Neural Latent Arbitrary Lagrangian-Eulerian Grids for Fluid-Solid Interaction
Authors: Shilong Tao
Zhe Feng
Shaohan Chen
Weichen Zhang
Zhanxing Zhu
Yunhuai Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Deep learning
Neural Operator
Primary area: learning on graphs and other geometries & topologies
TL;DR: We introduce a data-driven framework to handle the complex two-way fluid–solid interaction problems.

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 https://github.com/therontau0054/Fisale.

Published as a conference paper at ICLR 2026
NEURAL LATENT ARBITRARY LAGRANGIAN- EULERIAN
GRIDS FOR FLUID-SOLID INTERACTION
Shilong Tao', Zhe Feng', Shaohan Chen', Weichen Zhang
Zhanxing Zhu",; Yunhuai Liu
1 Schoool of Computer Science, Peking University
Global Innovation Exchange, Tsinghua University
3 School of Electrical and Computer Science, University of Southampton
ABSTRACT
Fluid-solid interaction (FSI) problems are fundamental in many scientific and engineering applications, yer effectively capturing the highly nonlinear two-way interactions remains 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 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 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 way FSI behaviors. Experimentally, Fisale excels in three reality-related challenging FSI scenarios, covering 2D, 3D and various tasks. The code is available at httpttts://// htttt:://gtthb..oom wwtp://twww..c.eeeeettttttueeeee
distinct
INTRODUCTION
Fluid-Solid Interaction (FSI) refers complex coupled phenomenon in which solids undergo motion or deformation under the action of surrounding flowing fluid, and reciprocally alter the pressure and velocity distribution within the fluid (Hou al. 2012). This kind of problems ubiquitous in real-world scenarios, spanning a wide range of practical applications. From biomedical engincering, such blood flow interacting with vessel valves (Bazigou & Makinen 2013 Enderle
& Bronzino 2012). to aerospace and civil engineering involving structural responses to fluid forces 201T). FSI plays a pivotal role in both analysis and design.
(Prasad & Wanhill 2017 Zhang
FSI problems encompass the interplay between fluid flow, solid deformation, and their intricate coupling dynamics. These interactions are typically governed by a tightly coupled system of partial differential equations (PDEs) (Belytschko) 1980). Accurately and efficiently solving them is crucial for practical applications (Kopriva] 20099; Roubíčck 2013). However, due to nonlinear materials, moving interfaces, and strong coupling relations, analytic solutions to these PDEs are typically intractable (Xing 2019 Génevaux et al. 2003). Thus, these FSI problems are generally discretized into meshes and solved using numerical methods such the Immersed Boundary Method (IBM) (Peskin 2002) and Arbitrary Seerrangian-Euleriiin (ALE) (Hirt et al.) 1974) method. These approaches handle the coupled system either monolithically (Heil al. 2008), treating fluid and solid unified domain, or through partitioned iteration (Degroote et al. 2009). where each subdomain is solved separately with interface data exchanged iteratively. While effective in many cases, both approaches
"Correspondence: yunhuai. liu @ @pw..edu.unnnnn.. 7.zhu @soton.ac.uk

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Published as a conference paper at ICLR 2026
partially open
wind
(a.1) cylinder flow
(b.1) rigid wing
Flexible
fully open
wind
defornation
(a.2) structure oscillation
(b.2) flexible wing
(c) venous valve open
Figure I: Fluid-solid interaction scenarios. (a.1) and (a.2) depict flow-around- body scenarios; (b.1) and (b.2) focus aerodynamic analysis of wings; (c) illustrates the periodic dynamies of venous valve; (a. 1) and (b.1) represent one-way FSI cases, while the others involve two-way FSI.
are computationally expensive (Umetani & Bickel 2018), strongly mesh-dependent Berzins Burkhart et al. 2013). and facing stability issues (Grétarsson al. 2011 Paidoussis 2005).
