Neural Petri flows for chemical reactions

Jose Eduardo Escrig Molina; Daniel Probst

cs.LGOct 6, 2026

Neural Petri flows for chemical reactions

Authors: Jose Eduardo Escrig Molina, Daniel Probst

Organizations: Bioinformatics Group Bioprocess Engineering Group Wageningen Unversity Wageningen, the Netherlands

Abstract

Petri nets have been used to describe chemical processes such as reactions.They map well to chemistry: Places are the bonds between atoms and the free valence of each atom, a token is a unit of bond order, a transition forms or breaks a bond, the conserved quantities are the valence budgets of the atoms, and the enabling rule is the valence rule. These semantics are not guaranteed by learned models of reactions or neural networks that are built on Petri nets that use the net as a scaffold for message passing. Here, we ask what architecture remains a Petri net for every value of its weights. We find the answer in the theory, where all semantics of a net share the firing form $m^\prime=m+Cσ$ , locality, as enabling reads only the inputs of a transition, and the enabling rule, and we prove that conservation forces the firing form and that non-negativity forces the enabling rule on local rate laws. This leaves free the rate law, which is the propensity of each transition to fire. We introduce Neural Petri Flow, which learns this rate law, or a readout for classification, and hard-wires the rest as parameter-free layers. On what we denote a valence net, atom mapping, reaction classification, and forward prediction become three tasks on one firing vector. Without training, the minimum firing vector maps 88.8% of the curated Golden set against 85.6% for RXNMapper, and 88.7 against 77.9% of the enzymatic reactions of EnzymeMap. On USPTO-480K, NPF trained on these firing vectors predicts 87.7% of the products and 67.4% when trained on a 1% subset of the training reactions. EC numbers of ECREACT are predicted at the third level for 90.2% of reactions, 5.6 points ahead of the best published method. With electrons as tokens, the same token game predicts 90.5% of the elementary steps of FlowER first, ahead of the published baseline, and every top-1 prediction is a valid molecule without a filter.

Figures & tables

Panel of Figure 1 — Figure 1: (a) A reaction as two markings of its valence net. (b) Only the rate law (magenta) is learned. (c) The minimum firing vector $\sigma$ maps atoms, trains the token game and feeds the classifier.

method	learned	Golden	NatComm	USPTO-3k	Recon3D	E. coli
RDT	rules	82.54	84.11	90.87	54.97	78.02
RXNMapper	yes	87.43	87.58	93.53	48.69	72.53
LocalMapper	yes	89.08	92.67	97.77	50.79	69.96
GraphormerMapper	yes	89.59	92.87	95.10	34.82	42.12
NPF, minimum firing vector	no	89.70	90.63	92.13	57.85	72.89

Table 1: Atom mapping on SynRXN ( Phan et al., 2026 ) . Best in bold , second underlined . Statistical tests and detailed numbers can be found in Table 9 .

Schneider50k (50 classes) $n$ accuracy macro-F1 DRFP + MLP 3 $95.52\pm 0.04$ $95.47\pm 0.04$ PGNN, readout 3 $97.67\pm 0.07$ $97.65\pm 0.07$ PGNN, firing vector 5 $98.64\pm 0.09$ $98.63\pm 0.09$ NPF, readout 5 $97.68\pm 0.09$ $97.67\pm 0.10$ NPF, firing vector 5 $98.80\pm 0.04$ $98.80\pm 0.04$
ECREACT, Enzyformer split	EC4	EC3	EC2	EC1	CARE task 2, easy	EC4	EC3	EC2	EC1
BEC-Pred	–	76.9	81.2	85.8	CLIPZyme	12.2	39.9	61.8	79.9
DRFP + CLM	–	81.7	85.6	88.3	DRFP similarity	59.3	77.1	85.2	90.6
Synthcoder-DistilBERT	–	84.4	86.7	90.9	CREEP	39.4	66.4	79.9	92.9
Enzyformer	–	84.6	88.2	90.9	NPF, no map	$67.85$ $\pm 0.39$	$82.27$ $\pm 0.29$	$89.74$ $\pm 0.39$	$95.76$ $\pm 0.59$
NPF	$75.75$ $\pm 0.18$	$90.20$ $\pm 0.15$	$92.45$ $\pm 0.23$	$95.40$ $\pm 0.23$	NPF + mapper	$70.23$ $\pm 1.59$	$84.31$ $\pm 0.64$	$92.03$ $\pm 0.78$	$96.78$ $\pm 1.03$

