PaperlayerView exact revision on arXiv

Source-linked AI summary

Fixed Points Without Fixed Diffusion: Implicit Neural Sheaves for Convergent Test-Time Computation

Rémi Bourgerie, Šarūnas Girdzijauskas, Viktoria Fodor

arXiv:2609.30277v1cs.LG

TL;DR

Implicit GNNs offer deep, flexible test-time computation, but convergence guarantees can restrict recurrent dynamics, while adaptive neural sheaf propagation creates state-dependent feedback. SheafDEQ combines scale-invariant adaptive sheaf propagation with a subhomogeneous equilibrium model, and improves over fixed-propagation implicit baselines on several tasks while retaining convergence guarantees.

  • Problem

    Implicit GNNs need conditions for a unique equilibrium reachable by iteration, and existing guarantees often constrain recurrent dynamics; fixed-propagation SubDEQ does not provide edge-dependent transformations.

    Source: §1 · pp. 1–2

  • Method

    SheafDEQ combines representation-dependent neural-sheaf propagation with scale-invariant restriction-map inference and a positive subhomogeneous update.

    Source: §1 · p. 2

  • Results

    SheafDEQ improves over fixed-propagation implicit baselines on Sums, MNIST Terrain, Coordinates, and community detection as connectivity becomes increasingly heterophilic.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Under explicit regularity conditions, fixed-point iteration reaches a unique equilibrium globally and linearly from every entrywise-positive initialization, with asynchronous convergence under bounded delays.

    Source: §1 · p. 2

    Limitations include an extreme initialization that does not settle within the tested iteration range, several transductive benchmarks, and parameter-unmatched comparisons.

    Source: §5 · p. 10

Abstract

from arXiv · show

Implicit Graph Neural Networks (IGNNs) define node representations as fixed points of message-passing operators, enabling effectively infinite-depth propagation, iteration-independent parameterization, and flexible test-time computation. Yet these benefits depend on the equilibrium being unique and attainable by fixed-point iteration. Existing constructions often impose constraints on recurrent updates to obtain these guarantees, limiting the transformations available at equilibrium. This raises a central question: can IGNNs gain expressiveness through richer, edge-dependent transformations while retaining the inherent strengths of their equilibrium formulation? We introduce SheafDEQ, a subhomogeneous deep-equilibrium architecture with adaptive neural-sheaf propagation. Its learned, matrix-valued sheaf restriction maps can align, mix, or reverse neighbouring representations. Under mild regularity conditions, we prove that SheafDEQ admits a unique equilibrium reached globally by fixed-point iteration from any positive initialization. Contractivity further guarantees convergence under bounded communication staleness. We evaluate SheafDEQ on distributed-inference tasks requiring repeated nonlocal aggregation and on community detection whose rewiring increasingly favours cross-community interactions. SheafDEQ improves over fixed-propagation implicit baselines on Sums, MNIST Terrain, and Coordinates, and on community detection as connectivity becomes increasingly heterophilic. Continued-iteration diagnostics show decreasing residuals and low prediction sensitivity after 100 iterations for initialization scales from $0.001$ to $10$, while delayed-update experiments show low sensitivity to bounded communication staleness.

1 Introduction

SheafDEQ combines equilibrium-based graph computation with adaptive, matrix-valued sheaf propagation. The paper establishes convergence guarantees for this state-dependent model and reports improvements over fixed-propagation implicit baselines on several tasks.

1 Introduction

  • Implicit GNNs reuse shared parameters across iterations, enabling effectively infinite-depth propagation and flexible test-time computation.The paper also describes a fixed-sheaf setting in which propagation can yield class-separating positive equilibria.Source: §1 · pp. 1–2
  • Existing implicit GNNs secure convergence through restrictions such as bounded recurrent operators, monotone or orthogonal parameterizations, or strongly convex learned energies.These mechanisms narrow the admissible recurrent dynamics.Source: §1 · pp. 1–2
  • SheafDEQ addresses the gap between convergent implicit models with fixed propagation and adaptive neural sheaves whose representation-dependent maps are repeatedly recomputed.SubDEQ uses fixed normalized adjacency, while the state-dependent feedback in implicit neural sheaves does not automatically preserve the homogeneity and regularity required by SubDEQ theory.Source: §1 · p. 2
  • SheafDEQ infers adaptive, signed, matrix-valued restriction maps with scale-invariant inference, preserving positive homogeneity despite state-dependent propagation.Sheaf propagation can align, mix, or reverse messages, unlike scalar convex averaging in graph attention, which can drive representations toward uniformity under repeated application.Source: §1 · p. 2
  • Under explicit regularity conditions, fixed-point iteration converges globally and linearly from every entrywise-positive initialization to a unique equilibrium.Asynchronous iterations also converge under bounded update and communication delays without spectral-radius or entrywise-sign constraints on learned restriction maps.Source: §1 · p. 2
  • SheafDEQ improves over fixed-propagation implicit baselines on Sums, MNIST Terrain, Coordinates, and community detection as connectivity becomes increasingly heterophilic.Continued-iteration and delayed-update experiments also report low sensitivity under the tested initialization scales and bounded communication staleness.Source: §1 · p. 2

