MeshOp: Operator Learning for Adaptive Mesh Refinement in Parametric PDEs

Adaptive finite-element methods (AFEM) construct problem-dependent meshes through repeated solve–estimate–mark–refine cycles, but this sequential mesh-discovery process must be repeated across PDE instances with different input functions or physical parameters. We propose MeshOp, an operator-learning framework for adaptive mesh refinement across parametric PDE families. MeshOp predicts AFEM-derived refinement fields directly from PDE inputs, thereby reducing the cost of sequential mesh discovery. The predicted field is converted into a conforming hierarchical mesh by a dedicated score-to-mesh realization procedure, while residual-based AFEM can continue from this learned initialization when higher accuracy is required. Across the three benchmarks, the mean exact-level disagreement between quantized predictions and AFEM targets ranges from $11.4\%$ to $27.4\%$, yet the realized meshes attain finite-element errors comparable to those of fixed-cycle AFEM. This empirical downstream tolerance suggests that effective discretization does not require exact pointwise recovery of the AFEM refinement field. MeshOp reduces online time by $2.34\times$ and $2.10\times$ relative to reference-error-matched AFEM on Burgers and disk convection–diffusion, respectively, and by $3.06\times$ relative to comparable-accuracy fixed-cycle AFEM on three-dimensional reaction–diffusion.