STCO: Conditional Neural Operators for Time-Dependent PDEs

Xingxin Yang; Zhan Zhang; Juan Li

cs.AIAug 20, 2026

STCO: Conditional Neural Operators for Time-Dependent PDEs

Authors: Xingxin Yang, Zhan Zhang, Juan Li

Organizations: Department of Engineering, King’s College London, London, United Kingdom

Abstract

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 $\bm{X}_{q}$ . FAGL provides regional coordinates $\bm{X}_{r}$ , 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 $\bm{X}_{r}$ 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.

Panel of Table 1 — Table 1: Regime-balanced accuracy and MCF sensitivity across twelve matched backbones. ID/OOD denote $\Delta n=1$ – $20$ / $21$ – $40$ ; errors macro-average three regimes. Positive $\Delta$ favors STCO, and bold marks the lower paired error. The Average row macro-averages backbone-level entries, including paired reductions. Mean $\Delta_{y}$ 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 $n_{r}=128$ , $\Delta n=10$ ; rows show $u$ , $v$ , and $p$ . (b,c) CFD GT, GAOT Base/STCO predictions, absolute $v$ errors, and independent-query loads for prescribed yaw ( $n_{r}=128$ ) and transverse gust ( $n_{r}=126$ ) at $\Delta n=20$ . Backgrounds mark ID/OOD-Lead, and vertical lines mark displayed fields. Annotations give relative- $L_{2}$ 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 $b$	$A_{0,b}$	$A_{1,b}$	$A_{2,b}$
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

A_{b}(1)=A_{\max}=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	$\tau$	$\psi$	$\Delta\psi$	$\bm{g}$	$\bm{u}_{\mathrm{bc}}$
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	0.386	0.239	0.156	0.172	0.510
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	$-11.6$	$-8.8$	$-2.5$	0.318	0.343	0.178	0.270	0.526
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,

\tau

is replaced by another same-band lead,

\psi

and

\Delta\psi

are drawn from fixed channelwise Gaussian marginals, and

\bm{g}

and

\bm{u}_{\mathrm{bc}}

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.

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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

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T_{\mathrm{in}}

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Abstract

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