2 Background and Related Work

The background contrasts equilibrium-based graph models and SubDEQ guarantees with adaptive neural-sheaf propagation. SheafDEQ combines these ideas by making sheaf convolution state-dependent while retaining an implicit equilibrium formulation.

Implicit GNNs

  • An implicit GNN defines node representations as the equilibrium of a graph neural operator, but existence, uniqueness, and convergence of Picard iteration require additional conditions.Prior guarantees use operator-norm or spectral-radius bounds, monotone-operator parameterizations, or strongly convex learned energies.Source: §2 · p. 3

Subhomogeneous deep equilibrium models

  • SubDEQs derive contractivity from positive subhomogeneous nonlinearities, leaving recurrent weights unrestricted and yielding global linear Picard convergence to a unique fixed point.The graph baseline uses fixed APPNP propagation; its update is made positive and sufficiently subhomogeneous through nonlinearities and a shift.Source: §2 · p. 3

Sheaf convolution

  • Sheaf propagation uses edge-specific matrix transformations of stalk representations before aggregation, rather than a scalar coefficient shared across feature coordinates.Neural sheaf networks infer incidence restriction maps from endpoint representations; stalk and channel mixing then form the sheaf convolution.Source: §2 · pp. 3–4

Relation to prior work

  • Table 1 compares evaluated graph architectures; finite-step models use a fixed computation budget, while equilibrium representations are obtained by recurrence, optimization, or direct solution.APPNP-tanh and SheafDEQ predictions approximate their recurrence limits with K steps.Source: Table 1 · p. 4
  • SheafDEQ extends SubDEQ by replacing fixed graph propagation with adaptive sheaf convolution whose restriction maps depend on the equilibrium state.Unlike finite-depth neural sheaf models, it repeatedly applies the same shared update; EnergyGNN instead derives uniqueness from a strongly convex PICNN-parameterized energy.Source: §2 · p. 4

3 SheafDEQ: adaptive neural-sheaf equilibria

SheafDEQ combines adaptive, representation-dependent sheaf propagation with a convergent equilibrium update. Its section establishes an oracle class-separation result, global convergence, and convergence under bounded staleness.

3.1 Adaptive sheaf equilibrium

  • SheafDEQ infers direction-specific, matrix-valued restriction maps from current representations and normalizes representations before inference to preserve scale invariance.The sheaf convolution acts on the original state, while the normalized representation determines the restriction-map geometry.Source: §3.1 · pp. 4–5

3.2 Expressivity at nonlinear equilibrium

  • The label-dependent fixed-sheaf construction yields positive equilibria whose coordinate orderings let a linear readout recover every node’s class.This is an oracle capacity result, not a learning guarantee; the construction uses one channel, no feature mixing, and no input injection.Source: §3.2 · p. 5

3.3 Well-posedness

  • Under the theorem’s regularity assumptions, SheafDEQ has a unique equilibrium and converges globally and linearly to it from every entrywise-positive initialization.The result assumes Bθ(X) is entrywise nonnegative, β ≥ 1.2, and PΘ is Lipschitz.Source: §3.3 · p. 6
  • A larger shift β gives a smaller contraction factor and faster guaranteed linear convergence, although these worst-case rates need not match observed speed.The text gives µ = 0.99 for tanh(t) + 1.2 and µ = 0.499 for tanh(t) + 1.603.Source: §3.3 · p. 6

3.4 Robustness to bounded staleness

  • Under the well-posedness assumptions, partially asynchronous iterations converge from every entrywise-positive initialization to the unique equilibrium.The stated setting bounds update intervals and communication delays, with each node using its current local state.Source: §3.4 · p. 6

4 Experiments

The experiments assess SheafDEQ’s predictive performance, finite-iteration recurrence behavior, and sensitivity to bounded communication staleness. Results cover distributed-inference tasks and community detection across rewiring levels.

