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Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria

Jialin Liu

arXiv:2609.35274v1cs.LGcs.AI

TL;DR

Many graph problems have multiple valid answers, but recurrent and equilibrium GNNs often use a unique fixed point or one designated target. This paper constructs and approximates recurrent dynamics whose different initializations reach different solutions, then tests energy- or residual-trained updates on scientific tasks. The learned models produce multiple high-quality predictions and outperform tested single-target, unique-equilibrium, and feedforward baselines on average solution quality.

  • Problem

    Many graph and combinatorial tasks have multiple valid solutions with no preferred target, while existing expressivity theory studies single-valued invariant or equivariant maps.

    Source: §1 · p. 2, §2 · p. 3

  • Method

    The paper constructs permutation-equivariant multi-attractor dynamics and trains recurrent updates with problem-specific energies or residuals, without solution labels.

    Source: §1 · p. 3, §2.1 · p. 3

  • Results

    Across Ising, protein-module, and chemical tasks, learned updates produce multiple high-quality predictions with high numerical convergence rates and better average solution quality than tested single-target, unique-equilibrium, and feedforward baselines; protein modularity is 0.5560 and chemical candidate hits are 83.77%.

    Source: §1 · p. 3, §3.2 · p. 9, §3.3 · p. 10

  • Takeaways & Limitations

    The findings support preserving access to multiple equilibria for graph tasks that admit multiple answers, with success depending on problem-specific objectives and optimization.

    Source: §4 · pp. 10–11

    The scientific tests do not certify asymptotic convergence or complete solution coverage, and inference computation is not matched across baselines.

    Source: §3 · p. 7

Abstract

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Recurrent and equilibrium graph neural networks (GNNs) often enforce a unique fixed point or use one training target per graph. Yet many combinatorial and scientific problems admit multiple valid solutions, with no preferred one. A designated target can then impose an arbitrary selection rule. For tasks invariant to node relabeling, a symmetric graph may have a symmetric solution set but no symmetric solution. We show that multiple equilibria enable one weight-tied message-passing GNN to represent set-valued equivariant maps: different initializations approach different valid solutions. Under stated regularity assumptions, we first construct globally Lipschitz, permutation-equivariant dynamics that converge almost surely to valid solutions and reach every solution branch with positive probability. We then establish approximate realization by recurrent message passing with continuous component maps, with arbitrarily small update and limiting errors and arbitrarily high probability. This goes beyond standard universality arguments: although message passing alone cannot distinguish symmetric nodes, the evolving state keeps nodes distinguishable at every finite step without auxiliary node identifiers. Such dynamics can be learned without solution labels using problem-specific energies. On Ising ground states, structural module detection in protein graphs, and chemical reaction steady states, the learned updates produce multiple high-quality predictions with high numerical convergence rates. They achieve better average solution quality than the tested unique-equilibrium, single-target, and feedforward baselines, while remaining competitive with much larger diffusion-based solvers.

1. Introduction

Recurrent GNNs reuse one message-passing update across steps, but many graph tasks have multiple valid answers and no preferred target. The paper studies whether one such update can represent multiple solutions while respecting graph symmetries.

Background

  • Recurrent GNNs apply a shared update repeatedly, while equilibrium variants can support implicit differentiation without storing every intermediate state.The usual recurrent goal is to approach a prescribed target from an initial state.Source: §1 · p. 1
  • Contraction makes iteration converge to the same fixed point from every initialization, ruling out initialization-dependent solutions regardless of the training objective.Uniqueness can also follow from strong monotonicity, though uniqueness alone does not ensure direct iteration converges.Source: §1 · p. 2

Motivation

  • Many graph tasks admit multiple valid solutions, so selecting one target can discard alternatives or conflict with input symmetries.On a symmetric four-node cycle, the two valid colorings are exchanged by a graph rotation.Source: §1 · p. 2

Contribution

  • The paper asks whether one shared recurrent update can represent many solutions and whether energy- or residual-trained dynamics can produce them without solution labels.Its theoretical results address set-valued representation and a limitation of unique equilibria; its applications include Ising ground states, protein modules, and chemical steady states.Source: §1 · p. 3

2. Main results

The main results establish that recurrent updates can represent multiple valid solutions through initialization-dependent limits, while contraction can obstruct even recovery of one valid solution. A toy diffusion task illustrates the distinct solution basins and the failure of two controls.