Table 2: Reaction classification, accuracy and macro-F1 in %,

\pm

: sample standard deviation over

n

seeds, three for NPF on ECREACT and CARE. ECREACT, split of Liu et al. (2026) , 4907 test reactions, baselines the mean over three runs. CARE task 2, easy split, 393 test reactions, baselines from Yang et al. (2024) . Best in bold , second underlined .

method	prediction modality	top-1	top-3	top-5
Molecular Transformer † ( Schwaller et al., 2019 )	SMILES	0.886	0.935	0.942
Graph2SMILES † ( Tu and Coley, 2022 )	SMILES	0.903	0.940	0.948
MEGAN † ( Sacha et al., 2021 )	Graph edits	0.863	0.924	0.940
GTPN † ( Do et al., 2019 )	Graph edits	0.832	0.860	0.865
NERF † ( Bi et al., 2021 )	Electron edits	0.907	0.933	0.937
MAELLE ( Xuan-Vu et al., 2026 )	Electron flow	0.872	0.930	0.939

Table 3: Top-

k

accuracy comparison on USPTO-480K. Entries taken from the original papers, as collected by Xuan-Vu et al. (2026) , are marked with † . NPF is scored as in Appendix B , one seed. With the enabling rule, every NPF prediction stays within the valence budgets of its atoms (Table 17 ).

		step, top- $k$				pathway, top- $k$
method	params	1	2	3	5	1	2	3	5
Molecular Transformer	12M	88.75	96.62	98.42	98.92	88.31	95.64	97.02	97.59
Graph2SMILES	18M	89.09	96.75	98.08	98.66	92.51	95.94	97.03	97.76
Graph2SMILES+H	18M	87.39	95.31	96.87	97.67	89.22	93.70	95.10	96.19
FlowER	7M	88.48	96.42	97.96	98.60	88.97	94.75	96.48	97.39
FlowER-large	16M	89.74	97.40	98.66	99.13	92.50	96.89	98.15	98.61

Table 4: Top-

k

accuracy (%) on the FlowER test set for individual steps (

162\,002

) and full pathways (

28\,049

reactions). A pathway counts as correct if some route to a recorded end product has every step in the top

k

. Steps from PMechDB and RMechDB are excluded. The upper limit reflects steps with more than one valid outcome. NPF ranks by beam probability. Baselines rank by sample frequency and come from Fig. 2 of Joung et al. (2025) . Best in bold , second underlined .

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 2 — Figure 2: A bromo Suzuki-type coupling as a token game of the model trained without precomputed mappings. (a) Showing the most probable order, each bond that changes in magenta, dashed if formed. $\odot$ is a token that a broken bond returns to the slack place of its atom. The order is valence bookkeeping, not the Pd mechanism, which would break C-Br first. (b) Forming C-C requires a token of free valence on both aryl carbons that can only be supplied by the two breaks, meaning enabling excludes the four orders that form C-C before the two bond breaks. The two remaining orders fire the same transitions and reach the same product.

method	within studies	new study	new carbon source	balance
NPF, GMA rate law	0.114 $\pm$ 0.002	0.153 $\pm$ 0.013	0.385 $\pm$ 0.002	0.006
NPF, rate law described in Section C.1	0.149 $\pm$ 0.007	0.157 $\pm$ 0.019	0.363 $\pm$ 0.023	0.019
PGNN, GMA rate law	0.119 $\pm$ 0.008	0.260 $\pm$ 0.019	0.798 $\pm$ 0.053	2 216
PGNN	0.144 $\pm$ 0.023	0.280 $\pm$ 0.019	0.858 $\pm$ 0.070	1 009
PGNN+ with the state equation, GMA	0.211 $\pm$ 0.017	0.208 $\pm$ 0.072	0.399 $\pm$ 0.025	0.008
random forest, projected onto the net	0.095	0.161	0.350	0.000