4 Experiments

  • The evaluation separates fixed-budget predictive benchmarks from recurrence and bounded-staleness diagnostics, covering five distributed tasks and community detection across rewiring levels.The distributed tasks include Chains, Counting, Sums, MNIST Terrain, and Coordinates; predictive results alone do not establish convergence.Source: §4 · p. 6, §4.1 · p. 7

4.1 Predictive benchmarks

  • On community detection, SheafDEQ strengthens as cross-community connectivity becomes informative, while the strongest adaptive model varies across rewiring regimes.DNSD leads at low rewiring, SheafDEQ at intermediate levels, and the Loop Transformer in one strongly rewired setting; intermediate-level differences warrant caution.Source: §4.1 · p. 7, Table 2 · p. 7
  • SheafDEQ improves over fixed-propagation baselines on Sums, MNIST Terrain, and Coordinates, and is competitive on Counting, while adaptive-model comparisons vary by task.DNSD is strongest on Counting, Sums, and MNIST Terrain; the Loop Transformer exceeds SheafDEQ on MNIST Terrain and Coordinates.Source: §4.1 · p. 8, Table 3 · p. 8

4.2 Empirical test-time convergence diagnostics

  • At iteration 100, normalized SheafDEQ’s prediction discrepancies remain below 1% for initialization scales 0.001–10, but not for scale 1000.Removing pre-restriction normalization increases residuals and initialization sensitivity; these finite-iteration diagnostics do not establish global convergence.Source: §4.2 · p. 8, Figure 1 · p. 9

4.3 Sensitivity to bounded communication staleness

  • Across tested bounded-staleness schedules, normalized SheafDEQ stays approximately 7.08×10^-4 from its synchronous reference, but this does not establish exact asynchronous convergence.Endpoint medians differ by only 3×10^-7, consistent with a shared numerical-reference floor rather than a resolved staleness trend.Source: §4.3 · p. 9

5 Conclusion

SheafDEQ combines recurrent equilibrium computation with adaptive, matrix-valued neural-sheaf propagation. Its reported strengths have boundaries: some benchmarks are transductive, comparisons are parameter-unmatched, and an extreme initialization does not settle within the tested iteration range.

5 Conclusion

  • SheafDEQ improves over fixed-propagation implicit baselines on tasks requiring nonlocal aggregation and when cross-community connectivity dominates.Pre-restriction normalization is not uniformly more accurate, but improves continued-iteration stability and reduces sensitivity to bounded communication staleness.Source: §5 · pp. 9–10
  • The conclusion flags an extreme initialization that does not settle within the tested iteration range, several transductive benchmarks, and parameter-unmatched comparisons.It identifies sharper convergence certificates and scalable asynchronous inference on inductive, real-world graphs as future work.Source: §5 · p. 10

A Notation

  • The notation appendix defines entrywise hidden-state matrix inequalities and the infinity norm as the largest absolute matrix entry.Tables 4 and 5 collect graph, sheaf, equilibrium, iteration, benchmark, and diagnostic notation.Source: A Notation · p. 11, Table 4 · p. 12, Table 5 · p. 13

B.1 Well-posed implicit graph neural networks

  • Implicit GNN guarantees can come from contraction constraints, controlled linear equilibria, monotone operators, or strongly convex energies, each tied to its construction.The appendix notes that strong convexity ensures a unique minimizer, while convergence depends on the optimization algorithm.Source: §B.1 · p. 14

B.2 Subhomogeneous equilibrium theory

Subhomogeneous equilibrium theory provides convergence guarantees without spectral constraints on recurrent weights, while SheafDEQ replaces fixed graph propagation with state-adaptive sheaf propagation. The appendix relates this design to sheaf diffusion and contrasts its class-margin behavior with ordinary graph propagation.