2.1. Can Recurrent GNNs Represent Set-Valued Equivariant Maps?

  • A globally Lipschitz, permutation-equivariant operator can converge almost surely to valid solutions and reach every solution branch with positive probability under the stated graph and branch regularity assumptions.The recurrent message-passing realization achieves arbitrarily small update and limiting errors with arbitrarily high joint probability; each branch is reached with positive joint probability over graphs and initializations.Source: §2.1 · pp. 3–4
  • Finite-step ideal trajectories retain distinct node states, letting recurrent message passing approximate the dynamics without auxiliary node identifiers.The realization uses continuous component maps, and the guarantee is representational rather than a guarantee about training.Source: §2.1 · pp. 4–5

2.2. Toy Example: Single-Source Graph Diffusion.

  • On the three-solution diffusion task, energy training forms attractors near all three valid sources, while single-target and contractive controls concentrate away from every valid solution.The figure follows 128 trajectories on one held-out graph; diffusion coordinates show the same qualitative pattern.Source: §2.2 · p. 5, Figure 3 · p. 6, Figure 2 · p. 6

2.3. Can a Unique Equilibrium Miss Every Valid Solution?

  • Continuous, permutation-equivariant contractive updates miss every valid solution with positive probability on independently weighted triangles, regardless of training or initialization.The failure probability depends on the operator; the result claims no uniform lower bound across models.Source: §2.3 · p. 7

3. Scientific applications

The scientific applications test whether energy- or residual-trained shared recurrent updates can produce multiple high-quality predictions with numerical convergence, without solution labels. Across Ising, protein-module, and chemical-reaction tasks, results show multiple predictions and favorable quality against tested baselines, subject to limits on convergence certification, coverage, and budget matching.

3. Scientific applications

  • Across the applications, tests evaluate multiple high-quality predictions and numerical convergence; they do not certify asymptotic convergence or complete solution coverage.Baseline inference computation is not matched.Source: §3 · p. 7

3.1. Ising Models.

  • The Ising task uses antiferromagnetic triangular lattices, where competing spin preferences and global spin-reversal symmetry create multiple low-energy configurations.Energy training uses ten shared updates without prescribing a solution for each initialization.Source: §3.1 · p. 7
  • 83.98% of Ising candidates are within 10% of optimum; the model averages 2.16 distinct optima per graph and finds at least two on 68.73% of graphs.It has the lowest mean energy and highest Hit10; convergence reaches 99.96% by step 200 and 100% by step 500.Source: §3.1 · p. 8, Table 1 · p. 8

3.2. Structural module detection in protein graphs.

  • For protein graphs, the shared recurrent update refines random soft partitions using a differentiable modularity-derived energy, then assigns each node its highest-weight module.Evaluation allows at most four modules and uses 20 initializations per graph.Source: §3.2 · p. 9
  • 77.49% Hit10 and mean modularity 0.5560 are the highest reported; the model averages 5.83 qualified partitions per graph and finds at least two on 80.78% of graphs.DIFUSCO (large) finds more qualified partitions but has lower mean modularity with approximately 155 times as many parameters.Source: §3.2 · p. 9, Table 2 · p. 10

3.3. Chemical reaction-network multistationarity.

  • The chemistry model updates species log concentrations with shared recurrence and trains on equation residuals without prescribed targets; the main comparison uses T = 200.The task seeks roots of the mass-action system, not necessarily states physically stable under the chemical ODE.Source: §3.3 · p. 9
  • 83.77% of chemical-reaction candidates are accepted hits, averaging 3.506 accepted clusters per graph; at least two occur on 41.67% of graphs.Numerical settling rises from 91.99% at T = 200 to 96.15% at T = 5,000; results do not certify distinct exact roots or complete root coverage.Source: §3.3 · p. 10

4. Conclusions

The paper concludes that multi-attractor dynamics let a shared update reach multiple high-quality solutions across three scientific tasks, while training objectives and optimization affect success. Its existence argument assumes a node-relabeling-closed graph domain, which need not be bounded, closed, connected, or compact.

