and column normalization over a cost matrix. Given an N×N permutation matrix, the complexity is approximately O (T N^2), where T is the number of Sinkhorn iterations. Additionally, the gradient computation for backpropagation through Sinkhorn updates introduces further memory and compute costs. For a multi-step reaction with n steps, the embeddings from each step are concatenated and passed through a transformer.\n2.Would a lighter-weight model maintain accuracy while improving efficiency?", "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": "BBNYhtGamX", "reviewer": "Reviewer_Vr41", "summary": "Predict permutation-matrix of a chemical reaction with GNN atom embeddings and iterative Sinkhorn interactions. Sampling from the permutation-matrix to get the atom-mapping which is used by a Transformer to predict the yield.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments, which we address below.\n\n> *results for atom mapping*\n\nAppendix E.3 contains the table with +- std error. Paired t-test in majority of cases revealed that the performance gain achieved by our method is statistically significant with $p=0.01$\n\n> *Approach is novel but memory intensive*\n\nOur method utilizes the most compute on computing the permutation matrix using Sinkhorn iterations, which has $O(N^2)$ complexity if $N$ is the total number of nodes. However, this is not a hard bottleneck and can be easily overcome using low rank OT.\n\nLow rank OT provides highly efficient Sinkhorn iterations (see Scetborn et al. Low-Rank Sinkhorn Factorization, ICML 2021). We first apporximate $M = \\sum _{i=1} ^d  \\max (H  _R[u,i], H  _I [u',i])\\approx AB^T$ where $A$ and $B$ are $N \\times d$ and $N \\times d$ low rank matrices. Similarly, we approximate the transport matrix $P$ as $P \\approx Q D(1/g) R^T$ where $Q, R$ are $N \\times r$ matrices, $g$ is $r \\times 1$, and $D(1/g)$ is a diagonal matrix with entries $1/g_i$. To ensure $P$ is doubly stochastic, we enforce:\n\n$$\nQ\\mathbf{1} = \\mathbf{1}, \\quad \\mathbf{1}^T Q = g^T\n$$\n\n$$\nR\\mathbf{1} = \\mathbf{1}, \\quad \\mathbf{1}^T R = g^T\n$$\n\n$$\ng^T \\mathbf{1} = 1, \\quad Q, R, g > 0\n$$\n\nInstead of minimizing $Tr(P^T M)$, we minimize:   $Tr((Q D(1/g) R^T)^T AB^T)$ using alterting minimization wrt $Q, R, g$. To optimize $Q$, we rewrite:\n$$\nTr((Q D(1/g) R^T)^T AB^T) = \\langle Q, AB^T R D(1/g) \\rangle\n$$\nWe apply Sinkhorn iterations to: $X = AB^T R D(1/g)$, computed in 3 steps. Note that complexity  of multiplying $m$ x $n$ and $n$ x $p$ matrices is $O(mnp)$ \n\n1. Compute $R D(1/g)$, which is $O(Nr)$ since $R$ is $N \\times r$, $g$ is $r \\times  1$.  \n2. Compute  $B^T R D(1/g)$ which is $O(dNr)$ since $B^T: d \\times N$ and  $R D(1/g): N\\times r$\n3. Compute  $X = AB^T R D(1/g)$ which is $O(Ndr)$  since $A: N \\times d$ and  $B^T R D(1/g): d\\times r$\n\nThus, complexity reduces from $O(N^2)$ to $O(Ndr)$. The same efficiency gain applies when optimizing $R$ and $g$.  \n\nMoreover, in terms of time efficiency,   our model is comparable with the some of the GNN based models (HGT and TAG) but not as high as YieldBERT.\n\n|Model|time (sec)|\n|-|-|\n|GCN|1.73|\n|HGT|4.68|\n|TAG|3.11|\n|GIN|1.69|\n|DeepReac+|0.9|\n|YieldBERT|84.98|\n|YIELDNET|3.44|\n\n\n> *USPTO ...not in the main Table 1?*\n\nWe agree with the reviewer. We would revise Table 1 to include the USPTO results in the main text.\n\n\n> *....activation energy directly impacts yield; what is the precise relationship*\n\nFor a single-step reaction assuming first order kinetics, the relationship is as follows.\nSuppose,  $\\Delta G^\\ddagger$ represents activation energy in Jmol$^{-1}$.   $k_B, h, R$ are the Boltzmann constant ($1.38\\times 10^{-23}$ JK$^{-1}$), Planck constant ($6.626\\times 10^{-34}$Js), and universal gas constant ($8.314$ JKmol$^{-1}$) respectively. Here, $t$ is the reaction time in sec. If \"Temp\" is the reaction temperature, then $k$, the first order rate constant is:\n$$  k = \\dfrac{k_B \\text{Temp}}{h} \\exp\\left(\\dfrac{-\\Delta G^{\\ddagger}}{R\\cdot\\text{Temp}}\\right),$$\n Yield is given by:\n$$yield = (1 - \\exp(-k t)) \\times 100$$\n\n\n\n\n\n\n\n> *conditions such as person executing the experiment, temperature, etc. .... I guess it's fair to distinguish between theoretical idealized yield and practical yield.