Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers

Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training has emerged as a critical alternative to enhance the robustness and scalability of these models. In this work, we introduce a pre-training framework tailored to both steady-state and transient regimes. For steady-state problems, we propose a geometry-driven strategy that leverages intrinsic shape descriptors to learn representations of complex 3D domains. For transient problems, we introduce a physics-driven approach based on online generation of synthetic PDE data, enabling scalable pre-training without reliance on expensive datasets. This framework learns transferable features for challenging downstream tasks. Across multiple experiments, our approach achieves up to 3$\times$ faster convergence, 2$\times$ greater data efficiency, and up to 25% higher accuracy during fine-tuning, particularly under realistic low-data regimes. This methodology provides a practical pathway toward data-efficient neural emulators for large-scale simulations.