A spectral–physical manifold for scalable knowledge transfer in partial differential equation families

High-fidelity simulation across geometrically heterogeneous physical systems is fundamentally constrained by the cost of resolving multiscale features over large design spaces. Although governing partial differential equations remain invariant, topological variations induce structured distortions in the solution space that existing neural operators and transfer learning methods fail to generalize across. Multi-task PDE learning is reformulated as navigation on a spectral–physical manifold, in which invariant differential operators are explicitly disentangled from geometry-specific adaptation. This formulation enables physically consistent knowledge diffusion across tasks with distinct topologies. Validated on 970 three-dimensional multi-physics problems with diverse geometries and canonical benchmarks in elasticity and heat conduction, the framework achieves fine-grid accuracy (R2>0.95) from coarse-grid supervision, converges faster with lower physical residuals than state-of-the-art operator-learning baselines, and reduces computational cost by approximately one to two orders of magnitude compared with conventional numerical simulation workflows, establishing a scalable and data-efficient paradigm for physics-based digital twins.