1999
Recently, deep learning has emerged as a powerful tool for solving PDEs (Li et al. 2021 Lu ct al. 202T). thanks its strong capacity to effectively capture nonlinear input-output mappings. Once trained, it can offer significantly faster inference than traditional solvers (Han et al. 2018 Tao et al. 6025a|b][aan et al. (0266). However, the context of FSI, most existing models primarily focus on one-way FSI scenarios, where the solid is typically treated static and rigid. For example, many studies have explored airfoil design tasks (Bonnet et al.) 2022 Valencia et al. (025). As shown in big.re([b.l), the aircraft wing is typically modeled static rigid body. This allows the solver to treat the wing region as a fixed, undeformable inner boundary and focus solely on the fluid domain, significantly reducing the complexity. However, shown in Figure b..2), real wings often made of flexible materials and may undergo noticeable deformation under aerodynamic loads (Aono et al., 2010; Eppler 2012). In such cases, the fluid-solid interface becomes dynamic, and the coupling relation grows more complex. Similar assumptions also appear in other benchmarks like cylinder flow (Pfaff et al. 2020 Li et al.) (2025) and car design tasks (Elrefaie al. 2024a b), where structural flexibility is ignored. Thus, how to effectively handle the evolution of fluid and solid, and capture their dynamic coupling interactions remain unanswered when learning two-way FSI problems.
Regarding existing two-way cases, GNN-based models have simulated several simple rigid motion scenarios Gonzalez et al. 2020 Liet al. 2019). While message passing underpins GNNs, it is inherently stateless and undifferentiated, making it difficult to distinguish inter- and intra-domain information in complex deformation scenarios (Hou al., 2019). Moreover, its local reception also falls short in global modeling (Li et al. (202d). A closely related work to ours is CoDA-NO (Rahman jt al. 2024), a multiphysics neural operator that tackles a classic two-way FSI problem, structure oscillation (Figure [a)2)) (Turek & Hron 2006). by partitioning the input domain along physical variable channels and learning the global mapping through podomain-wise attention. However, this variable-wise strategy maintains monolithic view and lacks explicit handling of the dynamic fluid-solid interface caused by structural deformation. More broadly, most neural operators struggle to simultaneously learn the distinct behaviors and bidirectional dependencies of fluid and solid domains under such monolithic modeling approaches. As result, two-way FSI still remains underexplored.
To effectively capture the evolution of solid and fluid states and their complex interactions, we propose Fisale, a purely data-driven framework inspired by classical numerical methods: ALE and partitioned coupling algorithm. ALE provides a unified representation for cross domains via mesh motion, while partitioned strategies decouple fluid and solid domains for separate processing and reduced nonlinearity. our design, recognizing the importance of the coupling interface, we explicitly model it as separate component on par with the solid and fluid. This enables the model to better capture the coupled dynamics. We then introduce multiscale latent ALE grids, onto which fluid, solid, and interface states are interpolated. These grids serve as unified multi-physics embeddings that encode spatial and physical quantities of the FSI system. Eventually, instead of monolithic updates, we introduce Partitioned Coupling Module (PCM) that mirrors the logic of classical partitioned solvers. By breaking the nonlinear problem into sequential substeps, it progressively captures evolutions and complex interdependencies through deep iteration. Our contributions are summarized as follows:
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We propose to solve two-way FSI problems by separately handling the evolution of different physical domains. In particular, we treat the coupling interface as an individual component, on par with fluid and solid, enabling a more effective capture of cross-domain interactions.
We propose Fisale, a purely data-driven framework inspired by ALE and partitioned coupling methods. It explicitly models fluid, solid, and their interface dynamics through multiscale latent ALE grids and stacked Partitioned Coupling Modules (PCM).
Fisale achieves consistent state-of arr across three reality-related challenging FSI tasks, particularly in scenarios involving large deformation and complex interaction.
Related Work Deep learning has recently gained traction in addressing scientific problems across various fields, including initial explorations into FSI problems. A common strategy is to hybridize traditional solvers with neural networks. For instance, in partitioned frameworks. deep learning models may replace either the fluid or solid solver (Xiao et al., 2024a; Zhu et al., 2019; Mazhar
ct al. 2023 Xu et al.] 2024, Liu al. 2024). while the other remains conventional. Alternatively, neura networks are integrated into solvers to accelerate costly steps like velocity estimation (Fan &
Wang 2024), interface force prediction (Zhang al.) 2022] Li al., 2023a) or control parameter
approximation (Takahashi et al. 2021). Different from hybrid, purely data-driven models bypass traditional solvers entirely and learn physical dynamics directly from data, sometimes with the guidance of physical priors. One prominent class is deep reduced-order models (ROMs) (Han jet al. 2022] |Ashwin et al.] (022)), which are built on the assumption that FSI dynamics evolve on a low-dimensional manifold (Lee et al. 2024). These methods use autoencoders (Zhai al. 2018)).
