Neural Homotopy System: Differentiable Predictor-Corrector for Homotopy Problems

The homotopy paradigm provides a general strategy for solving difficult problems. It connects a hard target problem to a simple source problem through a continuous interpolation, then tracks the resulting solution trajectory, typically with a predictor-corrector algorithm. This paradigm underlies methods across many domains, but the interpolation is usually hand-designed and task-specific. Existing learning-based attempts to replace it remain instance-specific, requiring retraining for every new problem instance. We propose Neural Homotopy System (NHS), a differentiable predictor-corrector framework in which the predictor and corrector are learned neural modules, trained once and amortized over problem instances within a given problem family. NHS parameterizes the predictor with a neural ordinary differential equation that learns the solution trajectory across homotopy levels, while the corrector uses a deep equilibrium model that drives the predicted solution toward a problem-specific equilibrium condition. A predictor-anchored proximal term makes this equilibrium explicitly depend on the prediction, allowing gradients to propagate across homotopy levels and enabling end-to-end training of the complete trajectory. Instantiating this shared architecture separately for unconstrained optimization, constrained optimization, and conditional probabilistic sampling, NHS consistently achieves higher solution accuracy than classical local and hand-designed homotopy methods, as well as existing learning-based baselines, across all three settings.