Neural Latent Arbitrary Lagrangian-Eulerian Grids for Fluid-Solid Interaction

Shilong Tao; Zhe Feng; Shaohan Chen; Weichen Zhang; Zhanxing Zhu; Yunhuai Liu

cs.LGFeb 28, 2026

Neural Latent Arbitrary Lagrangian-Eulerian Grids for Fluid-Solid Interaction

Authors: Shilong Tao, Zhe Feng, Shaohan Chen, Weichen Zhang, Zhanxing Zhu, Yunhuai Liu

Organizations: School of Computer Science, Peking University · Global Innovation Exchange, Tsinghua University · School of Electrical and Computer Science, University of Southampton

Abstract

Fluid-solid interaction (FSI) problems are fundamental in many scientific and engineering applications, yet effectively capturing the highly nonlinear two-way interactions remains a significant challenge. Most existing deep learning methods are limited to simplified one-way FSI scenarios, often assuming rigid and static solid to reduce complexity. Even in two-way setups, prevailing approaches struggle to capture dynamic, heterogeneous interactions due to the lack of cross-domain awareness. In this paper, we introduce Fisale, a data-driven framework for handling complex two-way FSI problems. It is inspired by classical numerical methods, namely the Arbitrary Lagrangian-Eulerian (ALE) method and the partitioned coupling algorithm. Fisale explicitly models the coupling interface as a distinct component and leverages multiscale latent ALE grids to provide unified, geometry-aware embeddings across domains. A partitioned coupling module (PCM) further decomposes the problem into structured substeps, enabling progressive modeling of nonlinear interdependencies. Compared to existing models, Fisale introduces a more flexible framework that iteratively handles complex dynamics of solid, fluid and their coupling interface on a unified representation, and enables scalable learning of complex two-way FSI behaviors. Experimentally, Fisale excels in three reality-related challenging FSI scenarios, covering 2D, 3D and various tasks. The code is available at \href{https://github.com/therontau0054/Fisale}.

Figures & tables

Figure 1: Fluid-solid interaction scenarios. (a.1) and (a.2) depict flow-around-body scenarios; (b.1) and (b.2) focus on aerodynamic analysis of wings; (c) illustrates the periodic dynamics of a venous valve; (a.1) and (b.1) represent one-way FSI cases, while the others involve two-way FSI.

Figure 2: The overview of Fisale. (a) describes the overall structure of Fisale; (b) depicts the pipeline of the processor; (c) shows the pipeline of the latent ALE grid initialization.

Dataset	Task	#Dim	#Mesh	#Input	#Output	#Split
Structure Oscillation	Single-Step Prediction	2D	1317	Solid: Geometry; Fluid: Geometry, Pressure, Velocity	Solid: Geometry; Fluid: Geometry, Pressure, Velocity	9561 1195 1196
Venous Valve	Autoregressive Simulation	2D	1693	Solid: Geometry, Stress; Fluid: Geometry, Pressure, Velocity	Solid: Geometry, Stress; Fluid: Geometry, Pressure, Velocity	720 90 90
Flexible Wing	Steady-State Inference	3D	37441	Solid: Geometry, Material; Fluid: Geometry, Attack Angle, Velocity	Solid: Geometry, Stress; Fluid: Geometry, Pressure, Velocity	1036 129 131

Table 1: Summary of experiment dataset. #Mesh records the average size of discretized meshes. #Split is organized as the number of samples in training, evaluation and test sets.

	Solid	Fluid	Interface	Mean ( $\downarrow$ )
Geo-FNO	0.0003	0.0387	0.0074	0.0155
GINO	0.0021	0.2536	0.0269	0.0942
LSM	0.0007	0.1951	0.0068	0.0675
CoDANO	0.0005	0.0703	0.0075	0.0261
LNO	0.0006	0.0244	0.0061	0.0104
Galerkin	0.0012	0.0507	0.0114	0.0211

Table 2: Performance on Structure Oscillation. Relative L2 is recorded. Second-best performance is underlined.

Figure 3: Local visualization of prediction results on solid displacement and fluid $x$ -velocity. Red circle indicates the domain with most solid displacement and sharp fluid velocity change.

	Relative L2 ( $\downarrow$ )
Geo-FNO	0.0730
LNO	0.0715
GNOT	0.0889
Transolver	0.0722
MGN	0.0742
AMG	0.0696

Table 3: Results of OOD test.