PCA (Maćkiewicz & Ratajczak] 1993). or POD (Berkooz et al. 1993) to reduce dimensionality, and
utilize neural networks to model the evolution in low-dimensional space efficiently. Another widely studied direction is dryssccsInffrmmn
Neural Networks (PINNs) (Raissi et al.] 2019). which embed the governing equations of the FSI problems directly into the loss function (Wang al. [021] Cheng et al. 2021 Chenaud et al. 2024). enabling the model to learn solutions that satisfy physical laws and serve as an instance- specific solver after trained. To address generalization across geometries and conditions, neural operators (Boullé & Townsend 2024) learn mappings between function spaces,
offering mesh-independent PDE solvers ( (Wu et al. 2024 Hao et al. 2023 Li et al. 2023d 2025). These methods have also been extended to multi-physics problems like FST (Rahman et al.) (024) by learning codomain- wise operator along physical variable channels. Complementarily, GNN-base
simulators leverage mesh (Plaff et al. 2020 Feng et al. (2266) or particle (Sanchez-Gonzalez et al.,
2020 Li et al. 2019) connectivity to learn local interactions and propagate across physical domains.
However, most of these studies remain limited one-way FSI scenarios, where the solid is assumed rigid and static, greatly simplifying the coupling dynamics. For the relatively few studies tackling two-way FSI, current methods, such GNN-based simulators (Pfaff et al., 2020) [Snncee--Gonzaler
et al. 2020 Li et al. 2019. Feng et al.] 2026) and multi-physics neural operators (Rahman et al. 2024 struggle to effectively capture coupling behavior due the undifferentiated and monolithic modeling. As result, effectively learning the evolution of deformable solids and surrounding fluid flows, and capturing their complex coupling relations remain an underexplored challenge.
2 PRELIMINARIES
Arbitrary Lagrangian-Euleriaan Method Lagrangian and Eulerian descriptions are two primary views for physical simulations. While the former tracks the motion trajectorics of material particles (Dym et al. 1973), the latter observes physical quantities fixed spatial locations (Morrison, 2013). Accordingly. y, solids are typically simulated using Lagrangian meshes that follow material deformation and motion, whereas fluids are often discretized using Dulerian grids accommodate complex flow behavior and topological changes, making their coupling both common in nature but difficult to simulate (Xie al.
2023 Axisa & Antunes 2006). The Arbitrary Lagrangian-EEllriann (ALE) method (Hirt et al. 1974) combines both descriptions and is widely used in FSI simulations (Takashi & Hughes| 1992 Donea et al. 1982). The core idea is defining a mesh velocity V. that moves independently of the material and tixed domain, enabling flexible mesh motion and improved stability. The mesh velocity v, is typically equal to the material velocity v in solid domains, while obtained by Laplacian smoothing (Field 1988) in fluid regions:
Vv
(1)
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Encoding
Attention-based PCM
Decoding
(ggregeticnn(FFM)
Quantities
Processor
Processor
Prros
pILd
L levels
Latent ALE
Quantities
Processor
Processor
(b) pipeline of processor
(ggregrtton(FFF)
Processor
Processor
Linear
offect for each
ALE Init
ALE Init
R
H pathways
regular gird seeding
geometry-aware offset
latent space projection
(a) overall model structure
(c) pipeline of latent ALE grid initialization
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.
where is a weighting coefficient and interface velocities serve as Dirichlet boundary conditions. This flexibility of mesh motion enables ALE to accommodate unified representations for heterogeneous domains, providing a strong foundation for learning-basee modeling of evolution and interaction.
Partitioned Coupling Algorithm The partitioned coupling algorithm is widely used in FSI simula- tions (ii et al. 2016, Degroote et al., 2008). By decoupling the fluid and solid fields, the coupled system is reformulated into smaller, typically better-conditioned subproblems that can be solved using standard solvers. In contrast to monolithic approaches, where a large, fully coupled nonlinear system solved simultaneously, the partitioned strategy reduces the dimension and complexity of the global system matrix and often yields improved numerical properties within each subdomain. A typical solution procedure (Plaezek et al. 2009) consists of: (1) solving the solid subproblem based on the current solid and fluid states; (2) updating the computational mesh via ALE mesh motion method to account for the solid deformation; (3) solving the fluid subproblem on the updated mesh and solid state; and (4) matching interface, repeating above procedures or advancing to the next step.
