Title: Flow-based Automatic Neural Operator with Hard Physical Constraints
Primary area: learning on time series and dynamical systems
Keywords: Operator Learning, Hard Constraint, Flow Matching

Abstract: Simulating physical systems governed by partial differential equations (PDEs) is crucial across science and engineering. Recently, generative models—exemplified by Flow Matching—have emerged as a highly competitive approach due to their ability to effectively model high-dimensional solution distributions. However, these models often struggle to ensure physical consistency, frequently violating fundamental conservation laws or boundary conditions. In this work, we propose Physics-Manifold Flow Matching (PMFM), a novel generative framework for PDE simulation that directly addresses this challenge. PMFM introduces two key innovations. First, it enforces strict, hard physical constraints by restricting the entire generative trajectory to a physical manifold defined by analytical equations, while employing a Geometric Guidance Mechanism (GGM) to maintain high-fidelity solutions. Second, to handle complex multi-physics problems, we introduce an Adaptive Constraint Projection Framework that learns to dynamically select and parameterize the currently active physical laws. We validate PMFM on several challenging systems that are highly sensitive to physical constraints, and the results show that our framework is significantly superior to state-of-the-art physics-informed generative models in producing physically valid, long-term-stable simulations.