4. Conclusions

  • Multi-attractor dynamics support set-valued graph tasks: one shared update can reach different valid solutions, while continuous equivariant contraction can obstruct recovery of even one.Across three scientific tasks, energy or residual training produces multiple high-quality predictions and high numerical convergence rates without solution labels.Source: §4 · p. 10
  • Success depends on problem-specific objectives and optimization, and understanding how training shapes convergence and solution coverage remains an important direction.Source: §4 · p. 11

Appendix A. Ideal Operator Existence

  • The appendix assumes a fixed node count and a graph domain closed under node relabeling; that domain need not be bounded, closed, connected, or compact.The graph and output spaces are equipped with Euclidean norms.Source: Appendix A · p. 16

A.1. Basic concepts and main result.

The section formalizes finite, permutation-equivariant solution sets and constructs one globally Lipschitz equivariant update whose trajectories converge to their solutions. The construction uses a selector and displacement on regular stable instances, while excluding exceptional and switching configurations.

A.1. Basic concepts and main result.

  • A finite-branch solution map represents a set of solutions by labeled branches, where node relabeling may permute branch labels while preserving the underlying solution set.Distinct labels may represent coincident solution values, which appear only once in the set S(G).Source: §A.1 · p. 17
  • The regular stable domain excludes graphs where branches fail local Lipschitz continuity or have unstable collapse patterns; symmetrizing branch labels can shrink this domain.Probability assumptions on the resulting domain must then be rechecked.Source: §A.1 · p. 17
  • On regular stable graphs, the constructed update is jointly globally Lipschitz and permutation equivariant, and it freezes on exceptional configurations and the switching set.Exceptional configurations include graphs in D or U; the update also freezes for states on the switching set.Source: §A.1 · p. 18
  • For almost every absolutely continuous initialization, iteration converges to the solution set; every solution is reached with positive probability if its strict Voronoi basin has positive initialization mass.The convergence rate has a trajectory-dependent constant, with no uniform positive lower bound claimed.Source: §A.1 · p. 18

A.2. Bad set and displacement functions.

  • The bad-set construction combines singular or unstable graph instances with switching states, then uses a selector and normalized displacement that are locally Lipschitz and equivariant off that set.An equivariant bounded-flattening argument yields a bounded, globally Lipschitz equivariant displacement factor.Source: §A.2 · pp. 18–20

A.3. Proof of the main result.

The proof constructs the update by extending a nearest-solution selector away from singular, unstable, and switching cases, then damping its displacement to obtain global regularity. It establishes almost-everywhere convergence, and the next appendix sets up realization with a recurrent message-passing block reused at every step.

A.3. Proof of the main result.

  • A nearest-solution selector is locally Lipschitz outside the joint bad set, and damping its normalized displacement yields a jointly globally Lipschitz update.The damping factor vanishes on the bad set, where the update freezes.Source: §A.3 · pp. 21–23
  • The proof preserves permutation equivariance because the damping and distance terms are invariant while the displacement is equivariant.The resulting update satisfies TπG(πy) = πTG(y).Source: §A.3 · p. 23
  • The proof concludes that almost every initialization converges to a solution, and every solution has positive probability of being reached when each strict basin has positive initialization mass.The switching set is negligible because it lies in a finite union of null affine hyperplanes.Source: §A.3 · pp. 24–25

Appendix B. Family-wide asymptotic realization by a recurrent MP-GNN

  • One message-passing GNN block is reused at every recurrent step to realize the ideal dynamics, with random initialization providing node addresses and the choice of solution basin.At later steps, the block receives only the graph and current state.Source: Appendix B · p. 25
  • The recurrent realization setup uses the random initial state both to distinguish nodes and to select a solution basin, without adding later node identifiers.The stated aim is a shared update whose almost-every trajectory converges to a solution and whose initialization reaches every solution with positive probability.Source: Appendix B · p. 25

B.1. Architecture and main result.

