Reliable Surrogate Modeling of PDEs via Physics-Informed Conformal Prediction on Graph Neural Networks

Deploying Deep Learning (DL) surrogate models in computational physics requires adherence to physical laws, generalization across diverse topologies of real-world applications, and the ability to quantify uncertainty. The high computational cost of traditional numerical solvers motivated the shift toward Artificial Intelligence (AI). Although Physics-Informed Neural Networks (PINNs) have enabled notable acceleration, the reliance on continuous automatic differentiation with respect to absolute coordinates removes spatial invariance and their ability to generalize to novel geometries. Graph Neural Network (GNN) and Geometric Deep Learning (GDL) surrogate models have addressed this by processing relative topological and graph-based features. Yet, the performance of standard message-passing architectures degrades under major distribution shifts in real-world deployment. Also, neither PINNs nor GNNs inherently quantify uncertainty, making them unreliable for safety-critical applications. To tackle these limitations, we propose the Physics-Informed Conformal Stacked Residual Network (PI-CSRN), based on a multi-resolution architecture with topology-preserving clustering and anisotropic, attention-driven message passing. PDEs are integrated into the training objectives of the surrogate models via discrete differential operators to preserve spatial invariance. Uncertainty estimation is provided via Conformal Prediction, with a residual learning mechanism to improve performance on novel manifolds. A custom benchmark of thousands of complex and diverse auto-generated 2D and 3D manifolds ensures the proper evaluation of surrogate models, using electrostatics and elasticity as the foundational physics. The proposed architecture achieves high accuracy, robustness to distribution shifts, and uncertainty quantification, thus offering a reliable blueprint for deployable AI physics surrogates.