Bifurcation Models: Learning Set-Valued Solution Maps with Weight-Tied Dynamics
A recent preprint on arXiv (2605.07277) presents bifurcation models, a method that ties weights in a dynamic framework for learning solution maps with multiple valid outcomes. In contrast to traditional supervised learning, which typically chooses a single solution at random, this approach employs various initial conditions to reach different stable equilibria, illustrating an attractor landscape. The authors establish that extensive set-valued maps with locally Lipschitz branches can be depicted through regular equilibrium dynamics, and that the selectors derived are mostly regular, whereas manual selectors may exhibit significant irregularity. Tests on frustrated Ising models reveal the capacity to identify several valid equilibria without branch labels, surpassing the performance of single-branch supervision. The approach is further confirmed through Allen–Cahn experiments.
Key facts
- arXiv preprint 2605.07277 introduces bifurcation models for learning set-valued solution maps.
- Standard supervised learning resolves ambiguity by choosing one solution as target, which can be arbitrary and discontinuous.
- Bifurcation models use weight-tied dynamics where different initializations converge to different stable equilibria.
- The model represents an attractor landscape rather than one chosen branch.
- Proof shows broad set-valued maps with locally Lipschitz branches can be represented by regular equilibrium dynamics.
- Induced selectors are almost everywhere regular, while manual selectors can be arbitrarily irregular.
- Experiments on frustrated Ising models show discovery of multiple valid equilibria without branch labels.
- Bifurcation models outperform single-branch supervision on frustrated Ising models.
- Allen–Cahn experiments further validate the approach.
- The paper is categorized as a cross announcement on arXiv.
Entities
Institutions
- arXiv