# STCO: Conditional Neural Operators for Time-Dependent PDEs

arXiv: https://arxiv.org/abs/2608.20477 · HTML: https://mlsanity.com/paper/2608.20477/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 37 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Semantic Image Synthesis with Spatially-Adaptive Normalization](https://mlsanity.com/paper/1903.07291/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Cosmos World Foundation Model Platform for Physical AI](https://mlsanity.com/paper/2501.03575/) (2025)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers](https://mlsanity.com/paper/2405.17527/) (2024)
- [Neural Operators for Bypassing Gain and Control Computations in PDE Backstepping](https://mlsanity.com/paper/2302.14265/) (2023)
- [RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains](https://mlsanity.com/paper/2501.19205/) (2025)
- [CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs](https://mlsanity.com/paper/2505.12944/) (2025)
- [GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioning](https://mlsanity.com/paper/2410.23889/) (2024)
- [Safe PDE Boundary Control with Neural Operators](https://mlsanity.com/paper/2411.15643/) (2024)

## Abstract (arXiv)

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.