The section constructs a shared recurrent message-passing block that approximates set-valued equivariant dynamics, then establishes when current node states let message passing realize equivariant operators exactly.

B.1. Architecture and main result.

  • The architecture shares continuous component maps across nodes and recurrent steps, recomputing hidden representations while carrying only the current state forward.Its only global operation is the final invariant sum; hidden layers do not form or broadcast a global aggregate.Source: §B.1 · p. 25
  • One deterministic, autonomous, weight-tied MP-GNN block works across the retained graph family and is permutation equivariant.Its continuous component maps are fixed across recurrent calls; the block receives the graph and current state at each call.Source: §B.1 · pp. 25–26
  • On a retained event, trajectories converge to limits ε-close to the solution set, with positive joint capture probability for every labeled branch.The limit need not belong exactly to the solution set, and branch capture is joint over graphs and initializations rather than guaranteed within every individual graph.Source: §B.1 · pp. 26–27

B.2. Exact realization using the current node states.

  • Current node states can provide distinct, robust addresses, enabling exact realization of continuous equivariant operators without auxiliary node identifiers.The address regions are fixed across the compact family, although node-to-region assignments may vary between inputs.Source: §B.2 · p. 27

2. The compact permutation-invariant family

Robust state-based node symmetry breaking makes exact message-passing realization possible on compact permutation-invariant families. Under absolutely continuous initialization in at least two state dimensions, the ideal dynamics also separate node states almost surely, allowing common address boxes to be chosen with arbitrarily small probability loss.

2. The compact permutation-invariant family

  • Exact realization depends on node states occupying distinct, positively separated address regions throughout the compact family.A connected full-circle family in the example does not satisfy this condition, despite the two node states remaining distance 2 apart.Source: §2 · p. 27
  • A compact family with robust state-based node symmetry breaking permits exact, simultaneous realization of continuous invariant scalar maps and equivariant node maps.A finite-dimensional representation reconstructs the graph, state, and receiving node up to relabeling, after which a continuous readout extends the target map.Source: §2 · pp. 27–29

B.3. Node separation and a common collection of boxes.

  • For dy ≥ 2 and conditionally absolutely continuous initialization, ideal-dynamics node states remain distinct at every finite time almost surely.The result assumes graphs lie almost surely in G⋆.Source: §B.3 · p. 30
  • A common collection of separated compact boxes can encode node addresses for finitely many state maps while controlling approximation error and losing arbitrarily little measure.The retained compact set can also preserve positive measure in specified positive-measure subsets and be permutation invariant.Source: §B.3 · pp. 31, 33

B.4. Proof of the main result.

The proof combines ideal updates with terminal dynamics that converge while preserving node distinguishability, then realizes both regimes with one fixed GNN. It establishes high-probability convergence near the solution set and positive joint capture probability for every labeled branch.

B.4. Proof of the main result.

  • The update follows ideal dynamics until entering a solution neighborhood, then moves each node halfway toward its current box center to ensure convergence while preserving distinct addresses.Small boxes and switching near the solution control both limiting error and deviation from the ideal update.Source: §B.4 · pp. 33, 36, 37
  • Every retained trajectory converges to a limit within ε of its selected solution, and each labeled branch has positive capture probability.The same fixed GNN generates all retained trajectories and is permutation equivariant.Source: §B.4 · p. 38

Appendix C. An obstruction to almost-sure validity for contractive equivariant dynamics

  • The contractive-dynamics obstruction is framed using each graph’s automorphism group, its fixed output subspace, and graph laws with full support.The supplied setup defines the fixed subspace as outputs unchanged by every graph automorphism.Source: Appendix C · p. 39

C.1. The operator-level obstruction.

Theorem 6 shows that continuous, equivariant, globally contractive dynamics can miss valid solutions with positive probability when a supported graph has solutions outside its automorphism-fixed subspace. Contraction forces a unique, initialization-independent limit, and continuity extends the mismatch to a graph neighborhood.

