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.
Figures & tables
Figure 1: Prescribed-condition operator learning in moving-body flow. The dashed frame marks the PCOL map from observed history and prescribed condition to future response. Examples include a vortical encounter, body motion, a convecting gust, and body-force actuation. Response panels show vorticity and pressure-derived load histories from CFD ground truth (GT). The predicted velocity–pressure field supports downstream control and optimization.
Figure 2: STCO architecture. (a) Overall pipeline from observed history and prescribed target-time conditions to the velocity–pressure response on Xq . FAGL provides regional coordinates Xr , while IN-DSFiLM and OUT-DSFiLM bracket a selected backbone core. (b) FAGL constructs an observed-vorticity-aware partition and aligns historical aggregates and target-time conditions on Xr by four-neighbor inverse-distance weighting. (c) DSFiLM applies lead-time scaling and state-gated motion, inflow, and force shifts, with signed distance localizing motion near the body.
Table 1: Regime-balanced accuracy and MCF sensitivity across twelve matched backbones. ID/OOD denote Δn=1 – 20 / 21 – 40 ; errors macro-average three regimes. Positive Δ favors STCO, and bold marks the lower paired error. The Average row macro-averages backbone-level entries, including paired reductions. Mean Δy averages six field gains; MCF reports lead-time and mean spatial-condition sensitivity.
Figure 4: Prescribed-condition velocity–pressure responses and pressure-derived loads. (a) CFD GT and matched Base/STCO absolute errors for Poseidon, PINO, and GEPS on a vortical encounter at nr=128 , Δn=10 ; rows show u , v , and p . (b,c) CFD GT, GAOT Base/STCO predictions, absolute v errors, and independent-query loads for prescribed yaw ( nr=128 ) and transverse gust ( nr=126 ) at Δn=20 . Backgrounds mark ID/OOD-Lead, and vertical lines mark displayed fields. Annotations give relative- L2 errors; color limits are shared within comparisons.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Label
Condition
Train
Val.
Test
Total
baseline
Undisturbed
4
1
0
5
vrt
Vortical encounter
15
2
7
24
cvg
Convecting gust
16
1
9
26
tg
Transverse gust
17
1
8
26
act
Body-force actuation
18
1
10
29
motion
Prescribed motion
19
1
4
24
Appendix
Table 2: Composition of the moving-body-flow benchmark. Training, validation, and test contain 92 , 8 , and 42 simulations. The test set covers all six prescribed-condition families.
Morphology b
A0,b
A1,b
A2,b
Anguilliform
0.0367
0.0323
0.0310
Carangiform
0.0200
−0.0825
0.1625
Subcarangiform
0.0200
0.0300
0.0500
Thunniform
0.0050
−0.0200
0.1150
Ostraciiform
0.0020
−0.0350
0.1330
Appendix
Table 3: Centerline-amplitude coefficients in Equation 10 . All five profiles satisfy Ab(1)=Amax=0.1 while varying the chordwise amplitude envelope.
Family
Varied benchmark factors
Profile or command
vrt
Circulation, lateral placement, and onset
Lamb–Oseen
cvg
Amplitude, streamwise placement, and onset
Gaussian, sine, one-minus-cosine
tg
Amplitude, streamwise placement, and onset
Top-hat, sine-squared, trapezoid
act
Amplitude, placement, onset, and duration
One-cosine, step, ramp-and-hold
motion
Surge, sway, and yaw magnitude
Smooth ramp
mix
Motion–disturbance pairing and relative onset
cvg + surge, act + surge, vrt + yaw, tg + sway
Appendix
Table 4: Prescribed-condition families and varied benchmark factors. Compound cases pair one motion command with one disturbance.
Regime
ID-Lead pairs/cases
OOD-Lead pairs/cases
Pre-event
834/38
834/8
Motion transition
834/4
834/4
External onset
834/37
834/37
Appendix
Table 5: Composition of the predefined 5,004 -pair evaluation set. Each regime–lead stratum contains 834 pairs. Metrics macro-average the contributing simulations within each stratum. A simulation may contribute pairs to more than one regime.
