Beyond Source Simulation: Pretraining with Proxy Governing Equations

Physical simulation is fundamental to aerospace, marine, and automotive engineering design, but obtaining high-fidelity solutions for diverse geometries and operating conditions remains computationally expensive. Although neural surrogates accelerate repeated prediction, their training still relies heavily on costly simulated fields for supervision. To overcome this limitation, we propose proxy governing equation pretraining, a new paradigm that learns transferable physical structure directly from unlabeled 3D geometries and governing laws, without precomputed source simulation fields. Specifically, we pair each geometry with randomly sampled physical conditions and pretrain a shared predictor using proxy equation residuals and boundary constraints. We adopt the classical Oseen system as a physically rich proxy that retains transport, pressure coupling, and viscous diffusion, but its second-order derivatives are difficult to estimate accurately on irregular 3D point sets. To address this, we introduce auxiliary momentum fluxes and reformulate the system into a provably equivalent first-order mixed form that preserves the proxy dynamics while requiring only first-order derivatives. Combined with boundary-aware operators and volume–surface pressure consistency, this formulation enables effective physical supervision on irregular 3D geometries. The pretrained model is then fine-tuned with high-fidelity target labels for diverse downstream tasks. Across five benchmarks spanning aerodynamic, hydrodynamic, and structural prediction, it transfers to substantially different target physics, including fluid-to-structure transfer. Our method matches full-data scratch accuracy with up to 60% fewer labeled simulations and further improves prediction accuracy on the data-rich DrivAerNet++ benchmark. These results show that proxy governing equations provide a principled way to lift geometry into transferable physical structure, extending physical-field pretraining beyond fixed collections of simulated solutions.