C.1. The operator-level obstruction.

  • At a supported graph whose valid branches lie outside its automorphism-fixed subspace, the unique equivariant limit cannot equal any valid branch.The theorem assumes continuous solution branches and a graph G0 in the support of the graph law.Source: §C.1 · pp. 39–40
  • A continuous, equivariant, globally contractive update has a unique limit for each graph, independent of initialization, and that limit is permutation equivariant.Its distance to the fixed point decays at rate κ^t.Source: §C.1 · pp. 39–40
  • The resulting positive-probability failure holds under any initialization law, without requiring independence, density, or full support.The graph law must assign the obstruction graph to its support; the proof does not assert a common finite convergence time over the unbounded state space.Source: §C.1 · pp. 39, 41
  • Continuity of the fixed-point map and solution branches extends this positive separation to a neighborhood of the obstruction graph.For every ε below the separation margin, that neighborhood has positive graph probability.Source: §C.1 · pp. 39–40

C.2. Application to independently weighted triangles.

For independently weighted three-node triangles, the symmetric equal-weight graph supplies an obstruction graph in the support of the graph law. Consequently, continuous, equivariant, contractive dynamics fail to reach valid diffusion solutions with positive probability.

C.2. Application to independently weighted triangles.

  • The valid outputs are single-source diffusion solutions with H_Gu=b, where b is a one-hot source indicator.The example uses three edge weights and three solution branches that remain distinct throughout the graph family.Source: §C.2 · pp. 41–42
  • Under independent uniform edge weights, continuous, equivariant, globally contractive dynamics have positive probability of ending farther than ε from the valid solution set.The conclusion holds for any joint law of graph and initialization with the stated graph marginal, and extends to any full-support Borel graph law on the triangle family.Source: §C.2 · p. 41
  • The obstruction graph is the equal-weight triangle, fixed by every node permutation, and it lies in the support of the independently weighted graph law.The edge-weight parameterization maps the weight cube homeomorphically onto the graph family, whose induced uniform product law has full support.Source: §C.2 · p. 42

C.3. Consequence for recurrent message-passing GNNs.

The operator-level obstruction applies to deterministic, autonomous, weight-tied recurrent message-passing GNNs when their complete update is continuous and contractive. In contrast, multi-attractor dynamics can reach each solution branch with positive probability.

C.3. Consequence for recurrent message-passing GNNs.

  • A deterministic, autonomous, weight-tied recurrent MP-GNN with continuous component maps and a contractive complete update inherits Theorem 6’s failure result.On the independently weighted triangle family, this applies for every finite number of hidden layers and every finite hidden width.Source: §C.3 · p. 43
  • The contractive-GNN consequence follows because the complete message-passing update is continuous and permutation equivariant under the stated architecture.The update’s contraction bound supplies the remaining theorem hypothesis.Source: §C.3 · p. 43
  • For the triangle family, multi-attractor dynamics have almost-sure validity and positive capture probability for every solution at each fixed graph, unlike contractive operators.A contractive operator has one limit per fixed graph and cannot reach two distinct solutions with positive probability; exact almost-sure validity is also ruled out under the independent-weight graph law.Source: §C.3 · p. 43

Appendix D. Single-source diffusion: protocol and additional results

The single-source diffusion experiment compares multi-attractor, supervised, and contractive recurrent models on randomly permuted weighted three-node graphs. The multi-attractor trajectories form three distinct clusters, while the controls concentrate near one point separated from the exact solutions.

