Bifurcation Models: Learning Set-Valued Solution Maps with Weight-Tied Dynamics

TL;DR

This paper introduces bifurcation models that learn set-valued solution maps via weight-tied dynamics, enabling the discovery of multiple equilibria without explicit labels, with applications in physics-inspired models.

Contribution

It demonstrates that equilibrium dynamics can represent complex set-valued maps and introduces methods to encourage solution diversity explicitly.

Findings

Models can discover multiple valid equilibria without branch labels.

Bifurcation dynamics outperform single-branch supervision in experiments.

Diversity can be explicitly encouraged with an accuracy-diversity tradeoff.

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.