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STCO: Conditional Neural Operators for Time-Dependent PDEs
Xingxin Yang, Zhan Zhang, Juan Li
TL;DR
Existing neural-operator predictions often use observed states and static descriptors, although control and optimization queries may prescribe future conditions. STCO provides a common PCOL interface for heterogeneous backbones, and across twelve matched backbones it reduces field and pressure-derived load errors.
Problem
Prescribed-condition operator learning predicts future PDE responses from observed history and target-time fields such as motion, inflow, or forcing.
Method
STCO combines vorticity-aware FAGL with DSFiLM to align observed and prescribed fields and route motion, inflow, and force conditions through heterogeneous backbone architectures.
Results
31.1% mean relative-L2 field-error reduction and 24.7% mean normalized pressure-derived load-error reduction were achieved across twelve matched backbones.
Takeaways & Limitations
STCO adds predictive value alongside retained observed-condition and lead-time pathways, lowering longer-lead field error for 11 backbones.
Takeaways & Limitations
The evaluation uses one seed and one observed frame in a two-dimensional moving-body Navier–Stokes system from one CFD solver, with viscous shear excluded from the load metric.
Abstract
from arXiv · showhide
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.
Introduction
Prescribed-condition operator learning predicts future PDE responses from observed history and target-time physical conditions, addressing queries in which future motion, inflow, or forcing is specified. STCO provides this conditioning through a shared interface for heterogeneous backbones and evaluates it in matched moving-body-flow comparisons.
- PCOL maps observed history, lead time, and target-time physical condition fields to the corresponding future response.
- Moving-body flow exposes representation alignment, selective condition modulation, and sparse-event supervision challenges across geometry, inflow, and forcing.
- STCO combines vorticity-aware FAGL, separate DSFiLM routes, and activity sampling while retaining each backbone’s core computation and existing context.
- The benchmark includes prescribed body motion, inflow disturbances, body-force actuation, and morphology, with predicted velocity–pressure fields supporting downstream control and optimization.
- STCO is evaluated through twelve matched baseline–STCO configurations with fixed data, backbone, decoder, and training protocols.
Related Work
Prior work addresses prescribed motion, boundary inputs, adaptive representations, and feature modulation, but STCO combines these ideas into a shared interface for heterogeneous operator backbones.
- Moving-boundary predictors infer future flow from recent fields and boundary histories, while related work predicts terminal states from prescribed motion sequences.
- Boundary-control operators map temporal boundary inputs to boundary-output trajectories, and other operators evaluate controls or predict controller inputs.
- Irregular-domain operators use coordinate deformation, graph–grid transfer, regional graphs, or latent tokens for adaptive representation.
- DSFiLM combines separate spatial condition routes, local state-dependent gates, and independently learned input and output modulation modules.
Methodology
The methodology formulates PCOL for incompressible moving-body flow and implements STCO by aligning observed and prescribed fields into adaptive regional slots before gated modulation around a backbone core.
- Problem Formulation: The nondimensional moving-body formulation uses velocity, pressure, geometry, motion, forcing, and inflow perturbation fields in a time-dependent fluid domain.
- Problem Formulation: PCOL maps an observed history and normalized query lead to the future response under the condition prescribed for that query.
- DSFiLM: STCO preserves backbone computation by connecting independently parameterized IN-DSFiLM and OUT-DSFiLM adapters around the selected core and readout.
- STCO Interface: Historical features and six target-time condition channels are standardized at regional coordinates, while signed distance is retained for query-frame masking and localization.
- FAGL: FAGL constructs exactly Nℓ adaptive regional slots using observed-frame vorticity and remaps history and target-time conditions to shared coordinates.
- STCO Architecture: Figure 2 summarizes the pipeline from observed history and prescribed conditions through FAGL and dual-site modulation to the velocity–pressure response.
- DSFiLM: DSFiLM encodes motion, inflow, and force separately, applies lead-time scaling, localizes motion near the body, and gates channels from current features.
- Sampling: External-activity sampling increases exposure to force and inflow events, emphasizing intervals crossing event onset while retaining nonzero sampling probability.
