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

Researchers present bifurcation models, a machine learning approach that uses weight-tied dynamical systems to learn multiple valid solutions for problems with set-valued outputs. Rather than forcing a single target label, the model represents an attractor landscape where different initializations converge to different stable equilibria, enabling discovery of diverse valid solutions without explicit branch labels.

This research addresses a fundamental challenge in supervised learning: many real-world problems have multiple correct answers, yet standard approaches arbitrarily select one as the target. Bifurcation models reframe this by leveraging dynamical systems theory, where a single neural network can represent an entire landscape of solutions. Different initializations act as natural explorers of this landscape, converging to different stable points that correspond to different valid solutions.

The theoretical contribution is substantial. The authors prove that broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics, and critically, that the selectors induced by the model are almost everywhere regular—far superior to arbitrarily chosen manual targets. This suggests the model discovers structure inherent in the solution space rather than imposing artificial discontinuities.

Experiments on frustrated Ising models and Allen-Cahn equations demonstrate practical viability. The approach outperforms single-branch supervision by discovering multiple valid equilibria without requiring labeled branches for each solution. However, experiments reveal that diversity requires explicit encouragement through loss function design, introducing an accuracy-diversity tradeoff that practitioners must navigate.

For machine learning researchers, this work opens pathways for handling inherently multi-solution problems in physics-informed learning, combinatorial optimization, and inverse problems. The connection between dynamical systems stability and solution multiplicity provides a principled framework beyond standard uncertainty quantification methods. The findings suggest that many supervised learning tasks may be artificially constrained by single-label conventions.

• →Bifurcation models learn entire solution landscapes through weight-tied dynamics rather than selecting arbitrary single targets.
• →Theoretically, induced selectors from the model are almost everywhere regular, outperforming manually chosen arbitrary selectors.
• →Experiments on Ising models show the approach discovers multiple valid equilibria without explicit branch labels.
• →Diversity in solutions requires explicit loss function design, revealing an inherent accuracy-diversity tradeoff.
• →This framework applies broadly to physics-informed learning, combinatorial optimization, and inverse problems with multiple valid solutions.