Appendix D. Single-source diffusion: protocol and additional results

  • The test uses complete undirected three-node graphs with independently sampled edge weights from Unif[0.5, 1.5], random node relabelings, and 500/100/100 train/validation/test instances.Inputs include only the three positive edge weights; node indices, drawing coordinates, prescribed sources, and exact diffusion fields are excluded.Source: Appendix D · p. 43
  • The shared recurrent block uses width-32, two-affine-layer SiLU MLPs, sum aggregation, and an output projection that clips source coordinates without imposing the mass constraint or choosing a source.The block and scalar skip are shared across recurrent time.Source: Appendix D · pp. 43–44
  • Energy-based training minimizes the diffusion energy at T=10, while supervised training uses a once-sampled source target retained across visits.The multi-attractor model has no weight constraints; contractive models use Lipschitz control during energy training.Source: Appendix D · p. 44
  • The contractive controls enforce contraction after initialization and every optimizer update, whereas the multi-attractor and supervised models have no such projection.The stated constraints certify contraction throughout training; single-target supervision alone does not guarantee a unique equilibrium.Source: Appendix D · pp. 44–45
  • Multi-attractor trajectories form three distinct diffusion-coordinate clusters, while supervised and contractive controls concentrate near one point separated from all three exact solutions.Figure 4 shows diffusion coordinates and complements Figure 3’s source-coordinate trajectories.Source: Appendix D · p. 45

Appendix E. Ising Experiments: Details and Additional Results

The Ising appendix details the data, model, baselines, and evaluation used to test solution quality, diversity, and convergence. On larger graphs, the fixed-capacity recurrent model retains strong quality and diversity, while longer test-time runs show that delayed convergence accounts for the few trajectories not stationary at t = 200.

Datasets and baselines

  • The study uses triangular-lattice graphs with four controlled size levels; each level has 2,500 training-source and 500 test graphs, and extra-large supervised training is omitted because labels can take over 5,000 seconds per instance.Certified test references are still used for extra-large graphs.Source: Appendix E · p. 45

Model architecture

  • The recurrent Ising model uses a graph-convolutional backbone with physical node features, including local interaction-field and coupling summaries inspired by Ising belief propagation.Those summaries are input features, not a separate solver or prescribed update direction.Source: Appendix E · pp. 45–46

Metrics

  • Evaluation generates 20 candidates per graph and reports mean energy, best-of-20 energy, Hit10, and distinct near-optimal configurations; duplicate configurations count once in the diversity measures.Quality and diversity are evaluated at the terminal step, without selecting intermediate trajectory states.Source: Appendix E · p. 49

Longer test-time trajectories

  • Of 30,000 Ising trajectories, 11 failed the fixed-point criterion at t = 200, but all met it by t = 500 and remained stationary through t = 2000.Energy, Hit10, D20, and G20 use outputs at t = 200; the extended runs are numerical observations, not asymptotic guarantees.Source: Appendix E · pp. 49–50, Table 6 · p. 50

Larger Graphs

  • Hit10 reaches 96.88%, 99.96%, and 100% on medium, large, and extra-large graphs, while mean distinct configurations within 10% of optimum are 19.376, 19.992, and 20.000, respectively.The width-16 model retains 3,153 parameters across scales; metrics use 20 candidates per graph.Source: Appendix E · p. 50
  • The 3-block feedforward model improves over one-shot prediction, but has worse mean and best-of-20 energies than the recurrent model; 10 untied blocks reduce Hit10 and repeat one rounded configuration across 20 initializations.Large DIFUSCO is competitive on medium and large graphs but also has worse mean and best-of-20 energies.Source: Appendix E · p. 50

Appendix F. Structural module detection in protein graphs: details and additional results

The protein-graph appendix specifies the data split, structural inputs, recurrent cell, modularity objective, and baseline training setup. Extended runs raise numerical convergence rates substantially beyond t = 200, while reported solution-quality and diversity metrics use outputs at t = 200.

