CoPiT: A conditional Position-induced Transformer for aerodynamic field prediction on unstructured meshes

Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models. Evaluating design changes through numerical simulation requires meshing the continuous geometry, a computationally expensive and often brittle process that can require manual intervention and replaces the continuous representation with a discrete approximation. Most neural surrogates accelerate the simulation, but inherit this representation gap by relying on meshes, point clouds, voxels, or other sampled approximations of geometry. We introduce CANTO, a transformer neural operator that maps directly from continuous CAD geometry to physical fields, without meshing the input geometry. We develop a theoretical framework for learning operators from geometric manifolds to function spaces of physical fields, representing geometry through sequences of parametric patches. CANTO instantiates this framework by directly tokenizing non-uniform rational B-spline (NURBS) patches from their control points, knot vectors, and weights, and predicts continuous surface and volume fields at arbitrary query locations. We evaluate CANTO on four automotive and aircraft aerodynamics industry benchmarks: AhmedML, WindsorML, DrivAerML, and HiLiftAeroML. CANTO achieves state-of-the-art accuracy on most evaluated surface and volume prediction tasks, including a 19.8% reduction in surface-pressure relative $L_2$ error compared with AB-UPT on HiLiftAeroML. Differentiability with respect to CAD parameters further enables gradient-based inverse design of designs. On AhmedML, CANTO identifies designs with 4.4 to 20.4% lower drag than the best dataset designs satisfying the same volume and lift constraints, with the improvements verified using the same CFD setup used to generate the original dataset.

Accurate aerodynamic prediction is critical for designing fuel-efficient and safe transportation systems such as aircraft and automobiles, yet traditional computational fluid dynamics (CFD) simulations remain computationally expensive and expertise-intensive, severely limiting their use in iterative design and real-time analysis. Existing deep learning surrogates suffer from two major limitations: (i) they are evaluated on datasets with narrow flow-condition ranges, leaving their performance under complex flow conditions undemonstrated; (ii) they treat global flow conditions as a single vector injected uniformly across all surface points, ignoring that different geometric regions experience distinct local flow phenomena, which degrades prediction accuracy under complex flow conditions. To address these limitations, we propose the Flow State Attention Network (FSAN). FSAN separately encodes point cloud and flow conditions, then partitions the geometry into multiple flow states via learnable soft assignments, and uses flow features to update these state representations, which in turn influence point cloud features through state changes. This enables fine-grained, state-specific interaction between geometry and flow information. Extensive experiments on two well-recognized aerodynamic benchmarks demonstrate that FSAN achieves the highest accuracy among the methods compared in this work at a higher computational cost. On Emmi-Wing, FSAN reduces the Relative L2 (REL-L2) error by over 20% compared to the strongest baseline (Transolver), and on DrivAerNet++, it achieves a 10% reduction compared to the strongest baseline (AdaField). These results establish FSAN as a promising neural surrogate on public benchmarks with diverse flow conditions and geometries.

Neural operators have emerged as efficient surrogates for time-dependent physical systems governed by partial differential equations (PDEs), but their future-state predictions are often conditioned only on observed states and static problem descriptors. For control or optimization, however, body motion, inflow, or forcing are prescribed for the query without being determined solely by the observed state. We introduce the Spatiotemporal Conditional Operator (STCO) for prescribed-condition operator learning (PCOL), a common interface that supplies prescribed target-time condition fields to heterogeneous backbone architectures while retaining their architecture-specific core computation and context pathways. Its condition interface combines Flow-Aware Graph Leaf (FAGL) with Dual-Site Feature-wise Linear Modulation (DSFiLM). Non-learned FAGL uses vorticity from the final observed frame to construct a fixed-cardinality adaptive partition, then co-locates the observed history and target-time condition fields at its regional coordinates. DSFiLM injects separate motion, inflow, and force routes before and after operator computation through current-feature-driven slot- and channel-wise gates. We evaluate twelve matched backbone architectures with different existing physical and temporal inputs. The immersed-boundary computational fluid dynamics (CFD) benchmark spans prescribed motion, inflow disturbances, body-force actuation, and morphology. Across twelve matched backbones, three regimes, and two lead ranges, STCO yields mean paired reductions of 31.1% in relative-L2 field error and 24.7% in normalized pressure-derived load error. It also lowers longer-lead field error for 11 backbones, while interventions on individual condition groups produce measurable prediction changes for every group evaluated.