🤖 AI Summary
Standard supervised learning introduces arbitrary and discontinuous implicit selectors in problems with multiple solutions by enforcing a single label assignment. This work proposes an equilibrium dynamics framework grounded in weight-sharing dynamical systems, where distinct initial states converge to different stable attractors, collectively representing the solution set through the attractor landscape. The framework employs generalized set-valued mappings to model locally Lipschitz branches and proves that the induced selector is almost everywhere regular—outperforming handcrafted alternatives. Coupled with a diversity control mechanism, the method successfully discovers multiple valid solutions in frustrated Ising models without requiring branch-specific labels, surpassing single-branch supervised approaches. Further experiments on the Allen–Cahn equation elucidate the inherent trade-off between solution accuracy and diversity.
📝 Abstract
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.