METHOD
Problem Setup Given bounded open domain D Rd and state space 24 C RV representing N physical quantities across fluid and solid fields (in Lagrangian, Eulerian, or hybrid form), we denote (g) E U and a+ E l as system states at time steps and + Al, evaluated at generalized spatial coordinate e Rd. The FSI prediction problem can be formulated as (g) J',, (g)) where Fg represents the learned mapping function. This basic single-step prediction be extended to several tasks like steady-state inference, discussed in Section
Notation For simplicity, we omit the time subscript and denote the fluid and solid observations at current time step u, i E xd 9, E BgNyxCy] and u, respectively. Here, g represents the generalized spatial coordinates and q denotes the associated physical quantities. As mentioned before, explicitly treat the interface as separate component and denote it as ig E N xd E NRN.C.. where Cj + C. C. and N₂ + N₂ + No, = N.
xd
NN qs E XC.)
Overall Framework As shown in Figure 2 and 2b, the pipeline of Fisale is formulated as:
Fo i O Brggrpps
(LAA) F.F.L.A)
(2)
PALIEINE
where o denotes operator composition. The full model Fo consists of H parallel latent pathways corresponding to different spatial scales. Each pathway begins with an individual ALE grid initial- ization [(h)] P which provides unified representation for different domains. Then each pathway proceeds through J. stacked processors. At the 1-th level of the h-th pathway, the processor consists
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of three sequential components: an encoding step from the original space to the latent ALE space F
a processing module F (a on the latent ALE grid, and a decoding step back to the rew
Onggetal I
original space F flt Le
To enable cross-scale communication, an aggregation module F(l)
is applied after each level 1, where features from all pathways are concatenated and passed through a Feed-Forward Network (FFN) aggregate information and produce updated features.
3.1 MULTISCALE LATENT ALE GRIDS
Instead of directly solving FSI in heterogeneous descriptions, we introduce a set of multiscale latent ALE grids, which provide unified representation for multiscale and cross-domain physics.
Latent ALE Grid Initiation The initiation of a single latent ALE grid includes two steps: regular grid seeding and geometry-aware offset. As shown in Figure we begin by initializing regular, axis-aligned Cartesian grid a E MRd over the d-dimensional Euclidean space [-3.5, 3.5 via uniform sampling. Notably, we normalize the input physical domain into a NN 1); according the 3-sigma rule (Huber 2018). the regular grid with interval 3-3.5, 3.5) covers more than 99.95 input mesh points. Here, the gri is flattened into sequence of length M, where M M1 x XM2..... x... X Ma denotes the total number of grid nodes, and each M2 represents the number of discretization points along the k-th spatial axis. This regular grid is treated as a reference grid, independent of any specific geometry. Its uniform structure enables the decoupling of spatial topology from geometric variation and facilitates geometry-aware deformation in the subsequent initialization stage (Li et al. 2023c).
To incorporate geometric awareness into the grid, we deform the regular reference grid by applying an offset field that reflects the spatial distribution of solid, fluid and coupling interface. We first compute direction vectors from each grid node a, to points in each domain. These vectors are fed into a linear layer and then weighted by a normalized radial basis kernel to prioritize closer points. For example, the offset contributed by the fluid geometry g, is given by:
(y(a:)= = Σ exp(Linear( ||a, gı, 113)
Ny
j=1 اj=1 exp(Linear( (aaincarr-|laa -,, 12)) (g/, - aa)
where the subtraction is implemented by broadcast. With this offise,, the weights assigned to fluid points are optimized based on distances, leading to geometry-aware aggregation. The total offset is obtained by summing contributions from all domains: A(a) (a) + A + Aa(a). Finally, the latent grid is obtained by applying the total geometry- offset followed by a linear projection: offset encourages grid nodes to move closer to regions
g. = Linear(a a + A(a)). This geometry- of geometric interest, such as fluid-solid interfaces, by aggregating directional influences through distance-weightel kernels. Moreover, the spatial decay of the kernel attenuates the influence of distant regions, leading to weaker and more uniform updates in non-critical regions, which helps preserve the grid's s smoothness and regularity. We then build edges by performing k-nearest neighbor (k-NN) search on the latent grid ga E R X D i.e., E = (g.) The edge set E defines a latent interaction graph over the deformed grid, which supports grid's 's update in subsequent modules.