Experiments
Experiments evaluate STCO across twelve matched operator backbones on a two-dimensional moving-body CFD benchmark, using shared accuracy, load, sensitivity, and latency measures. STCO improves field and pressure-derived load accuracy across regimes and lead ranges, retains advantages at longer leads, and achieves comparable inference latency.
- Experimental Setup: The benchmark contains 142 two-dimensional incompressible moving-body simulations at Re = 5000, spanning five morphologies, an undisturbed baseline, and six prescribed-condition families.The simulations were generated with WaterLily.jl and contain 36,352 stored frames.
- Experimental Setup: Experiments compare twelve matched baseline and STCO configurations spanning graph, spectral, geometry-aware, transformer, and latent operator architectures.Backbone cores, data, decoders, sampling, and training protocols are matched within each pair; Base retains original inputs while STCO activates the complete interface.
- Evaluation: Evaluation uses 5,004 fixed reference–query pairs across three regimes and ID/OOD leads, scoring relative-L2 field error, normalized pressure-derived load error, and batch-one H200 latency.Field error macro-averages u, v, p and simulations; load error uses joint pressure-derived drag–lift RMSE, while latency excludes FAGL preprocessing.
- Evaluation: MCF measures functional dependence on lead time and spatial condition inputs by comparing original and intervened inverse-standardized full-mesh predictions under controlled interventions.Interventions hold the observed history, other condition groups, coordinates, fluid mask, and random state fixed, complementing accuracy metrics against CFD ground truth.
- Results: STCO lowers field error in 68/72 regime–lead comparisons, with a 31.1% mean reduction across all twelve backbones, and reduces pressure-derived load error by 24.7% on average.Field gains are 40.4% in ID-Lead and 23.1% in OOD-Lead; load reductions are 19.3% and 20.4%, respectively.
- Results: STCO lowers longer-lead field error for 11 backbones, while resolved motion and inflow examples show condition-specific velocity and pressure-derived load improvements at ∆n = 20.For prescribed yaw and transverse gust, transverse-velocity error decreases by 62.8% and 70.4%, while lift and drag RMSE reductions span 33.6%–75.1%.
- Results: Accuracy gains occur at comparable inference latency across matched Base/STCO pairs, and MCF is nonzero for every evaluated spatial condition group and STCO backbone.Spatial sensitivity and accuracy need not rank identically: MCF peaks for GINO, whereas field gain peaks for GAOT.
- Limitations: The evaluation is limited to one seed, one observed frame, a two-dimensional moving-body Navier–Stokes system, and one CFD solver, with viscous shear excluded from EFp.Complete condition trajectories, cross-time-consistent prediction, broader PDEs, and closed-loop action selection remain future work.
Conclusion
STCO extends prescribed-condition operator learning across heterogeneous backbones and reduces field and load errors across matched evaluations.
- STCO couples vorticity-aware FAGL, DSFiLM, and activity sampling across heterogeneous backbones.
- Across twelve matched pairs, STCO reduces mean field and load error within and beyond the training lead range.
- Condition dependence remains measurable at comparable inference cost.
Supplementary Material
The supplement provides additional details on the benchmark, FAGL implementation, training, evaluation, and extended results.
- The supplement details the benchmark, FAGL implementation, training, evaluation, and extended results.
Benchmark
The benchmark uses immersed-boundary CFD simulations of moving-body flow with prescribed disturbances, actuation, motion, and morphology, evaluated through fields, loads, and varied conditions.
- Each simulation contains T = 256 stored frames generated with the immersed-boundary CFD solver WaterLily.jl.
- The reference and query indices satisfy nq = nr + ∆n, with normalized lead τ = ∆n/(T −1).
- All simulations use Re = 5000, a NACA 0012 section, and a 514×514 grid resolving one chord with Lpix = 128 lattice units.
- The benchmark varies five amplitude envelopes while fixing St = 0.4, λ+ = 1, and ϕ = 0.
