Predicting Sparse Observables in Parameterized Unsteady Flows Using Coarse-Mesh Graph Neural Networks

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Full-order surrogates for high-dimensional nonlinear flows are often unnecessary when applications require only selected degrees of freedom. We propose reduced-order models that bypass full-state reconstruction by deploying graph neural networks (GNNs) directly on coarse meshes. Applied to parameterized 2D/3D confined cylinder flows and an unevenly heated cavity over Reynolds/Rayleigh-number ranges, the models generalize to unseen conditions and accurately predict sparse probe locations and integrated quantities. Coarse-mesh GNNs match or outperform finer-mesh surrogates while reducing training time and memory, even for 3D chaotic dynamics. A dynamics-informed adaptive sampling strategy guided by frequency-spectrum variations reduces maximum errors by an order of magnitude versus uniform sampling at negligible cost. Coarse-mesh training also makes noise-augmented multi-step unrolling practical, improving long-horizon autoregressive predictions. The approach outperforms parametric DMD, sparse MLPs, U-Nets, and Fourier neural operators at lower cost, and enables system identification and state estimation from sparse observations. These results show that coarse-mesh representations, adaptive sampling, and noise-augmented unrolled training form an efficient framework for reduced-order modeling of parameterized unsteady flows without solver-resolution reconstruction.