多吸引子图神经网络：超越唯一均衡的集值表达性

中文摘要 AI 辅助

循环和均衡图神经网络（GNN）通常强制一个唯一的不动点，或为每个图使用一个训练目标。然而，许多组合和科学问题允许多个有效解，且没有偏好哪一个。指定的目标可能强加任意的选择规则。对于对节点重新标记不变的任务，对称图可能有对称的解集，但没有对称解。我们证明，多个均衡使一个权重共享的消息传递GNN能够表示集值等变映射：不同的初始化接近不同的有效解。在所述的规则性假设下，我们首先构造全局Lipschitz、置换等变的动力学，该动力学几乎必然收敛到有效解，并以正概率到达每个解分支。然后，我们通过具有连续分量映射的循环消息传递建立近似实现，具有任意小的更新和极限误差以及任意高的概率。这超出了标准的普适性论证：尽管仅消息传递无法区分对称节点，但演化状态在每个有限步骤中保持节点可区分，而无需辅助节点标识符。这种动力学可以在没有解标签的情况下，使用问题特定的能量进行学习。在伊辛基态、蛋白质图中的结构模块检测和化学反应稳态上，学习到的更新产生多个高质量预测，具有高数值收敛率。它们在平均解质量上优于测试的唯一均衡、单目标和前馈基线，同时与更大的基于扩散的求解器保持竞争力。

英文摘要

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.

发表机构

• University of Central Florida（中佛罗里达大学）

机构由 AI 辅助整理，请以论文原文为准。