Appendix

Table 5: Relative error of the fluxes of held-out cultures of E. coli , mean

\pm

sample standard deviation over three seeds for the learned models, within the studies, on a held-out study and on other carbon sources, and the balance residual

\lVert C\sigma-(m_{B}-m_{A})\rVert_{1}

within the studies, in mmol/gDW/h.

method	within studies	new study	new carbon source	balance
NPF, GMA rate law	0.170 $\pm$ 0.013	0.245 $\pm$ 0.085	0.387 $\pm$ 0.020	0.011
PGNN, GMA rate law	0.120 $\pm$ 0.003	0.256 $\pm$ 0.020	0.979 $\pm$ 0.438	2 052
PGNN+ with the state equation, GMA	0.265 $\pm$ 0.088	0.355 $\pm$ 0.141	0.348 $\pm$ 0.016	2.512
random forest, projected onto the net	0.117	0.154	0.333	0.000
random forest	0.112	0.175	0.869	1.314
ridge regression, projected onto the net	0.144	0.176	0.649	0.000

Appendix

Table 6: The comparison of Table 5 with every carbon by-product that a study did not measure closed, carbon dioxide the only free carbon exchange. Mean

\pm

sample standard deviation over three seeds for the learned models. NPF with the rate law of Appendix C.1 and PGNN without read arcs were not run under this rule.

Panel of Figure 3 — Figure 3: The firing vector. (a) A map $\pi$ seats each product atom on the precursor atom with its number, and grey atoms leave unrecorded. (b) The state equation gives $\sigma(\pi)$ , one firing per changed bond place, and the grey entry follows as $m_{A}+C\sigma$ . The H that moves from N to Cl passes from $S_{5}$ to $S_{4}$ , while cycles in $\ker C$ , such as the C=O of the tetrahedral intermediate, and the order of the firings stay unseen. (c) The mapper ranks maps by the bonds formed and broken, then by bond orders and charges, and returns $\hat{\sigma}=\sigma(\pi_{1})$ .

cost	choice	held out	all	interval	proved
bonds made and broken only	83.0	81.03	81.25	79.4 to 83.1	99.94
+ second level, all hydrogen	83.5	82.44	82.56	80.8 to 84.3	99.89
+ hydrogen on heteroatoms not counted	85.0	83.40	83.58	81.8 to 85.3	99.89
+ C–H as bond places (the cost)	86.0	83.65	83.92	82.2 to 85.6	100.00
+ third level (the mapper)	91.0	88.53	88.81	87.3 to 90.2	100.00
RXNMapper	85.5	85.58	85.57	83.9 to 87.2	–

Appendix

Table 7: Atom mapping on the Golden set. Scored by the full condensed graph of reaction, for every level of the cost, on the 200 reactions of the random choice, the 1 560 held out, and all 1 760 usable reactions with a 95 % bootstrap interval. Of the held out reactions, 112 are only mapped correctly by the best variant of our mapper, and 66 only by RXNMapper. Proved is the share of reactions where the search finishes within 60 s.

places	reactions	NPF	RXNMapper	only one	$p$	not minimal
1 or 2	772	95.1	92.7	38 / 20	0.025	0.4
3 or 4	658	87.8	84.8	46 / 26	0.024	3.6
5 or 6	212	82.1	74.5	30 / 14	0.023	3.8
7 or more	118	65.3	62.7	16 / 13	0.711	12.7

