Cross-Physics Mapping: Learning Operators Between Fundamentally Different Physical Dynamics

Neural operators have demonstrated remarkable success in learning solution mappings within individual physical systems, yet whether they can learn transformations between fields governed by fundamentally different partial differential equations remains largely unexplored. We introduce Cross-Physics Mapping (CPM), an operator-learning framework for translating heterogeneous physical fields through compatible latent representations without assuming equivalence between their governing dynamics. We formulate conditions under which cross-physics operators are well-defined and introduce a dimensionless scaling principle for constructing physically compatible training pairs. To investigate this problem, we generate 1,000 paired diffusion and wave realizations with shared latent geometries, material heterogeneities, and excitation configurations, while restricting network inputs to surface-accessible spatiotemporal observations. We systematically benchmark seven architectures, including six neural operators and a convolutional baseline, in both mapping directions. Our experiments reveal a pronounced directional asymmetry: diffusion-to-wave mapping is substantially more challenging because diffusion attenuates the high-frequency and phase-sensitive information required for reconstructing propagating wavefronts. The U-shaped Neural Operator achieves a relative ${\ell }_2$ error of 0.307 and an $R^2$ of 0.905 in this direction, whereas the Galerkin Neural Operator achieves 0.154 and 0.935, respectively, for wave-to-diffusion mapping. These results establish cross-physics operator learning as a feasible task under shared latent structure while revealing that its achievable accuracy depends critically on the information preserved by the source dynamics. CPM provides a foundation for investigating heterogeneous operator learning, physics-aware modality translation, and the identifiability of mappings between distinct physical systems.