Definition of Latent ALE Grid During the solving process, this grid is maintained and iteratively updated in latent space according to the learned dynamics. Crucially, it adheres to the principle of the ALE method, namely, evolving with a motion that is decoupled from both the frames of material points and spatial grids, thereby enabling an intermediate behavior between Lagrangian and Fulerian viewpoints Hiit et al. 1974). We therefore refer to this representation as the Latent ALE Grid.
Multiscale FSI phenomena are inherently multiscale (Steinhauser 2018). For example, a flexible wing interacting with airflow involves large-scale aerodynamic forces and small-scale local defor- mations. Capturing such behaviors requires the ability model and propagate information across different resolution levels. Fortunately. it is straightforward and natural in our framework to construct grids at multiple scales, simply by varying the number of grid nodes M during initialization. By operating in parallel, each grid records physical context at a different level of detail. This allows coarse grids to efficiently model global structures, while fine grids focus on local interactions, together forming hierarchical representation well-suited for multiscale FSI problems.
Supplementary Notation
We define x (1.M) e XD to be the output feature of the I-th level 0.h x corresponds to the input emhedding of the observation uy, i.e.,
in the h-th pathway. The
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..A Linear(uy)) The solid and interface adhere the same manner. We denote g as the state of x the latent ALE grid at the 1-th level of the h-th pathway. Since the encoding, decoding, and PCM operations are shared across levels and pathways, we omit and in the following descriptions.
3.2 PHYSICAL QUANTITIES ENCODING AND DECODING
Before each coupling process, we project physical quantities onto the grid to enable unification across heterogeneous regions. After coupling learning, we project them back for multiscale aggregation. Both processes applied by weighted interpolation.
Physical Quantities Encoding For each domain, we compute the projection weight through an attention-like manner. The interpolation weight for fluid domain w MxN, is defined as:
f QK" where Q = Linear(ga) and K = Linear(xy)
Then the fluid projection Py E M is conducted by: P, = Softmax(wy)xx-
Solid and coupling interface domains follow the same procedure with projection weights w, and Ws, projected features P, and Ps. respectively. As a result, we extend the latent ALE grid ga as (ga; Ps, Py, p. This latent tuple encodes both the grid's geometry-aware position and the surround- ing physical context, serving as the input to subsequent modules. Unlike traditional discretizations that partition physical domains rigidly, ously at each latent node. This allows the model to naturally capture cross- interactions and dynamics near interfaces, forming a flexible and expressive multi-physic embedding for learning.
represent fluid, solid, and interface features simultane-
Physical Quantities Decoding Let (&a. P.. py. p PCM(1ga, Ps. Py, }) denote the updated features. The decoding is similar to the encoding as: & Softmax (w; )os-
Here, the direction of Softmax(-) is different from the encoding to keep the sum of the interpolation weight equal 1. To fuse features in different scales, we concatenate the decoding feature from different pathways and adopt a Feed- werd-Fo Network (FFN) for aggregation. Then the fused features are split and taken back to their own pathways.
3.3 PARTITIONED COUPLING MODULE
We design Partitioned Coupling Module (PCM) that follows the process of partitioned coupling algorithm learn the evolution of fluid, solid and their complex coupling an iteratively deep manner. Each PCM includes four forward steps outlined in Section
Update Solid State We first employ the cross-attention mechanism to update the solid state. Attention mechanisms (Vaswani al., (017)) are well-suited for modeling PDE-related physical systems due to their strong capacity to capture nonlinear dependencies, long-range interactions, and perform spatial
aggregation across irregular domains (Hao et al. 2023 Wu et al. 2023 Li et al. 2023b). Recent
studies have further shown that attention itself can be interpreted as an integral operator, capable of approximating complex mappings across function spaces (Cao 2021 Wu et al. 2024 Li et al.