	Solid		Fluid		Interface
	Geometry	Stress	Pressure	Velocity ( $x$ )	Geometry	Stress	Pressure
Geo-FNO	0.3687	3252.51	124.27	0.1304	0.3948	5471.54	110.35
LSM	0.4788	4166.96	145.03	0.1419	0.4635	6547.88	122.53
CoDANO	0.6843	4385.24	171.57	0.1713	0.7806	6843.06	143.65
Galerkin	0.3471	3226.86	109.52	0.1025	0.3213	5093.68	113.34
GNOT	0.3833	4207.17	100.59	0.1147	0.3679	5384.59	128.57

Table 4: Performance on Venous Valve. We record RMSE-all (

\downarrow

), the average RMSE of the whole rollout trajectory and all samples. Results of other physical quantities are listed in Table 20 .

Figure 4: Visualization of ground truth and prediction results. The red circle indicates the distortion of solid shape, where Fisale can effectively handle.

Panel of Table 5 — Table 5: Performance on Flexible Wing task. Relative L2 is recorded. Second-best performance is underlined. (a) the relative L2 error of the prediction results; (b) the visualization of prediction errors.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The physical domain of venous valve simulation model.

Leaflet Thickness ( $mm$ )	Inlet Blood Velocity ( $m/s$ )	Valve Material
		$C_{1}$ ( $MPa$ )	$C_{2}$ ( $MPa$ )
range(0.5,1.0,0.1)	range(0.1,0.6,0.1)	{0.01,0.05,0.1,0.15,0.2}	{0.001,0.005,0.01,0.02,0.05}

Appendix

Table 6: Parameter settings of venous valve simulation model.

Figure 6: The physical domain of flexible wing simulation model.

Thickness	Taper Ratio	Sweep Angle ( ∘ )	Free-stream Velocity ( $m/s$ )	Attack Angle ( ∘ )	Wing Material
{10,12,14,16}	{0.5,0.6,0.7}	{10,25,40}	{50,75,100}	{-12,10,14,18}	Carbon Fibers Titanium Alloy Al-Zn-Mg Alloy

Appendix

Table 7: Parameter settings of flexible wing.

Material Name	Density ( $kg/m^{3}$ )	Young’s Modulus ( $MPa$ )	Poisson’s Ratio
Carbon Fibers	$2.30\times 10^{-6}$	$2.30\times 10^{5}$	0.210
Alpha-Beta Titanium Alloy	$4.43\times 10^{-6}$	$1.13\times 10^{5}$	0.342
Al-Zn-Mg Alloy	$2.83\times 10^{-6}$	$7.20\times 10^{4}$	0.327

Appendix

Table 8: Material parameters of wings.

Datasets	Training Configuration			Model Configuration
Datasets	Epochs	LR	Batch	Level ( $L$ )	Pathway ( $H$ )	Grid Shape ( $M$ )	Channels ( $D$ )
Structure Oscillation	100	$1\times 10^{-3}$	50	2	2	$[16,16]$ $[8,8]$	$[64,64]$
Venous Valve	100	$1\times 10^{-3}$	50	2	2	$[16,16]$ $[8,8]$	$[64,64]$
Flexible Wing	100	$5\times 10^{-4}$	1	3	2	$[5,5,5]$ $[4,4,4]$	$[96,128]$

Appendix

Table 9: Training and Model Configurations of Fisale. The definition of batch size differs between autoregressive simulation task and other two tasks. For single-step prediction and steady-state inference tasks, the batch size refers to the number of samples in a batch. For autoregressive simulation, only one sample is processed during each forward and backward pass, and the batch size corresponds to the number of time steps in the sample. Since GPU memory usage varies across different models, the batch sizes of baseline models in the autoregressive simulation task are dynamically adjusted to avoid GPU memory overflow while maintaining performance.

	Structure Oscillation	Venous Valve	Flexible Wing
Geo-FNO	1.60 M	1.60 M	5.22 M
GINO	1.72 M	/	5.17 M
LSM	2.23 M	2.23 M	/
CoDANO	1.83 M	1.83 M	5.01 M
LNO	1.62 M	/	5.34 M
Galerkin	1.74 M	1.74 M	3.18 M

Appendix

Table 10: Parameter count of baseline models and Fisale.

	Solid ( $\times 10^{-3}$ )	Fluid ( $\times 10^{-2}$ )	Interface ( $\times 10^{-2}$ )
Geo-FNO	$\pm 0.01$	$\pm 0.05$	$\pm 0.01$
GINO	$\pm 0.50$	$\pm 2.02$	$\pm 0.78$
LSM	$\pm 0.01$	$\pm 0.72$	$\pm 0.02$
CoDANO	$\pm 0.07$	$\pm 0.05$	$\pm 0.07$
LNO	$\pm 0.03$	$\pm 0.29$	$\pm 0.12$
Galerkin	$\pm 0.08$	$\pm 0.03$	$\pm 0.02$

Appendix

Table 11: Standard deviations on Structure Ocsillation experiment.