ID-Lead gain (%)
OOD-Lead gain (%)
Route-level MCF
Backbone
Metric
Pre-event
Motion
External
Pre-event
Motion
External
τ
ψ
Δψ
g
ubc
MGN
Field
63.9
24.4
43.0
21.2
−7.4
15.9
0.386
0.239
0.156
0.172
0.510
Load
77.3
8.4
9.6
48.2
3.9
30.5
RIGNO
Field
11.2
8.2
19.1
8.8
3.8
17.3
0.318
0.343
0.178
0.270
0.526
Load
11.3
−3.3
2.2
−11.6
−8.8
−2.5
PINO
Field
70.1
39.1
54.0
61.2
39.1
58.1
0.211
0.339
0.186
0.401
0.458
Appendix
Table 6: Stratified paired gains and route-level MCF on the predefined 5,004 -pair evaluation set. Positive gains indicate lower STCO error, bold marks negative gains, and “External” denotes external onset. STCO improves 68/72 field cells and 63/72 pressure-derived load cells, with mean gains of 31.1% and 24.7% . For MCF, τ is replaced by another same-band lead, ψ and Δψ are drawn from fixed channelwise Gaussian marginals, and g and ubc use active benchmark fields. Deterministic draws are shared across models; unchanged-input repeats define the numerical floor. GAOT has the largest mean field gain, and GINO the largest spatial MCF.
We study data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, we apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, we propose a spatiotemporal neural operator mapping a resolved history slab on Ω×[−Tin,0] to a resolved future slab on Ω×[0,Tout]. Spatial mixing uses Fourier convolution, while temporal mixing uses a causal kernel operator with position-attention weights on time lags. This causal temporal operator encodes finite-memory effects in the resolved dynamics while preserving the directionality of the history-to-future map. To improve rollout robustness and suppress nonconservative artifacts, we embed a flux-form inductive bias by parameterizing the windowed update in explicit divergence form. We also provide a data-driven guideline for selecting the memory length Tin via the decorrelation time of a closure-injection diagnostic computed from filtered trajectories. We validate on the coarse-grained viscous Burgers' equation, the Kuramoto-Sivashinsky equation, and two-dimensional turbulent flows, obtaining stable autoregressive rollouts with improved long-horizon accuracy and statistical fidelity.
Junfeng Chen
Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China
Neural operators learn mappings between function spaces, but are typically developed with dense input-output training fields and fully observed inputs at inference. Many scientific problems require instead predicting solution fields from sparse, irregular, or partial observations under uncertainty. We introduce Neural Operator Processes (NOPs), a framework that unifies neural-process conditioning with neural-operator decoding to predict full output fields from limited context. NOPs condition on sparse joint input-output observations and support deterministic and probabilistic prediction within a shared encoder-decoder architecture. We study two conditioning strategies, convolutional pooled summaries and query-aligned attention, and analyze how their interaction with latent stochastic variables depends on PDE geometry. Across function regression and three PDE benchmarks, we find that sparse conditional operator learning is viable and can match dense-grid behavior in several regimes, that preserving local context-query geometry is essential in non-periodic settings but less so in spectrally smooth periodic regimes, and that uncertainty-aware operator learning succeeds when latent conditioning complements rather than overwrites the local geometric pathway. These results provide a basis for probabilistic operator learning under partial observations and help bridge operator learning and probabilistic meta-learning in function space.
Partial differential equation (PDE) simulation holds extensive significance in scientific research. Currently, the integration of deep neural networks to learn solution operators of PDEs has introduced great potential. In this paper, we present UniFluids, a conditional flow-matching framework that harnesses the scalability of diffusion Transformer to unify learning of solution operators across diverse PDEs with varying dimensionality and physical variables. Unlike the autoregressive PDE foundation models, UniFluids adopts flow-matching to achieve parallel sequence generation, making it the first such approach for unified operator learning. Specifically, the introduction of a unified four-dimensional spatiotemporal representation for the heterogeneous PDE datasets enables joint training and conditional encoding. Furthermore, we find the effective dimension of the PDE dataset is much lower than its patch dimension. We thus employ x-prediction in the flow-matching operator learning, which is verified to significantly improve prediction accuracy. We conduct a large-scale evaluation of UniFluids on several PDE datasets covering spatial dimensions 1D, 2D and 3D. Experimental results show that UniFluids achieves strong prediction accuracy and demonstrates good scalability and cross-scenario generalization capability. The code will be released later.
Haosen Li, Qi Meng, Jiahao Li +4
Academy of Mathematics and Systems Science, Chinese Academy of Sciences · Microsoft Research Asia · Renmin University of China +1