APPSolver: Adaptive Patch Partitioning for Point-Wise Ship Flow Prediction on Unstructured Meshes

Wenhua Huo; Fenglei Han; Wangyuan Zhao; Xiao Peng; Chunhui Wang; Jialin Wu; Jiayi Han

cs.AIAug 29, 2026

APPSolver: Adaptive Patch Partitioning for Point-Wise Ship Flow Prediction on Unstructured Meshes

Authors: Wenhua Huo, Fenglei Han, Wangyuan Zhao, Xiao Peng, Chunhui Wang, Jialin Wu, Jiayi Han

Organizations: College of Shipbuilding Engineering, Harbin Engineering University

Abstract

Large non-uniform point sets make direct attention-based surrogate modeling costly for ship hydrodynamics. We introduce APPSolver, a point-wise flow-prediction framework built around Adaptive Patch Partitioning (APP), a deterministic quadtree representation for fixed two-dimensional horizontal slices extracted from ship CFD simulations. APP assigns finer patches near the hull and coarser patches farther away, downsamples patch contents, and recovers predictions to the full reference point set. Under a corrected protocol that constructs natural $(t,t+1)$ pairs before splitting, reuses training-set normalization statistics, and reports three model seeds, learned tokenizers are more accurate than APP-Transformer, and a persistence baseline has lower one-step MAE on all three ShipBench hulls. The supported benefit of APP is therefore computational rather than universal predictive superiority: on a representative DTC input, APP-Transformer requires 1.815 GFLOPs and 1.309 ms per model forward, while a matched ablation shows that adaptive partitioning reduces MAE by 16.4-24.9% relative to a uniform partition augmented with learned slicing. Condition encoders provide setting-dependent gains in leave-one-hull-out evaluation, but the current absolute next-state objective does not establish accurate long-horizon dynamics. These results characterize APP as a compact spatial representation with an explicit accuracy--efficiency trade-off. Code is available at https://github.com/wenhuahuo/APPSolver .

Figures & tables

Figure 1: Overview of APPSolver on a fixed non-uniform 2D reference slice. APP applies quadtree partitioning, distance-based refinement, padding, and downsampling to construct patch tokens. A condition encoder optionally maps hull geometry and operating parameters into a prefix token. The Transformer predicts downsampled patch outputs, which are recovered to the full reference point set with distance-weighted $k$ -nearest-neighbor interpolation.

	DTC		KCS		KVLCC2
Model	MAE	Rel. $L_{2}$	MAE	Rel. $L_{2}$	MAE	Rel. $L_{2}$
APP-Transformer	.09254 $\pm$ .00016	.22406 $\pm$ .00046	.12910 $\pm$ .00125	.26102 $\pm$ .00163	.04600 $\pm$ .00085	.96909 $\pm$ .00881
APP-DPT	.16703 $\pm$ .00221	.32745 $\pm$ .00240	.21707 $\pm$ .00377	.38297 $\pm$ .00585	.05513 $\pm$ .00107	.98120 $\pm$ .00486
Transolver	.04512 $\pm$ .00077	.18113 $\pm$ .00043	.08942 $\pm$ .00154	.22394 $\pm$ .00269	.03507 $\pm$ .00201	.93869 $\pm$ .03573
UPT	.04405 $\pm$ .00093	.18467 $\pm$ .00074	.08861 $\pm$ .00115	.22546 $\pm$ .00042	.03894 $\pm$ .00320	.95292 $\pm$ .00995
GNOT	.14898 $\pm$ .00111	.31130 $\pm$ .00035	.20378 $\pm$ .00628	.37454 $\pm$ .00730	.07015 $\pm$ .00265	.98318 $\pm$ .00175

Table 1: Corrected ShipBench comparison. Each hull reports normalized MAE and relative

L_{2}

; learning models are mean

\pm

sample standard deviation over three model seeds. Persistence is deterministic. Best values in each metric are bold.

Hull	Overall	$u$	$v$	$w$	$p_{\mathrm{rgh}}$
DTC	.09254 $\pm$ .00016 / .03753	.06094 $\pm$ .00136 / .02072	.09679 $\pm$ .00069 / .04346	.09394 $\pm$ .00074 / .03798	.11849 $\pm$ .00076 / .04795
KCS	.12910 $\pm$ .00125 / .07707	.06667 $\pm$ .00040 / .02950	.11234 $\pm$ .00008 / .06080	.10171 $\pm$ .00098 / .05625	.23567 $\pm$ .00453 / .16174
KVLCC2	.04600 $\pm$ .00085 / .03196	.05351 $\pm$ .00239 / .03288	.04762 $\pm$ .00244 / .02923	.05403 $\pm$ .00131 / .03330	.02883 $\pm$ .00054 / .03241

Table 2: APP-Transformer versus persistence normalized MAE by channel. Cells show APP mean

\pm

standard deviation / persistence. Lower values are bold.