*\n\nThere are several factors that could influence yields. We do incorporate both catalyst and solvent, when present, into the reactant set R.   Temperature was excluded as it is invariant in the case of high-throughput experimentation datasets (e.g., DF and NS datasets), which maintain consistent reaction conditions throughout, or varies mildly for others (e.g., SC). Other variabilities (e.g., purification procedure) are lacking in the state-of-the-art datasets, which is why we were unable not consider them. We discussed these factors in the conclusion. We will elaborate them further, if our paper gets accepted.\n\n\n\n> *why not incorporate the whole reaction and perform attention over that*\n\nYes, $H_{CGR}$ captures the signals from atom mapping. We experimented with the transformer model where we incorporate whole reaction and perform attention over all reaction components, without explicit CGR modeling.  Following MAE values show that our method performs significantly better.\n\n| Method|NS1|NS2|NS3|SC|\n|-|-:|-:|-:|-:|\n|Whole-reaction-attention|11.356|9.459|9.342|10.686|\n|**YIELDNET**|**9.245**|**8.387**|**7.914**|**8.751**|\n\nDespite the increased model size, whole-reaction-attention performs worse. We believe this is due to the fact that attention is non-injective in nature, i.e., multiple atoms from reactants can be mapped one single atom in product. In contrast, permutation is injective-- they provide one-to-one mapping between atoms. One way to mitigate the problem of transformer  is to design permutation induced transformer, which we believe has a strong potential in this direction.", "claims_and_evidence": "- propose differentiable approximation of the atom-mapping: I would argue that e.g. RXNMapper is also a \"differentiable approach to atom mapping\" but in their evaluation they show that they outperform RXNMapper (Table 2). Here also +- standard deviation as in Table 1 would be nice as well as statistical significancance tests (further state which test, and assumtion has been used). Their approach is novel and performant but seems memory intensive. \n- proposed YieldNet outperforms several baselines: experimentally validated in Table 1; code and data provided to reproduce as supp. across multiple datasets via 10-fold cross-validation; (why is USPTO in the supp but not the main Table 1?)", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "Not checked.", "experimental_designs_or_analyses": "All evaluations are sound.\nFurther evaluations on different splits might be interesting but overall the analysis is done in a just manner.", "relation_to_broader_scientific_literature": "Key contributions could be demarcated better from scientific literature.", "essential_references_not_discussed": "OK", "other_strengths_and_weaknesses": "Original - yes\nSignificance - yes \nClarity - yes - paper is very well explained (minor suggestions follow)", "other_comments_or_suggestions": "Suggestions to add to clarity: \n- you mention several times that activation energy directly impacts yield; what is the precise relationship\n- it seems that the topic of yield in reality is much more nuanced then described in the manuscript: e.g. conditions such as person executing the experiment, temperature, .... I guess it's fair to distinguish between theoretical idealized yield and practical yield.\n- For reaction-path encoder - how is the atom mapping encoded here / how does it find it's way to the final yield? it seems the H_CGR is an embedding that holds this information for one step --> why not incorporate the whole reaction and perform attention over that", "questions_for_authors": "no further questions", "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": "Me6IKUB40w", "reviewer": "Reviewer_TZPS", "summary": "The paper introduces YIELDNET, a neural model designed to predict chemical reaction yields by learning a condensed graph representation of reactions. Unlike traditional methods that rely on quantum chemistry-based molecular descriptors, YIELDNET approximates atom mapping—the correspondence between reactant and product atoms—using a differentiable node alignment network. This mapping allows the construction of a Condensed Graph of Reaction (CGR), a supergraph