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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed operator learning · 0 citations (OpenAlex)

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Proposes bifurcation models, weight-tied dynamics whose attractors represent set-valued solution maps, with theory on regular representation and Lipschitz selectors, and experiments on frustrated Ising and Allen-Cahn problems.

Problem

Supervised learning of multi-solution (set-valued) maps forces a single branch label via a hidden selector that can be arbitrary and irregular; the question is whether weight-tied dynamics can represent the full solution set regularly and discover multiple solutions without branch labels.

Approach

Represent a set-valued map as the attractor set of a weight-tied update g(y,x) whose different initializations converge to different branches; prove a globally input-Lipschitz construction and that induced selectors are locally Lipschitz almost everywhere, while manual selectors can be arbitrarily irregular. Train unsupervised on physical energy, with optional explicit diversity regularization.

Key results

• On frustrated Ising, energy-trained recurrent GNN reaches test energy -0.983 vs -0.750 for vanilla GNN energy-trained and -0.092 for Gurobi-label recurrent GNN, finding ~13.6 of 20 solutions per graph.
• On Allen-Cahn, energy-only training yields residual MSE 9.19e-4 with 1.00 solution cluster; diversity weight 1.2 raises clusters to 16.11 but residual rises to 5.74e-3.
• Manual alternating selectors on toy problems: test error grows from 0.0181 (2 intervals) to 10.60 (64 intervals) for toy example 1, versus 0.0028 for bifurcation model.

Surprising findings

• Energy-only Allen-Cahn training collapses to a single dominant branch despite multi-equilibrium capacity.
• Explicit diversity promotion yields an accuracy-diversity tradeoff in Allen-Cahn.

Limitations

• Multi-branch discovery is problem-dependent; Allen-Cahn energy minimization collapses to one branch without diversity term.
• Diversity regularization degrades physical accuracy (residual rises roughly sixfold).
• Theory assumes finitely many locally Lipschitz branches and bounded domain; construction is nonparametric and not directly a trained architecture.

Experimental setup

Datasets: frustrated Ising graph instances (antiferromagnetic) Allen-Cahn forcing fields on 2π-periodic torus

Benchmarks: frustrated Ising test graphs Allen-Cahn test forcing fields

Metrics: rounded-spin Ising energy steady-state PDE residual MSE number of distinct solutions/clusters perturbation sensitivity percentiles test error for selector regularity

Baselines: Gurobi label fitting vanilla GNN IMEX label fitting energy-only training

Builds on: implicit neural operator deep equilibrium models Fourier neural operator

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