CoupledFlow: One-Step Neural Operators for Coupled Multi-Physics PDEs

We investigate the problem of learning neural operators for coupled multi-physics PDEs, where multiple physical processes co-evolve and influence one another via bidirectional coupling. For example, in subsurface CO$_2$ storage, fluid flow, heat transfer, and geo-mechanical deformation evolve in a tightly coupled manner. Prior work, including neural operators and generative PDE solvers, mostly focuses on single-physics settings or relies on iterative inference, limiting their effectiveness in coupled PDEs. This paper introduces coupled-flow, a one-step generative neural operator that revisits coupled operator learning as solving a distributional transport problem. It parameterizes process-specific average velocity fields conditioned on the full coupled state, enabling cross-process interactions to emerge naturally from the transport dynamics. We further establish a Wasserstein-2 generalization bound that ties the prediction error of coupled-flow to its training losses through the regularity of the induced flow. Experiments on a wide variety of multi-physics benchmarks demonstrate that coupled-flow achieves significant performance improvements over coupled neural-operator and generative PDE baselines.