that serves as a surrogate for the transition state, a crucial determinant of reaction yield. The model then processes CGR embeddings through a transformer-guided reaction path encoder to predict yields. A key advantage of YIELDNET is that it operates under distant supervision, meaning it does not require explicit atom mapping or transition state labels during training, yet it outperforms baseline models. This approach enhances inductive bias by integrating a differentiable approximation of the transition state, improving the accuracy of yield predictions in multi-step chemical reactions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments, which we address as follows.\n\n> *CGR meaningfully approximates TS?*\n\nDuring rebuttal, we performed quantum chemical validation analogous to that reported by Choi, in 'Prediction of transition state structures of gas-phase chemical reactions via machine learning'  Nature Commun. 2023.\n\nHere, we have used our CGR based TS as the initial guess for quantum chemical optimization for reactions in a GP dataset. We observed that 80% of these converged to TS. Subsequent QM calculations could produce correct reactant and product from the TS for 60% cases.\n\nIn addition, we also compare CGR-based TSs with true (QM-computed) TSs using  two metrics:\n\n(1) Accuracy: The fraction of total connections in the CGR-based TS which match with the  QM computed TS.  \n(2) RMSD:  Root mean squared deviation (RMSD -lower the better) between atom positions obtained from CGR-based TS and the QM computed TS. Results for the GP dataset for our method and other interpolation based methods as follows. We observe that our method performs better. \n\n ||Accuracy |RMSD|\n |-|-|-|\n |0.8R+0.2P|0.51|1.50|\n |0.5R+0.5P|0.71|1.24|\n |0.2R+0.8P|0.91|1.33|\n |Our|**0.95**|**0.71**|\n\n \nOur CGR serves as a surrogate for TS to enhance yield prediction, not to provide an accurate TS prediction.  More accurate TS prediction requires TS supervision during training, which most yield datasets lack.  Complex datasets like SC contain ~200 atoms per reaction. Here, QM calculations on TSs take a couple of days per reaction--- assuming a good initial TS guess, which is even more time-consuming given the complexity of the dataset. Hence, general evaluation for our CGR was infeasible.\n\nHowever, as the reviewer suggested, we performed experiments on reaction instances in the GP dataset, which shows that despite absence of supervision, our CGR shows better proximity to the TS than its alternatives.\n\n> *compare to quantum-ML models*\n\nWe  feed the QM-optimized-TS (TS*) in our model to compute yield prediction error $E^*(r) = |y^*(r)-y _{TS^*}(r)|$ for each reaction $r$ and then compute yield prediction error $E(r) = |y^*(r)-y _{CGR}(r)|$ provided by our CGR based TS. The average difference between these errors is $<$ 8% of QM-optimized-TS based yield prediction error.\n\n> *generalize to different reaction classes*\n\nWe compute MAE on test examples that include reactions from SC dataset, whereas the model is trained on DF dataset. We observe that our method shows better performance.  \n\n|||\n|-|-|\n|GCN|13.40|\n|HGT|25.38|\n|TAG|18.52|\n|GIN|17.03|\n|YB |23.64 |\n|Our |10.60 |\n\n> *ignore reaction conditions?*\n\nWe do incorporate both catalyst and solvent, when present, into the reactant set R. We will explicitly mention them in the revised paper. Temperature was excluded as it is invariant in the case of high-throughput experimentation datasets (e.g., DF and NS datasets), which maintain consistent reaction conditions throughout, or varies mildly for others (e.g., SC). Integrating temperature into the node/edge features might enhance CGR or yield prediction quality.\n\n> *a catalyst as a part of the reagent*\n\nIn i-th step of a multi-step reaction,  the reactant set $R_i$ is a disjoint graph of all the participating species including catalyst, base, solvent, etc.  \n\n>*Failure mode*\n\nUpon close inspection, we observe that our method couldn't do well in reactions that are very unusual/rare in terms of bond formation and breaking, due to their under-representation in training samples. Poor predictions on such  reactions are unlikely to make any adverse impact, since they are too rare. If the dataset labels (yield values) carry large measurement errors, then also our model (and the baselines) would perform poorly. \n\n>*lack yield visualization*\n\nyhist.pdf in https://bit.ly/rbynet  shows that our predicted yield distribution mimics true yield distribution, very closely.