Beyond Equality: Inequality-Constrained Flow Matching for Physically Admissible Solutions

Generative models for PDE solutions must produce \textit{physically admissible} samples, yet state-of-the-art physics-constrained samplers enforce only equality constraints ($h(u)=0$) and ignore the inequality constraints ($g(u)\leq 0$) that equally govern admissibility---e.g., entropy conditions and invariant-interval bounds. We introduce Inequality-aware Physics-Constrained Flow Matching (I-PCFM), a framework that \textit{jointly handles equality and inequality constraints} at inference time without retraining. Within I-PCFM we systematically compare three general-purpose projection strategies grounded in classical constrained optimization---(A)~slack-variable reformulation, (B)~log-barrier augmentation, and (C)~active-set projection---together with (D)~Frank-Wolfe guidance, a projection-free strategy with closed-form linear-minimization oracles (LMOs) for linear constraints and a Riemannian extension for nonlinear settings. Across four PDE benchmarks, including the 1D Heat and 1D Reaction-Diffusion equations, we establish three key findings: (1)~equality-only projection \textit{degrades} inequality satisfaction by $5$--$20\times$ relative to unconstrained generation, (2)~among general-purpose strategies, active-set projection attains both the highest joint feasibility ($97\%$ on Reaction-Diffusion) and the lowest runtime, and (3)~Frank-Wolfe guidance matches active-set projection's feasibility while running up to $40\times$ faster. The code can be found at \url{https://anonymous.4open.science/r/ipcfm-icml-workshop/}.