Dataset and splits

  • The PROTEINS task uses 1,113 graphs, split by complete graph-isomorphism groups into 891 training, 111 validation, and 111 test graphs, with no exact topology shared across splits.The original graph-classification label is retained for audit but never supplied to models.Source: Appendix F · p. 52

Model

  • The recurrent state is a soft four-module partition, updated by a width-32, three-layer PNA-style message-passing cell with a graph-level readout and time-tied parameters.The model has 35,784 trainable parameters; nodes receive structural features, and the unsupervised model never receives Gurobi labels.Source: Appendix F · pp. 53–54

Training and objectives

  • At resolution γ = 1, training minimizes negative modularity; its differentiable soft-partition energy equals the expected discrete energy up to a graph-dependent constant.The modularity terms jointly favor modules with more internal connectivity than expected from node degrees under the null model.Source: Appendix F · p. 54

Baselines and training

  • The PROTEINS comparisons include single-target and contraction-constrained models, untied feedforward depths, and distributional or diffusion-based baselines under the reported training configurations.The recurrent model and baselines use three training seeds; the recurrent model trains for 3,500 updates using four initial partitions per sampled graph.Source: Appendix F · pp. 54–55, Table 9 · p. 55

Longer test-time trajectories

  • The convergence rate rises from C200 = 88.05 ± 2.23% to C800 = 98.75 ± 0.62% and C3000 = 99.49 ± 0.29%; by t = 5000, all trajectories are convergent or classified as periodic.The extended checks indicate that most trajectories not converged at t = 200 were longer transients, while a small remainder became stably periodic.Source: Appendix F · p. 55
  • Although training unrolls only T = 5 iterations, reported modularity, Hit10, and discovery metrics use terminal outputs at t = 200, without waiting for every trajectory to converge.Longer runs examine convergence; they do not replace the t = 200 outputs used for solution-quality and diversity results.Source: Appendix F · p. 55

Appendix G. Chemical reaction networks: details and additional results

The appendix details a constructed chemical-reaction benchmark and a physics-assisted recurrent solver, then situates the work against prior approaches to recurrent GNNs, multiple solutions, and symmetry breaking. Its extended-horizon analysis reports increasing numerical settling while cautioning that the diagnostics do not establish asymptotic attraction.

Appendix G. Chemical reaction networks: details and additional results

  • The benchmark contains 1,242 training, 266 validation, and 236 test instances, each with two numerically verified stationary witnesses that need not exhaust its solution set.Instances are constructed from reaction networks labeled Multi, with 2–5 species and 6–19 reactions.Source: Appendix G · pp. 55–56, Table 11 · p. 56
  • The chemistry model is a physics-assisted learned solver with a tied recurrent update, equivariant operations, and initialization from random node vectors; it has no auxiliary carried hidden state and is not guaranteed contractive.The state is the species log-concentration vector, and subsequent calls receive no identifier forcing.Source: Appendix G · pp. 57–58
  • Accepted predictions are tested against original-parameter residuals, but numerically separated predictions may still approximate the same root, so cluster counts do not certify distinct stationary solutions.The recurrent settling test measures one-step log-state change and does not certify asymptotic convergence.Source: Appendix G · pp. 59–60
  • The extended test-time analysis extends frozen models and initialization seeds to t = 20,000, while quality and diversity remain evaluated at T = 200.The longer run changes neither model weights nor initialization distributions or physical acceptance thresholds.Source: Appendix G · p. 60
  • Settling rises from 91.99 ± 0.35% at t = 200 to 96.15 ± 2.01% at t = 5,000, then plateaus; at t = 20,000, 96.12 ± 2.01% are step-settled, 2.77 ± 2.09% oscillatory, and 1.11 ± 0.31% unresolved.These are learned-solver diagnostics, not evidence of physical kinetics or asymptotic attraction; the table defines the categories across three training seeds.Source: Table 13 · p. 60, Appendix G · pp. 60–61

Appendix H. Related Work

  • Unlike prior set-valued attractor work, this paper establishes permutation-equivariant dynamics and their message-passing realization, including cases where contractive updates miss every valid solution with positive probability.The related-work discussion distinguishes this account from prior multi-solution discovery with recurrent GNNs that did not establish equivariant dynamics or their message-passing realization.Source: Appendix H · p. 61
  • The proposed symmetry breaking occurs through the evolving state without auxiliary inputs; randomness enters once through the initial state, which preserves node distinguishability at every finite step.The discussion notes that augmented features can also seed the initial state, but augmentation alone does not provide a mechanism for sampling multiple valid outputs.Source: Appendix H · pp. 61–62

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