\n\n>*yield for rearrangements, pericyclic processes*\n\nOur datasets include rearrangements (present in GP), pericyclic processes (present in GP), metal-catalyzed bond formations (all reactions in SC), etc. \nFollowing are MAE based comparisons.\n\n|-|Our|Closest baseline|\n|-|-|-|\n|rearrangement|21.82|DeepReac+: 28.79|\n|Pericyclic|23.26|GCN: 24.16|\n|Metal-catal.|8.75|DeepReac+: 10.44|\n\n> *experimental design has limitations, dataset, benchmarking*\n\nWe evaluated over 10 SOTA datasets against six baselines across ten splits for all experiments, including ablation study, to maintain consistency. We believe that our work conducts experiments with more rigor and comprehensiveness than any previous yield prediction baselines.\n\nWe found that the pre-final layer embedding $z_r$ (Eq. 19) correlates with activation energies (corr.pdf in https://bit.ly/rbynet). Extracting mechanistic signals from yield alone is challenging and no prior yield prediction work attempts it-- they only focus on black box yield predictors. Hence, our work makes notable progress, paving the way for future research.", "claims_and_evidence": "Several claims are stated in this paper, including:\n\n1) The CGR serves as a surrogate for the TS -- The claim is not rigorously validated. The claim relies on structural similarity but ignores electronic structure and activation energy, which are critical in TS determination. Transition states require potential energy surface calculations, which the authors do not provide. A surrogate for the TS must be benchmarked against computed TS structures to be credible. Even though in Figure 8 the author shows how TS is represented, for more complicated reactions, the middle state can also be intermediate. This simplicity of the whole reaction scheme should be emphasized.\n\n2) The differentiable atom mapping is an effective approximation of the true atom mapping. The authors state that YIELDNET approximates atom mapping via a differentiable node alignment network (using Sinkhorn iterations). Experiments proved that the model outputs a doubly stochastic alignment matrix that relaxes hard permutation constraints, which mitigates NP-hard graph matching challenges. However, the improvement compared to other mapping methods is marginal. Test cases where the mapping fails are not mentioned —does it work for rearrangements, symmetry-driven reactions, and pericyclic processes? What about other edge cases (e.g., radical shifts, metal-catalyzed bond formations)?\n\n3) The author claims that yield prediction can be learned w/o explicit TS information, giving experiments of empirical comparison with existing ML models for yield prediction. But thermodynamically, the yield is decided by: 1) reaction duration + activation energy height; 2) activation energy barriers comparison of multiple reaction pathways that compete with each other. The interpretation of the inherent state does not guarantee any physical understanding or explanation of its performance.\n\n4) The author states that YIELDNET generalizes well across different reaction types\n.", "methods_and_evaluation_criteria": "The author performs proper methods to design the model and organized a reaction dataset for the problem.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experimental design and analysis in YIELDNET focus on reaction yield prediction using differential atom mapping, and condensed reaction graphs (CGRs). While the methodology is innovative, the experimental design has limitations in terms of chemical validity, dataset composition, benchmarking, and mechanistic interpretability.\n\nThe paper claims to predict reaction yields by modeling CGR embeddings and atom mappings, but it does not explicitly explain the relationship between TSs defined by the energy barrier and the connectivity graphs used in their model. Besides, other terms that will influence yield (reaction time, solvent, catalysts...) are ignored. Since yield is highly condition-dependent, the author should consider design and analyze the limitations.