Appendix

Table 8: The Golden set by the number of places that the curated map changes, high numbers may be one-pot or multi-step reactions. Accuracy is given in %, the reactions that only our mapper or only RXNMapper map correctly with the exact sign test, and the share of curated maps that are not minimal, as they change more places than the cheapest map. Our mapper is ahead in every bin. Curated maps that are not minimal become more common the more places change, as a mechanism with more steps is likelier to move atoms along a costlier path, and a bond that forms and breaks again is invisible for every map.

set	RDT	RXNMapper	LocalMapper	GraphormerMapper
Golden	175 / 49, $<$ 0.001	109 / 69, 0.003	119 / 108, 0.51	122 / 120, 0.95
NatComm	41 / 9, $<$ 0.001	24 / 9, 0.01	21 / 31, 0.21	21 / 32, 0.17
USPTO-3k	78 / 40, $<$ 0.001	3 / 45, $<$ 0.001	4 / 173, $<$ 0.001	17 / 106, $<$ 0.001
Recon3D	15 / 4, 0.02	35 / 0, $<$ 0.001	30 / 3, $<$ 0.001	90 / 2, $<$ 0.001
E. coli	17 / 31, 0.06	30 / 29, 1.00	22 / 14, 0.24	90 / 6, $<$ 0.001

Appendix

Table 9: Atom mapping on SynRXN, our mapper against each mapper from the original publication. Reactions that only our mapper maps correctly / only the other mapper, and the exact McNemar

p

. The accuracies are in Table 1 . Only 5 of the 382 Recon3D references are clean injections, the others reuse atom-map numbers, which caps every bijective mapper there, and 257 of the 273 of E. coli are.

criterion	NPF, minimum firing vector	RXNMapper	only one	$p$
reaction centre (SynRXN)	88.70	77.92	$5\,213$ / $741$	$<0.001$

Appendix

Table 10: Atom mapping of enzymatic reactions on EnzymeMap (

41\,510

reactions), against its validated maps. Correct reaction centre under the validator of SynRXN, as in Table 1 in %. Only one: reactions mapped correctly by our mapper only / by RXNMapper only, with the exact sign test.

method	$n$	accuracy	macro-F1
DRFP + MLP	3	$95.52\pm 0.04$	$95.47\pm 0.04$
generic readout	3	$97.74\pm 0.01$	$97.72\pm 0.02$
generic head, firing vector	5	$98.64\pm 0.09$	$98.63\pm 0.09$
NPF, state-equation readout	5	$97.68\pm 0.09$	$97.67\pm 0.10$
NPF, firing vector of the mapper	5	$98.80\pm 0.04$	$98.80\pm 0.04$

Appendix

Table 11: Reaction classification on Schneider50k, published split with 200 training and 800 test reactions per class. Mean

\pm

sample standard deviation over

n

seeds. Best in bold , second underlined .

			1 000 labelled		250 labelled
model	full training set	larger molecules	alone	+ firing task	alone	+ firing task
DRFP + MLP	95.52 $\pm$ 0.04	78.43 $\pm$ 0.63	86.62 $\pm$ 0.77	–	65.03 $\pm$ 1.58	–
generic readout	97.74 $\pm$ 0.01	71.76 $\pm$ 1.08	78.59 $\pm$ 1.27	90.02 $\pm$ 0.16	32.14 $\pm$ 1.64	52.64 $\pm$ 1.33
NPF, state-equation readout without gate	97.47 $\pm$ 0.14	67.43 $\pm$ 1.03	91.26 $\pm$ 0.27	94.17 $\pm$ 0.32	61.19 $\pm$ 4.24	81.65 $\pm$ 3.21
NPF, state-equation readout	97.68 $\pm$ 0.09	71.55 $\pm$ 3.65	91.57 $\pm$ 0.64	94.76 $\pm$ 0.20	57.01 $\pm$ 3.19	86.77 $\pm$ 1.26
generic head on the firing vector	98.64 $\pm$ 0.09	82.83 $\pm$ 1.03	93.83 $\pm$ 0.51	–	48.99 $\pm$ 1.31	–

Appendix

Table 12: Classification on Schneider50k, accuracy on the

39\,994

test reactions of the published split, mean

\pm

sample standard deviation over 5 seeds for the readout, the firing vector and its generic head with all labels and 3 otherwise. With