- Convecting and transverse gusts use multiple prescribed profile families, including Gaussian, sine, one-minus-cosine, top-hat, sine-squared, and trapezoidal profiles.
- The benchmark includes Lamb–Oseen vortical inflow, localized body-force actuation, prescribed motion, and compound conditions.
- The benchmark allocates 92, 8, and 42 simulations to training, validation, and test, respectively.
- With 216 reference indices and 20 training leads, the training set provides 397,440 admissible training pairs.
FAGL Implementation
FAGL builds a fixed-cardinality, vorticity-aware regional representation from the observed frame and aligns prescribed target-time fields to shared regional coordinates.
- FAGL derives observed vorticity ωr = ∂xvr −∂yur and its magnitude µr = |ωr| from the observed response.
- Observed and prescribed fields are represented through shared regional slots, with prescribed target-time fields interpolated to slot centroids.
- Graph-edge dissimilarity combines grid distance, observed-vorticity contrast, and contrast in ∥∇µr∥2.
- Edges are scanned once in ascending dissimilarity during component merging.
- The merge threshold κ is frame-adaptive.
- If merging yields fewer than Nℓ regions, refinement adds regions through prioritized splitting.
- Refinement stops at Nℓ regions or when no eligible split remains, then normalizes the collection to exactly Nℓ slots.
- Each nonempty slot stores the mean coordinate and observed fields of its member points.
Training and Evaluation
The study uses matched Base–STCO pairs, activity-aware sampling, adaptive FAGL representations, and a shared evaluation set to measure field and pressure-derived load prediction errors. Evaluation covers 5,004 reference–query pairs across regimes and lead ranges on the complete mesh.
- Matched comparison: Each Base–STCO pair shares the same data, representation, backbone context, decoder, query geometry, optimization, and training protocol.Base bypasses DSFiLM, whereas STCO activates the complete interface with target-time conditions and lead time.
- Sampling: External-activity sampling first selects a simulation, then samples a reference–query pair, weighting active intervals and onset-crossing pairs while retaining quiescent intervals.Prescribed body motion is excluded from the sampling score and remains represented by ψ and ∆ψ.
- Representation: At Nℓ= 12,000, FAGL reconstruction mean errors are 1.5%, 11.8%, 6.9%, and 7.6% for u, v, p, and ωr, respectively.FAGL uses ωr for partitioning, while the operator predicts (u, v, p).
- Evaluation set: Evaluation uses all target-frame fluid points on the complete 514×514 mesh across 5,004 shared reference–query pairs.The set is divided into six regime–lead strata, with ID-Lead at ∆n = 1–20 and OOD-Lead at ∆n = 21–40.
- Metrics: Field error uses masked norms over velocity and pressure, while pressure-derived drag and lift loads use the prescribed query-frame body contour and shared quadrature.Aggregation equally weights contributing simulations and regime strata; paired gain is 100(EBase −ESTCO)/EBase.
Extended Results
Across matched backbones and evaluation settings, STCO improves field and pressure-derived load predictions, including longer-lead field accuracy. The extended results also show performance variation across prescribed-yaw, gust, inflow, and motion configurations.
- Aggregate results: 31.1% and 24.7% are the mean paired gains in field and pressure-derived load error, with improvements in 68/72 and 63/72 cells, respectively.Positive gains indicate lower STCO error on the predefined evaluation set.
- Prescribed yaw: 18.9% is the mean reduction in Ev for prescribed-yaw responses, with STCO improving Ev for 9/12 backbones.The figure compares matched Base/STCO fields, errors, and pressure-derived lift and drag over leads ∆n = 1–40.
- Transverse gust: 58.5% is the median Ev reduction across twelve backbones for transverse-gust responses, while pressure-derived load error falls by 34.3% for 11/12 backbones.The load comparison spans independent target-time queries over forty leads.
- Response cases: The median Ev is 0.280 for prescribed-yaw transitions and 0.210 for compound vortical-inflow and motion responses.Errors concentrate near the moving body and wake in the prescribed-yaw case, and near the moving body and disturbed wake in the compound case.