让模型同时学习多个正确解，突破传统单解限制。
Bifurcation Models: Learning Set-Valued Solution Maps with Weight-Tied Dynamics
- 用权重共享的动态系统表示多解吸引子景观
- 在自旋玻璃模型中发现多个有效平衡态，性能优于单解监督
- 可显式控制解的多样性与精度的权衡，适合多解问题
许多科学和组合问题存在多个正确解，而非单一标签。标准监督学习通过选定一个解作为目标来处理这种歧义，但该隐含选择可能任意、不连续，且比真实解集更难学习。本文研究分岔模型，一种权重共享的动力学视角：不同初始化可收敛至不同稳定平衡点，从而模型表征的是吸引子景观而非单一分支。我们证明，具有局部Lipschitz分支的广泛集合值映射可通过常规平衡动力学表示，且其诱导的选择器几乎处处光滑，而人工选择器可任意不规则。在受挫伊辛模型上的实验表明，此类动力学无需分支标签即可发现多个有效平衡态，性能优于单分支监督。Allen-Cahn实验进一步显示，多样性并非自动出现：可通过显式激励实现，但伴随精度与多样性之间的权衡。
原文摘要 · Abstract (English)
Many scientific and combinatorial problems admit multiple correct solutions, not a single label. Standard supervised learning resolves this ambiguity by choosing one solution as the target, but this hidden selector can be arbitrary, discontinuous, and harder to learn than the underlying solution set. We study bifurcation models, a weight-tied dynamical view in which different initializations can converge to different stable equilibria, so the model represents an attractor landscape rather than one chosen branch. We prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics and that the induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular. Experiments on frustrated Ising models show that such dynamics can discover multiple valid equilibria without branch labels and outperform single-branch supervision. Allen--Cahn experiments further show that diversity is not automatic: it can be encouraged explicitly, but with an accuracy--diversity tradeoff.
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