Homomorphic Generator Learning: Compositional PDE Neural Operators for Multi-Physics Surrogates

PDE surrogates may be asked to predict systems whose physical mechanisms were observed separately, but never together, during training. Although the corresponding equation terms add, their solutions generally do not; a learned solution map therefore has no explicit rule for combining those observations. We propose (HGL), which learns a reusable generator for each mechanism and constructs an unseen system by adding its active generators. Each generator combines a role-constrained principal operator with a gated correction. Known symbols provide operator priors; otherwise, the symbols are learned from data. Mechanism coefficients are supplied by the equation or estimated from a short observed trajectory. A task-specific evaluator converts the combined generator into a rollout or steady-state solution. On unseen advective Fisher–KPP dynamics, HGL improves long-horizon prediction with either supplied or estimated coefficients relative to the evaluated solution-operator baselines. It also performs well as training coverage expands from individual mechanisms to partial combinations and when generators learned from different equation families are reused in a steady-state problem. These results support generator-level composition as an inductive bias for multi-physics generalization.