B.2 Subhomogeneous equilibrium theory

  • When the square update is positive, Lipschitz, and subhomogeneous with degree below one, Picard iteration converges globally and linearly to a unique fixed point.Strong subhomogeneity is preserved by the compositions used in SubDEQ; positive subhomogeneous activations set the contraction degree without a spectral constraint on W.Source: §B.2 · p. 15
  • SubDEQ's graph realization uses a fixed normalized adjacency operator, whereas SheafDEQ replaces it with propagation inferred from the evolving representation.The fixed operator is not inferred from the state.Source: §B.2 · p. 15

B.3 Cellular sheaves, sections, and neural sheaf diffusion

  • Neural sheaf propagation uses state-dependent restriction maps, whose matrix products transform messages before aggregation; fixed-sheaf continuous diffusion can preserve nonuniform structured limits.In an equilibrium model, the state determines the maps, which determine the propagation operator that acts again on the state.Source: §B.3 · pp. 15–16

B.4 Extended related work

  • SheafDEQ differs from variational implicit models by defining a fixed point of adaptive, matrix-valued propagation rather than minimizing a scalar energy.The related-work discussion also distinguishes its learned state-dependent maps from fixed-sheaf asynchronous diffusion.Source: §B.4 · p. 16

C.1 A graph–sheaf PPR margin comparison

  • Signed sheaf propagation preserves the balanced block mode's margin exactly, while ordinary graph PPR retains a homophily-dependent fraction that declines under extreme heterophily.The comparison is an oracle construction: its restriction maps use class labels, so it shows representational potential rather than guaranteed recovery through learning.Source: §C.1 · p. 17

C.2 Proof of Proposition C.1

The proof compares graph and signed-sheaf PPR through a margin bound, then shows that the signed sheaf Laplacian is related to the graph Laplacian by a sign change of basis. The SBM proof uses the signed block mode to establish margin preservation.

C.2 Proof of Proposition C.1

  • Because y ⊥1 and pα(λ) decreases in λ for fixed 0 < α < 1, taking the supremum over w and b gives an APPNP bound that is strict because λ2 > 0.Source: §C.2 · pp. 17–18
  • The signed-sheaf Laplacian equals a label-sign change of basis of the graph Laplacian, so both operators have the same eigenvalues.The proof uses this relation to compare their resolvents.Source: §C.2 · p. 18

C.3 Proof of Corollary C.2

  • In the two-block SBM proof, the signed adjacency maps the block-label vector to a scalar multiple of itself, preserving its class-aligned direction.The preceding margin calculation gives γy(ϑya⊤) = ϑ for every ϑ > 0.Source: §C.3 · p. 18

C.4 Proof of Proposition 3.1

The proof establishes a unique fixed point and derives positive homogeneity of adaptive sheaf propagation from scale-invariant restriction-map inference. Under stated regularity and positivity conditions, the complete update is subhomogeneous.

C.4 Proof of Proposition 3.1

  • The update maps a compact box into itself, yielding existence, and the proof establishes that its fixed point is unique.The proof invokes Brouwer’s theorem for existence and a contradiction argument for uniqueness.Source: §C.4 · pp. 19–20

C.5 Scale invariance of normalized geometry

  • Pre-restriction normalization makes the representation-derived geometry and restriction maps invariant to positive rescaling of the state.The resulting sheaf operators, including the propagation operator, are unchanged under that rescaling.Source: §C.5 · pp. 20–21

C.6 Homogeneity of adaptive sheaf propagation

  • Scale-invariant restriction maps make the adaptive propagation positively 1-homogeneous, without sign or spectral conditions on W1, W2, or the restriction maps.Lipschitz continuity remains an explicit assumption, not a consequence of scale invariance or pointwise nonvanishing.Source: §C.6 · p. 21

C.7 Application of Sittoni and Tudisco’s Lemma 4.1

  • Composing the homogeneous adaptive propagation with a positive, strongly subhomogeneous shifted-tanh branch yields a positive, Lipschitz, subhomogeneous complete update.Strong subhomogeneity is required of the outer branch; the signed adaptive operator is concluded to be ordinarily subhomogeneous.Source: §C.7 · pp. 21–22

C.8 Consequence of Sittoni and Tudisco’s Theorem 3.7

The complete update satisfies the conditions of the subhomogeneous equilibrium result, which gives uniqueness and global linear convergence of Picard iteration. The proof also establishes convergence under bounded asynchronous delays.