Model	Cavity	Tube	Dam	Cylinder
APP-Transformer	.05659 $\pm$ .00166	.04915 $\pm$ .00291	.07452 $\pm$ .00350	.08088 $\pm$ .00129
APP-DPT	.12221 $\pm$ .00555	.12593 $\pm$ .00083	.16252 $\pm$ .00232	.18833 $\pm$ .00718
Transolver	.01188 $\pm$ .00336	.02188 $\pm$ .00281	.02968 $\pm$ .00253	.01880 $\pm$ .00273
UPT	.00827 $\pm$ .00249	.02005 $\pm$ .00194	.03406 $\pm$ .00410	.02029 $\pm$ .00482
GNOT	.09395 $\pm$ .00392	.25438 $\pm$ .00169	.14753 $\pm$ .01018	.16611 $\pm$ .00489
FNO	.07837 $\pm$ .00172	.17955 $\pm$ .00867	.33922 $\pm$ .00635	.24454 $\pm$ .01226

Table 3: Normalized MAE on CFDBench, reported as mean

\pm

sample standard deviation over three seeds. Best values are bold.

Figure 2: Illustrative APP-Transformer prediction on a DTC 1Re sample. Rows correspond to $u$ , $v$ , $w$ , and $p_{\mathrm{rgh}}$ ; the inset identifies the locally refined near-hull region. Quantitative conclusions are based on full-reference metrics rather than apparent error intensity in this rendering.

Strategy	DTC	KCS	KVLCC2
Adaptive	.09254 $\pm$ .00016	.12910 $\pm$ .00125	.04600 $\pm$ .00085
Uniform	.12431 $\pm$ .00152	.16564 $\pm$ .00085	.04890 $\pm$ .00184
Uniform + learned	.12324 $\pm$ .00026	.16086 $\pm$ .00022	.05502 $\pm$ .00046

Table 4: Production APP-Transformer partition ablation. Values are normalized MAE, mean

\pm

sample standard deviation over three seeds.

Hull	Best Transolver	Best UPT	APP-Transformer
DTC	32 slices: .04512 $\pm$ .00077	64 tokens: .04395 $\pm$ .00065	.09254 $\pm$ .00016
KCS	8 slices: .08855 $\pm$ .00115	8 tokens: .08804 $\pm$ .00118	.12910 $\pm$ .00125
KVLCC2	4 slices: .03458 $\pm$ .00160	8 tokens: .03496 $\pm$ .00229	.04600 $\pm$ .00085

Table 5: Best token-budget setting and normalized MAE for Transolver and UPT, compared with APP-Transformer.

Scenario	Zero	Numeric MLP	Fourier MLP	FiLM	Qwen2.5	Qwen3.5
Full ShipBench	.03979	.04048	.04133	.04036	.04069	.04128
Leave out DTC	.02690	.02623	.02859	.17942	.02717	.02609
Leave out KCS	.04395	.04779	.04273	.23264	.04262	.04199
Leave out KVLCC2	7.84960	8.24206	8.00103	8.50888	7.89234	7.77022

Table 6: APP-Transformer condition-encoder ablation. Entries are mean normalized MAE over seeds 42, 43, and 44.

Hull	$h=1$	$h=10$	$h=20$	$h=50$	Cumulative ratio
DTC	.08295/.02751	.18484/.07201	.29028/.09217	1.33241/.10816	6.01 $\times$
KCS	.11286/.05944	.22500/.09099	.33339/.09919	.76954/.13835	3.81 $\times$
KVLCC2	.01679/.00378	.05459/.00856	.06893/.01004	.07087/.01173	6.44 $\times$

Table 7: APP-Transformer / persistence normalized MAE during a 50-step autoregressive rollout. The cumulative ratio is cumulative APP MAE divided by cumulative persistence MAE through

h=50

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Model	Params (M)	FLOPs (G)	GPU median (ms)
APP-Transformer (Ours)	0.930	1.815	1.309
APP-DPT (Ours)	0.939	2.047	2.329
Transolver	0.989	53.644	24.767
FNO	0.990	7.482	3.785
Fusion-DeepONet	0.997	23.598	7.751
UPT	1.000	18.936	6.112

Appendix

Table 8: Model-forward profiling on a representative DTC input at batch size 1 on an NVIDIA RTX A6000. GPU latency is the median of 200 CUDA-Event measurements after 50 warm-up iterations. Timing uses final checkpoints; accuracy tables use best-validation checkpoints.

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