2025). Moreover, the flattened sequence of the latent ALE grid aligns naturally with the input format required by attention mechanisms, making attention seamless and effective choice for updating physical states across fluid, solid, and interface domains. Given current embedding on the latent ALE grid (ga: Ps, Py, ps the update formulated as:
Q = Linear(Concat( P. + ta. Ds + Bg)), K, V Linear(Concat(g) + ga. Py + ga. Ps + g.))
p'. P/ = Chunk Q D-1)) where Q = Softmax (Q) and K Softmax(K)
The query Q is constructed by complete solid representation and the key K and value V include the information of the whole system. The geometry of the latent ALE grid ga serves as positional embedding that provides spatial prior. We concatenate the components along the length direction and adopt the linear attention which has been approved as a kind of neural operator (Cao 2021). Through cross-attention, each solid node selectively attends to the entire system, allowing it to update its state based on both internal structural cues and external influences from the fluid and interface. The Chunk(-) divides the output to recover the updated solid state p', and the interface state P/--

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Update Grid Coordinate We next adopt velocity-based Laplacian smoothing strategy, defined in [ to update the latent ALE grid coordinates in response to solid motion and deformation while preserving grid quality. go the spatial diffusion of mesh velocity Vg and can be interpreted as steady-state flux balance over the grid. Upon discretization, the divergence and gradient operators naturally translate into local neighbor interactions (Han et al. (023a): ΣjeN() (Vg.j j Vg.i) = 0. The velocity at each mesh point is updated based on weighted combination of its neighboring nodes:
gi ΣSEN() Viy
where (i) denotes the neighbors of the i-th node, and "iy is diffusion-like weight that reflects local mesh connectivity. Coincidentally, this update scheme is naturally aligned with the local message passing (Gilmer et al. 2017) used in graph-based models.
update the grid based on the neighbor relation (i) C E, which is built in the initialization of ALE. In latent space, treat [0, as learnable parameters, where ΣjEN(O) Vij 1, and update the mesh velocity as:
gi Ejex() Yi CiinearrCoonat(p.... Py, P))))))
Then the latent grid coordinates are explicitly updated by ga + ga + At Vg. In our stacked- module architecture, we set At = 1 grid quality, without grid distortion, we conduct geometry smoothing over the local neighborhood Sa, E))N()) BiyBa.j. where 3 E [0, 1] and Ejen(() Biz =
abstract timestep per module. Finally, to control the
Update Fluid State Similar the update of the solid state, update fluid state
Q Linear(Concat(p/ + Ba. Pb + 8a)), K, V = Linear(Conca((p'' + B's Py + B'a Pb + g.))
P'. P = Chunk (O(K"V-D")
Update Interface Influence Finally, we use self-attention mechanism to align the information across the solid, fluid, and their coupling interface regions
Q,K, V L.iear(CCnncat/i + +g.. ply + +g)),, P". Chunk (Q(KTV -D1))
The self-attention operation enables mutual interaction among the three domains, allowing the model to capture dependencies and reconcile inconsistencies across the solid-fluid interface by attending to relevant features globally. Finally, these four steps form single PCM and stack the modules to enhance the represent capacity of the model.
EXPERIMENTS
Table 1: Summary of experiment dataset. #Mesh records the average size of discretized meshes. #Split is organized as the number samples in training, evaluation and test sets.
Dataset
Task
#Dim
#Mesh
#Input
#Output
#Split
Solid: Geometry:
Solid: Geometry;
9561
Structure Oscillation
Single-Step Prediction
2D
1317
Fluid: Geometry,
Fluid: Geometry,
1195
Pressure, Velocity
Pressure, Velocity
1196
Solid: Geometry, Stress;
Solid: Geometry,
720
Venous Valve
Autoregressive Simulation
Stress;
2D
1693
90
Fluid: Geometry,
Fluid: Geometry,
90
Pressure, Velocity
Pressure, Velocity
Solid: Geometry, Material;
Solid: Geometry,
Stress;
1036
Flexible Wing
Steady-State Inference
3D
37441
Fluid: Geometry,
Fluid: Geometry,
129
Attack Angle,
Pressure,
131
Velocity
Velocity
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We evaluate Fisale three reality-related rellttdd challenging FSI scenarios. These scenarios have different dimensions, targets, scales, and complexity. Detailed benchmark information listed in Table !
Baselines We compare Fisale with over ten advanced learning-based solvers, including Neural
Operators: GeoFNO (Li ct al. 2023c), GINO (Li ct al. 2023d), CoDA-NO (Rahman et al. 2024).