\n\nThe authors collected several datasets for the training and test of YIELDNET, but it lack detailed yield distribution visualization and reaction introductions to help readers understand the difference between the dataset and the potential bias involved.", "relation_to_broader_scientific_literature": "YIELDNET could be a potentially useful model for yield prediction. However, the paper still lacks sufficient connection to fundamental chemistry, including physical organic chemistry principles such as transition state theory, kinetic control, and thermodynamics. To make it a more persuasive model with chemistry understanding, the author should report the validation of model generalization on unseen reaction classes.", "essential_references_not_discussed": "Related and up-to-date papers are properly cited.", "other_strengths_and_weaknesses": "Strengths: The introduction of differentiable atom mapping for reaction yield prediction is an innovative step. Unlike traditional rule-based or SMILES-based atom mapping methods, YIELDNET relaxes atom mappings into a soft differentiable space using the Sinkhorn algorithm.\n\nWeakness: The model is only tested on a limited dataset and the generalization ability of the model is still a mystery. Besides, The model does not analyze when and why YIELDNET fails and how missing reaction condition embeddings influence the performance.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. How does the Condensed Graph of Reaction (CGR) meaningfully approximate the transition state? The paper claims that the CGR serves as a surrogate for the transition state (TS), but it does not include quantum mechanical validation of this claim. If I can surrogate the \"intermediate\" with 0.8*R+0.2*P; 0.5*R+0.5*P,0.2*R+0.8*P would that be helpful?\n\n2. How does YIELDNET compare to quantum-ML hybrid models (with QM-optimized TS) for yield prediction? If it does lend the TS information, there should be similarities.\n\n3. How does YIELDNET generalize across different reaction classes?\n\n4. Why does YIELDNET ignore reaction conditions (temperature, solvent, catalyst effects)? How you would design the model to incorporate this situation as the condensed graph may introduce a catalyst as a part of the reagent during a specific step of the reaction?\n\n5. What are the major failure modes of YIELDNET?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This work investigates the problem of predicting yield for (potentially multi-step) reactions, based on structures of the molecules involved alone (without transition state or atom mapping information). This is approached by designing a model which predicts a soft matching between reactant and product atoms, then uses it to create a surrogate for the transition state, and finally processes that to predict yield. Authors show that training this end-to-end with only yield supervision leads to accurate predictions and even learning to predict atom mapping as a by-product.

After the rebuttal period, all three reviewers were voting to accept this paper, with varying degrees of confidence. On the positive side, reviewers praised sensible model design (which draws upon assumptions appropriate for the domain, e.g. transition state influencing yield), strong empirical performance across datasets, and real-world significance (as yield prediction is an important problem in chemistry and drug discovery). On the negative side, some reviewers were initially worried about resource requirements (both time and memory) and generalization; however, these concerns were well-addressed by authors during the discussion period.

Having read the manuscript in detail, I agree with the unanimous reviewer decision: this paper is a solid contribution to ICML. It reads very well, and shows a thoughtfully selected array of experiments and ablations. I recommend acceptance (of medium strength). My main rationale for not going for a strong accept recommendation was that this work may have limited applicability outside of yield prediction (and even there, it's only applicable to reactions with fully specified side products), thus expected audience may be slightly smaller than for an average accepted paper.
Decision: Accept (poster)