Neural PDEs have emerged as inexpensive surrogate models for numerical PDE solvers. While they offer efficient approximations, they often lack robust uncertainty quantification (UQ), limiting their practical utility. Existing UQ methods for these models typically have high computational demands and lack guarantees. We introduce a novel framework for calibrated physics-informed uncertainty quantification to address these limitations. Our approach leverages physics residual errors as a nonconformity score within a conformal prediction (CP) framework. This enables data-free, model-agnostic, and statistically guaranteed uncertainty estimates. Our framework utilises convolutional layers as finite difference stencils for gradient estimation, our framework provides inexpensive coverage bounds for the violation of conservation laws within model predictions. In our experiments, we utilise CP to obtain marginal coverage for each cell and joint coverage over the entire prediction domain of various PDEs.
Add the full text or supplementary notes for the publication here using Markdown formatting.