1\,000

or 250 labelled reactions, 20 or 5 per class, a stratified subsample drawn with the seed of the run, alone trains on these reactions only and + firing task also on the maps of the unlabelled rest. Larger molecules trained on reactions up to the median size and tested on the largest quarter. Firing task trained with the auxiliary task on the training reactions without a label (Appendix B ). A dash where the task does not apply, as DRFP and the CGR D-MPNN have no net to attach it to, and the firing vector already holds what it predicts. The CGR D-MPNN is Chemprop 2.3.1 on the condensed graph of reaction with the maps of the mapper, 5 seeds with all training reactions and 3 otherwise, and with a label budget and matched steps it trains for the optimiser steps of its default run on all training reactions.

atom maps	accuracy	macro-F1	$p$ (accuracy)	$p$ (macro-F1)
none, state-equation readout	90.36 0.48	66.80 0.77	–	–
RXNMapper	90.47 1.00	67.23 1.96	0.87	0.75
minimum firing vector (the mapper)	91.04 0.74	67.51 1.54	0.26	0.53
EnzymeMap’s curated maps	91.81 0.12	69.63 0.58	0.03	0.008

Appendix

Table 13: EC classification on EnzymeMap at EC level 3 (200 classes;

41\,816

training and

4\,715

test reactions,

10

% of every class held out), accuracy and macro-F1 in %, mean

\pm

sample standard deviation over three seeds. The same classifier reads the firing vector of each source of atom maps: without maps, the state-equation readout reads the two sides of the reaction.

p

from Welch’s test against the classifier without maps.

	1 %		10 %		25 %
model	top-1	top-5	top-1	top-5	top-1	top-5
MT	0.16	0.37	74.16	84.19	82.39	90.37
NPF, supplied maps	67.24 $\pm$ 0.24	80.36 $\pm$ 0.36	78.85 $\pm$ 0.04	88.73 $\pm$ 0.08	–	–
NPF, targets of the mapper	67.44 $\pm$ 0.19	80.45 $\pm$ 0.18	78.83 $\pm$ 0.14	88.78 $\pm$ 0.16	81.73 $\pm$ 0.13	90.64 $\pm$ 0.06
NPF width 256, supplied maps	–	–	–	–	82.95 $\pm$ 0.26	91.04 $\pm$ 0.12
NPF width 256, targets of the mapper	–	–	–	–	83.23 $\pm$ 0.20	91.00 $\pm$ 0.07

Appendix

Table 14: Forward prediction on the nested 1, 10 and 25 % subsets of the USPTO-MIT training file (

4\,090

40\,900

and

102\,259

lines), top-1 and top-5 in % of all

40\,000

test reactions for the major product, the largest molecule of a prediction against the recorded major product, from beam search. NPF: mean

\pm

sample standard deviation over three seeds, 1.6M parameters at width 128 and 9.8M at width 256. The Molecular Transformer (MT) is trained from scratch for

5\,000

50\,000

and

75\,000

steps, one run each.

targets	training reactions	greedy top-1	beam top-1
atom maps supplied in the data set	$39\,936$	78.57 $\pm$ 0.04	79.05 $\pm$ 0.04
mapper, set of tied vectors	$39\,381$	79.72 $\pm$ 0.03	80.08 $\pm$ 0.05
mapper, one vector	$39\,381$	79.15 $\pm$ 0.26	79.54 $\pm$ 0.25

Appendix

Table 15: Targets of the forward model on 10 % of USPTO-MIT, greedy and beam top-1 in % on all

40\,000

test reactions under the rule of Appendix B , mean

\pm

sample standard deviation over three seeds. Training reactions are those of the subset with a target that decodes to the product.

