Solving parametric variational inequalities via flow matching

Variational inequalities (VIs) provide a unified framework for modeling optimization, equilibrium, and multi-agent decision problems. In many applications, the VI depends on exogenous parameters and must be solved repeatedly across varying instances, giving rise to parametric variational inequalities (PVIs). When solution sets are non-singleton, a single prediction can discard alternatives useful for downstream criteria such as welfare or robustness. To address this, we propose a conditional flow-matching approach that learns distributions over PVI solutions, enabling solution sampling and post-hoc selection for unseen instances. Theoretically, we establish VI-specific guarantees that translate flow-matching error into Wasserstein distributional, feasibility, and stationarity errors, together with finite-sample high-probability bounds that account for approximation, optimization, and generalization errors. Empirically, we evaluate the framework on three complementary benchmarks: Braess traffic equilibria, where it recovers diversity along a non-singleton equilibrium set; nonconvex AC optimal power flow, where it exhibits near-theory-consistent feasibility-error scaling as an out-of-assumption stress test and competitive time–accuracy tradeoffs; and non-monotone bimatrix games, where best-of-$M$ and welfare analyses evaluate flexible equilibrium selection.