	Solid		Fluid		Interface
	Geometry ( $\times 10^{-2}$ )	Stress ( $\times 10^{0}$ )	Pressure ( $\times 10^{0}$ )	Velocity ( $x$ ) ( $\times 10^{-2}$ )	Geometry ( $\times 10^{-2}$ )	Stress ( $\times 10^{0}$ )	Pressure ( $\times 10^{0}$ )
Geo-FNO	$\pm 3.88$	$\pm 319.62$	$\pm 12.61$	$\pm 1.69$	$\pm 4.19$	$\pm 483.77$	$\pm 11.81$
LSM	$\pm 4.26$	$\pm 216.70$	$\pm 13.08$	$\pm 2.12$	$\pm 4.56$	$\pm 286.48$	$\pm 11.13$
CoDANO	$\pm 6.67$	$\pm 552.81$	$\pm 18.19$	$\pm 2.87$	$\pm 6.30$	$\pm 603.29$	$\pm 16.16$
Galerkin	$\pm 0.71$	$\pm 79.86$	$\pm 4.24$	$\pm 0.86$	$\pm 1.02$	$\pm 107.30$	$\pm 2.62$
GNOT	$\pm 1.66$	$\pm 258.56$	$\pm 8.69$	$\pm 1.04$	$\pm 1.89$	$\pm 236.67$	$\pm 3.87$

Appendix

Table 12: Standard deviations on Venous Valve experiment.

	Solid ( $\times 10^{-2}$ )	Fluid ( $\times 10^{-2}$ )	Interface ( $\times 10^{-2}$ )
Geo-FNO	$\pm 0.01$	$\pm 0.04$	$\pm 0.24$
CoDANO	$\pm 0.18$	$\pm 0.15$	$\pm 0.27$
LNO	$\pm 0.02$	$\pm 0.02$	$\pm 0.04$
Galerkin	$\pm 0.09$	$\pm 0.19$	$\pm 0.25$
GNOT	$\pm 0.02$	$\pm 0.04$	$\pm 0.01$
Transolver	$\pm 0.01$	$\pm 0.04$	$\pm 0.03$

Appendix

Table 13: Standard deviations on Flexible Wing experiment. Some baselines are not included due to severe degradation, where standard deviations are too unstable to provide meaningful reference.

	Parameters (M)	Running Time (S)	GPU Memory (GiB)	Mean Relative L2 ( $\downarrow$ )
Geo-FNO	5.22	71.97	0.66	0.0524
GINO	5.17	223.74	22.95	0.4745
CoDANO	5.01	1167.47	17.48	0.1096
LNO	5.34	50.4	1.60	0.0235
Galerkin	3.18	129.40	4.15	0.0577
GNOT	4.64	246.42	13.04	0.0289

Appendix

Table 14: The efficiency comparison on Flexible Wing dataset with more than 35,000 mesh points for each sample in average. The Running Time is measured by the time to complete one epoch. Since the number of mesh points varies across samples, we report the peak GPU memory usage observed within a single epoch. Additionally, the recorded runtime includes the time spent on constructing graph edges. As a result, models like HOOD ( Grigorev et al., 2023 ) and AMG ( Li et al., 2025 ) , which involve multiple rounds of dynamic edge construction during iteration steps, become particularly time-consuming.

	Solid	Fluid	Interface	Mean Relative L2 ( $\downarrow$ )	Decrease
Fisale	0.0042	0.0155	0.0211	0.0136	-
$w/o$ explicit interface	0.0061	0.0212	0.0251	0.0175	28.68%

Appendix

Table 15: Ablation study of the explicit interface component.

Ordering	Solid	Fluid	Interface	Mean Relative L2 ( $\downarrow$ )
fluid-grid-solid-interface	0.0043	0.0155	0.0206	0.0135
grid-solid-fluid-interface	0.0042	0.0155	0.0206	0.0134
grid-solid-interface-fluid	0.0041	0.0157	0.0216	0.0138
grid-interface-solid-fluid	0.0041	0.0155	0.0212	0.0136
solid-fluid-interface-grid	0.0044	0.0161	0.0213	0.0139
solid-grid-fluid-interface	0.0042	0.0155	0.0211	0.0136

Appendix

Table 16: Ablation study of the update ordering within PCM.