model	$1\,995$	$7\,973$	$37\,220$	top-1	top-5	valid
	greedy top-1			beam
NPF token game	84.31 $\pm$ 0.71	90.34 $\pm$ 0.34	93.34 $\pm$ 0.28	93.39 $\pm$ 0.18	98.08 $\pm$ 0.07	100.00 $\pm$ 0.00
without the enabling mask	84.19 $\pm$ 0.18	89.82 $\pm$ 0.64	93.06 $\pm$ 0.11	93.13 $\pm$ 0.12	98.07 $\pm$ 0.04	99.98 $\pm$ 0.02
one-shot, greedy valence repair	–	–	81.60 $\pm$ 0.50	–	–	100.00 $\pm$ 0.00
one-shot, generic	66.28 $\pm$ 0.51	75.02 $\pm$ 0.53	79.72 $\pm$ 0.22	–	–	97.21 $\pm$ 0.30

Appendix

Table 16: Forward prediction on our split of Schneider50k,

4\,667

test reactions and 3 seeds. Greedy top-1 against the number of training reactions with a target, beam top-1 and top-5 with all of them, and the share of valence-valid predictions.

NPF model	valence rule	valid molecules	invalid	dropped fragments
aromatic, recorded maps	100	99.875	50	–
aromatic, targets of the net	100	99.897	41	0.130
aromatic, targets of the net, no enabling	99.987	99.890	44	0.138

Appendix

Table 17: Validity of the top-1 predictions on USPTO-480K (

39\,994

test reactions that RDKit parses), one seed each. Valence rule : share of predictions whose markings respect the enabling rule of the valence net. Valid molecules : share of predictions whose products RDKit sanitises, with the number of invalid predictions. Dropped fragments : share of predictions in which a touched molecule could not be decoded and was left out. Invalid products of the aromatic models come from the half-token tolerance on aromatic bonds.

method	valid SMILES	heavy atoms	atoms, protons, electrons
Molecular Transformer	70.2	39.1	33.0
Graph2SMILES	76.3	30.7	17.2
Graph2SMILES+H	78.8	27.7	19.0
FlowER	95	by construction	by construction
NPF, arrow net	100	by construction	by construction
NPF, electron net	100	by construction	by construction

Appendix

Table 18: Validity and conservation of the predictions on the FlowER test set, in %. Valid SMILES of the top-1 prediction, and the share of predictions that conserve heavy atoms, and heavy atoms, protons and electrons together. For NPF and FlowER, conservation holds by construction, as every firing keeps the P-invariants of the net (Proposition 1 ) and FlowER’s electron changes sum to zero. For NPF we report mean sample standard deviation over three seeds, all

162\,002

steps. Baselines from the text of Joung et al. [2025] , single runs, where FlowER’s validity is given as approximately 95 %.

component	criterion	Petri	generic	$p$
one bond change at a time	top-1, all training reactions	93.34 $\pm$ 0.28	79.72 $\pm$ 0.22	$<$ 0.001
same	top-1, 1 995 training reactions	84.31 $\pm$ 0.71	66.28 $\pm$ 0.51	$<$ 0.001
enabling (Prop. 4 )	valence-valid products	100 (all weights)	97.21 $\pm$ 0.30	–
enabling mask in the game	top-1	93.34 $\pm$ 0.28	93.06 $\pm$ 0.11	0.22
readout (Prop. 3 )	% changed by a far methyl	0.00	0.27	–
same	accuracy, 1 000 / 250 training reactions	91.26 / 61.19	78.59 / 32.14	0.002 / 0.003

Appendix

Table 19: Each Petri component against a counterpart of equal size without it on Schneider50k, with

4\,667

forward test reactions, 3 seeds for forward prediction, and 3 to 5 seeds for classification. Mean

\pm

sample standard deviation where shown,

p

from a Welch test, and a dash where a proposition guarantees the value. The rows of Proposition 3 are the pure readout,

w\equiv 1

, for which it holds. The subsamples of

1\,000

and 250 training reactions, 20 and 5 per class, are stratified and drawn with the seed of the run, the same for all models of that seed. Italic , not load-bearing, as the Petri component is not significantly better,

p\geq 0.05

, or is worse.

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