Neural operators have become efficient surrogates for solving time‑dependent partial differential equations (PDEs), yet conventional approaches predict future states solely from observed states and static problem descriptors. In control or optimization scenarios, one often needs to prescribe body motion, inflow, or external forces at query time, which are not uniquely determined by the observations. To address this, we introduce the Spatiotemporal Conditional Operator (STCO) for Prescribed‑Condition Operator Learning (PCOL), offering a unified interface that feeds target‑time condition fields into heterogeneous backbones while preserving each network’s core computation and context pathways.
STCO’s condition interface combines Flow‑Aware Graph Leaf (FAGL) with Dual‑Site Feature‑wise Linear Modulation (DSFiLM). FAGL, without learning, uses vorticity from the final observed frame to build a fixed‑cardinality adaptive partition and co‑locates the observed history and target‑time condition fields at the regional coordinates. DSFiLM injects separate motion, inflow, and force routes before and after operator computation, employing slot‑ and channel‑wise gates driven by current features for feature‑level modulation.
We evaluate twelve matched backbone architectures covering various physical and temporal inputs. The benchmark is an immersed‑boundary computational fluid dynamics (CFD) testbed that includes prescribed motion, inflow disturbances, body‑force actuation, and morphology changes. Results show that across twelve backbones, three regimes, and two lead times, STCO reduces the relative $L_2$ field error by an average of $31.1\%$ and the normalized pressure‑derived load error by $24.7\%$. Moreover, STCO lowers long‑lead field error for 11 backbones, and interventions on each condition group produce measurable prediction changes.
Blogger's Review: By providing a flexible condition‑injection mechanism, STCO markedly improves the accuracy and controllability of neural operators for time‑dependent PDEs, offering a practical pathway for real‑time control and optimization in engineering applications.