LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning

Abstract: Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed-loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.