	Params (M)	Solid	Fluid	Interface	Mean Relative L2 ( $\downarrow$ )	Decrease
Fisale	4.88	0.0042	0.0155	0.0211	0.0136	-
Simpler Module	4.96	0.0045	0.0178	0.0223	0.0149	9.56%

Appendix

Table 17: Ablation study of replacing PCM with a simpler attention module.

$M^{1/3}$	Params (M)	Solid	Fluid	Interface	Mean Relative L2 ( $\downarrow$ )
3	2.48	0.0047	0.0171	0.0226	0.0148
5	2.63	0.0047	0.0169	0.0224	0.0147
7	3.66	0.0044	0.0171	0.0221	0.0145
8	5.11	0.0045	0.0172	0.0223	0.0147
9	7.81	0.0046	0.0167	0.0219	0.0144
10	12.50	0.0049	0.0172	0.0227	0.0149

Appendix

Table 18: Scaling results of mesh resolution

M

with a single pathway.

$H$	$M^{1/3}$	Channels	Params (M)
1	[5]	[128]	2.63
2	[4, 5]	[128, 96]	4.88
3	[4, 5, 6]	[128, 96, 64]	6.99
4	[3, 4, 5, 6]	[128, 128, 96, 64]	11.99
5	[3, 4, 5, 6, 7]	[128, 128, 96, 64, 64]	15.69

Appendix

Table 19: Scaling settings of pathways

H

Panel of Figure 7 — Figure 7: Scaling results of pathways $H$ .

Figure 8: Showcase comparison of structure oscillation task. The solid deformation and fluid velocity are shown.

Figure 9: Showcase comparison of structure oscillation task. The solid deformation and fluid pressure are shown.

	Fluid		Interface
	Geometry	Velocity ( $y$ )	Velocity ( $x$ )	Velocity ( $y$ )
Geo-FNO	0.2850	0.0225	0.0780	0.0174
LSM	0.4022	0.0323	0.0825	0.0196
CoDANO	0.6203	0.0378	0.1042	0.0265
Galerkin	0.2811	0.0195	0.0658	0.0130
GNOT	0.2971	0.0182	0.0891	0.0157

Appendix

Table 20: Performance comparison on Venous Valve. Supplemetary results of Table 4 . We record RMSE-all, the average RMSE of the whole rollout trajectory and all samples. A smaller value indicates better performance.

Figure 10: Showcase comparison of venous valve task. The solid deformation and fluid pressure are shown.

Figure 11: Showcase comparison of venous valve task. The solid stress and fluid velocity are shown.

Figure 12: Showcase comparison of venous valve task. The wing deformation, wing stress and surface pressure are shown.

Figure 13: Attention pattern in last layer PCM modules of Fisale. The top row corresponds to the low-resolution latent ALE grid, and the bottom row corresponds to the high-resolution latent ALE grid. Regions with higher activation intensity are highlighted with red boxes. (a.1) and (a.2) represent the updating of the solid component; (b.1) and (b.2) represent the updating of the fluid component; (c.1) and (c.2) represent the interaction on the interface component.

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PDE Surrogate Modeling Graph Neural Operators Fluid-Structure Interaction Physics-Based Simulation Neural Operators Computational Fluid Dynamics

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We propose an arbitrary Lagrangian-Eulerian (ALE)-consistent machine learning framework for long-term fluid-structure interaction (FSI) prediction on deforming unstructured meshes. Specifically, the fluid dynamics are modeled by a surrogate that combines a graph neural operator (GNO) with a vision Transformer (ViT) for spatiotemporal prediction, while a lightweight long short-term memory (LSTM) network predicts structural kinematics at the interface. The two surrogates are coupled through a standard partitioned procedure. Most importantly, kinematic compatibility at the moving interface is enforced via an ALE-consistent boundary-correction step that updates the fluid-side interface velocity with the predicted structural velocity at each coupling update, thereby improving near-interface accuracy and long-term rollout stability. To mitigate autoregressive error accumulation, a two-stage training strategy is adopted, consisting of single-step supervised pretraining followed by long-term autoregressive fine-tuning. The proposed framework is validated on the benchmark problem of a flexible beam vibration in the wake of a cylinder. Results demonstrate accurate phase-consistent predictions over long rollouts and robust generalization under inlet-profile variations in both interpolation and extrapolation settings. Systematic ablation studies further assess the respective contributions of the ViT module, ALE-consistent boundary correction, and long-term training to predictive accuracy and rollout robustness.

Abstract

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