C.8 Consequence of Sittoni and Tudisco’s Theorem 3.7

  • The complete update is positive, Lipschitz, and subhomogeneous, implying a unique equilibrium and global linear convergence of Picard iteration in the entrywise infinity norm.The guarantee follows by applying Sittoni and Tudisco’s Theorem 3.7 after vectorizing the state.Source: §C.8 · p. 22

C.9 Proof of Corollary 3.3

  • Under bounded delays and the condition that every node updates within B_async steps, the maximum error is nonincreasing and contracts after all nodes update.The proof bounds stale views after the initial delay window using the compact set K_X.Source: §C.9 · p. 22

D Baseline Architectures

The appendix specifies baseline architectures and their forward and backward computation, contrasting equilibrium or closed-form solvers with finite unrolled models. It also documents the implementation choices used for comparisons.

D.1 Feature-only MLP; D.2 APPNP family

  • The feature-only MLP serves as a graph-agnostic lower bound, while APPNP uses a fixed iteration budget and teleport weight.APPNP’s standard recurrence also has a closed-form equilibrium used as an alternate diagnostic solver.Source: §D.1 · p. 23, §D.2 · p. 23, U (k), APPNP · p. 23

D.3 IGNN

  • IGNN enforces a unique fixed point by projecting recurrent weights to maintain contraction, then uses Picard iteration and implicit differentiation.The forward solve uses a tolerance of 10^-6 with a maximum of 50 iterations.Source: §D.3 · p. 23

D.4 EIGNN

  • EIGNN solves a linear fixed point in closed form using a learned positive-semidefinite feature-propagation matrix.Its backward pass differentiates through the small feature-side eigendecomposition rather than unrolling Picard iterations.Source: §D.4 · pp. 23–24

D.5 EnergyGNN (edge-wise)

  • EnergyGNN represents embeddings as the minimizer of a strongly convex energy and uses L-BFGS with implicit differentiation through the energy Hessian.The appendix reports that the solver may fail to converge within its iteration budget on some graphs.Source: §D.5 · p. 24

D.6 Deep Neural Sheaf Diffusion

  • DNSD is a finite-depth sheaf model with 12 untied layers and learned directional restriction maps, using either full or diagonal maps and tanh or ReLU.Its propagation uses sheaf adjacency rather than a sheaf-Laplacian residual update.Source: §D.6 · p. 24

D.7 GraphAttnDEQ

  • GraphAttnDEQ replaces learned sheaf propagation with attention recomputed from the current iterate, but claims no contraction or other well-posedness guarantee.It is trained by ordinary backpropagation through a finite K-step unroll.Source: §D.7 · p. 25

D.8 Loop Transformer

  • The Loop Transformer repeats a graph-masked Transformer block with tied weights for K steps, without an equilibrium guarantee.It includes residual connections, post-layer normalization, and a GELU feedforward network, and is trained by unrolled backpropagation.Source: §D.8 · p. 25

D.9 Summary of solver/gradient mechanisms; E Experimental Details

  • IGNN, EIGNN, and EnergyGNN avoid backpropagation through an unrolled trajectory, unlike APPNP, GraphAttnDEQ, and the Loop Transformer.The appendix also lists datasets, splits, configurations, parameter counts, training, and compute details needed to reproduce the experiments.Source: Table 6 · p. 26, E Experimental Details · p. 26

E.1 Datasets

The appendix defines distributed-inference and community-detection benchmarks, then documents data splits, model configurations, training, and evaluation choices. It also records important limits on interpreting community results and baseline coverage.

E.1 Datasets

  • Community detection uses 1,500-node graphs with three balanced communities and progressively rewires within-community edges to cross-community endpoints.Each rewiring level adds 10% of the original within-community edges; level 10 is fully rewired and is not a monotone difficulty index.Source: §E.1 · p. 26

E.2 Data Splits

  • The distributed tasks use node-level splits except Coordinates, which uses held-out graph-level splits; community detection trains on six graphs and tests on three separate graphs.The appendix describes the community training and validation node masks separately from its held-out test graphs.Source: §E.2 · p. 27, Table 7 · p. 27

E.3 Model Configurations

  • SheafDEQ uses embeddings of dimension c=dq and learns directional restriction maps from endpoint representations, with optional pre-restriction normalization.The normalized variant is covered by the conditional contraction guarantee; the unnormalized ablation breaks scale invariance.Source: §E.3 · pp. 27–28

