Harmonic-Mapping Operator: A scalable graph-free and geometry-agnostic surrogate for thermal simulation with a moving heat source

Motivated by additive manufacturing (AM), we study surrogate modeling of transient heat conduction on complex geometries with a moving spot heat source. Existing surrogates are often limited to rectangular domains, require retraining for each new geometry, or rely on graph representations that add significant overhead and hinder scalability. To address these gaps, we introduce the Harmonic-Mapping Operator (HMO), a scalable, graph-free, and geometry-agnostic surrogate that learns thermo-physical responses without retraining for new part geometries. HMO maps arbitrary shapes onto a canonical square domain via harmonic maps, solves the transient heat equation with a single neural operator trained once, and projects results back through the inverse map. Geometry influences the model only through a metric tensor, enabling consistent generalization to unseen shapes. To scale to industrial components, a domain-decomposition inference (DDI) scheme applies the operator in parallel across subdomains while maintaining flux continuity. Compared with GeoFNO and MeshGraphNet baselines, HMO achieves lower errors, superior long-horizon stability, and faster inference, providing a reusable and real-time surrogate.