E.4 Parameter Counts

  • SheafDEQ’s parameters are tied across iterations, so increasing K increases computation without increasing its parameter count.The appendix contrasts this with DNSD’s 12 untied layers and the untied hidden maps of the depth-parity MLP.Source: §E.4 · p. 28

E.5 Training Procedure

  • Predictive SheafDEQ results use the Kth iterate after unrolled backpropagation, so those tables compare predictions at a shared iteration budget rather than convergence to equilibrium.Reported test metrics average five parameter-initialization seeds, with standard deviation across those seeds.Source: §E.5 · p. 29

E.6 Hyperparameter Selection

  • SheafDEQ and GraphAttnDEQ sweep βshift over {1.2, 1.3, 1.4, 1.6}, selecting configurations by mean validation metric across five seeds.The same validation-based selection rule is used for the explicitly evaluated APPNP and DNSD configurations.Source: §E.6 · p. 29, §E.5 · p. 29

E.7 Compute Infrastructure

  • Benchmark runs use AMD MI250X GPUs, while continued-iteration and bounded-staleness diagnostics use a local CPU machine.Diagnostic checkpoints are selected by reaching at least 80% validation accuracy at the trained horizon on Lhetero=7.Source: §E.7 · p. 30

E.8 Community Detection: Additional Notes

  • Community results show high seed variability at intermediate rewiring levels, and aggregate statistics cannot distinguish optimization variability from convergence to different numerical states.EIGNN and EnergyGNN are excluded from community detection; the appendix says dense spectral computations were impractical for EIGNN and EnergyGNN’s solver did not converge within budget.Source: §E.8 · pp. 30–31

E.9 Benchmark Datasets: Additional Notes

The appendix records benchmark solver and coverage limits, then specifies how initialization trajectories and the Loop Transformer control were evaluated.

E.9 Benchmark Datasets: Additional Notes

  • EnergyGNN did not converge on Coordinates across five initialization seeds, and its reported loss is marked for completeness rather than as a valid converged result.The available diagnostics do not identify the numerical cause.Source: §E.9 · p. 31
  • EIGNN is reported only on Chains and Counting; its dense eigendecomposition exceeds the node-count guard for Sums and MNIST Terrain, and no valid Coordinates result was completed.The available logs do not establish why the Coordinates runs failed; dashes denote configurations without valid evaluation results.Source: §E.9 · pp. 31–32

E.10 Protocol for Figure 1

  • Initialization trajectories compare the default row-wise normalized all-ones state with five positive random states at scales from 10^-3 to 10^3 against finite references continued to Kref = 100.The default endpoint is the reference by construction and is not assumed to be an exact equilibrium.Source: §E.10 · p. 32
  • The Loop Transformer is extended beyond its trained 20 loops, but only its one-step logit update is reported because its recurrent state is not exposed.Inference uses dropout-disabled evaluation mode.Source: §E.10 · p. 32

E.11 Bounded-staleness evaluation details

The bounded-staleness evaluation compares asynchronous schedules with finite synchronous references and finds nearly unchanged normalized-model endpoints, unlike the unnormalized ablation.

E.11 Bounded-staleness evaluation details

  • Each variant uses one selected checkpoint on the fixed Lhetero = 7 graph, with schedules varying update frequency and maximum neighbour-timestamp delay across 20 realizations.All schedules share the graph, input, parameters, and initialization; states are recorded after complete slot commits.Source: §E.11 · p. 32
  • The normalized model's synchronous reference is its Kref = 420 Picard iterate with hidden-state residual 8.95 × 10^-7; the synchronous control is omitted from endpoint comparisons beyond that finite reference.The control's later discrepancy would reflect the finite numerical reference rather than delayed communication.Source: §E.11 · pp. 32–33
  • 3 × 10^-7 separates normalized SheafDEQ endpoint discrepancies across the delayed conditions and schedules, consistent with a numerical-reference floor rather than a resolved staleness trend.The tested conditions are (2, 1), (5, 2), and (8, 4).Source: §E.11 · p. 33, Figure 2 · p. 33
  • The ablation without pre-restriction normalization has discrepancies near 4 × 10^-2 with materially wider schedule ranges.Its synchronous reference had not converged after 500 steps and retained residual 1.04 × 10^-2.Source: §E.11 · pp. 32–33

Related papers

Loading 2609.30277v1…