LSM (Wu et al. 2023 LNO (Wagg& & Wang 2024 Transformers: Galerkin Transformer (Cao
2021), GNOT (Hao et al. 2023). ONO (Xiao et al.
2024b) Transolver (Wu et al. 2024); GNNs:
MGN Pfaff et
2020). HOOD (Grigorev al. 2023), AMG (Li et al. 2025).
Implementation and Metries Fisale and all baseline models are trained and evalu- ated under the same protocol. For fair comparison, we match the parameter count of each baseline to that of Fisale. For loss functions and evaluation metrics, we use Relative L2 error for the single-step predic- tion and steady-state inference tasks, and Root Mean Squared Error (RMSE) for the autoregressive simulation task. For experi- ments involving multiple physical domains, the loss term is calculated for each physical domain and combined with equal weights. All experiments are conducted on single NVIDIA RTX 3090 GPU (24 GB) and re- peated multiple times. We provide more
Table 2: Performance on Structure Oscillation. Relative L2 is recorded. Second- beoond-best performance is underlined.
Solid
Fluid
Interface
Mean (4)
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
GNOT
0.0006
0.0361
0.0076
0.0148
ONO
0.0012
0,0732
0,0126
0.0290
Transolver
0.0004
0.0265
0.0075
0.01
MGN
0.0007
0.0282
0.0112
0.0134
HOOD
0.0006
0.0277
0.0109
0.0131
details in Appendix and D
AMG
0.0004
0.0211
0.0051
0.0089
Fisale
0.0003
0.0148
0.0047
0.0066
4.1 STRUCTURE OSCILLATION
0_016
The structure oscillation prob- lem, also called "FLUSTRUK- A", is a famous benchmark for validating classical fluid-solid in- teraction (FSI) solvers (Turek & Hron 2006). It involves a thin elastic beam immersed in an in- compressible, dynamic fluid that develops self-sustained, material-dependent periodic oscillations. Unlike the widely studied Cylinder- Flow benchmark (Pfaff et al. 2020) which involves no structural flexibility, FLUSTRUK-A introduces strong two-way coupling and nonlinear deformation, making it more challenging to solve accurately.
0.002 Ground Truth
Fisale
AMG
Figure 3: Local visualization of prediction results on solid dis- placement and fluid z-velocity, Red circle indicates the domain with most solid displacement and sharp fluid velocity change.
The dataset is proposed in CoDA-NO (Rahman et al. 2024) with 1000 frames and different Reynolds number Re. Following the convention, we train Fisale and baselines learn the mapping from current u, to the next At- We set At as 4 frames let the oscillation evolve adequately. The results shown in Table 2 From the results, we can observe that Fisale achieves advanced ability, particularly in fluid domains. Compared with other baselines who model different domains in homogeneous way, Fisale can better capture the bidirectional interactions between fluid and solid at the interface and predict the motion and shape more accurately (the tail part of the structure as shown in Figure 3 This in turn enhances the accuracy of fluid predictions. The unified latent ALE representation, combined with iterative partitioned coupling, allows Fisale to progressively resolve the nonlinear interactions between fluid and solid, resulting in more accurate and stable predictions under strongly coupled two-way FSI.
Table 3: Results of OOD test.
Relative L2 (L)
Geo-FNO
0.0730
LNO
0,0715
Following the convention in CoDA-NO, we explore the out-of- distribution (OOD) performance of Fisale. In previous exper- iments, we trained models with data that contain the Reynolds number Re E {200, 400. 2000}. We directly test trained mod- els on data with Re E ( (4000). As presented in Table Fisale consistently performs best over strong baselines OOD sam-
GNOT
0.0889
Transolver
0.0722
MGN
0.0742
AMG
0.0696
Fisale
0.0637
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ples. This better generalization be attributed to its latent ALE representation, which captures flow patterns across multiple spatial scales, and the partitioned coupling design, which provides modular and robust way update interdependent physical states. Furthermore, the explicit modeling of the coupling interface enhances the model's ability to extrapolate dynamic interface interactions under stronger nonlinear coupling, which is prevalent in Rer-Reed----mbbeee regimes.
4.2
VENOUS VALVE
The venous valve problem models the opening and closing dynamics of valves in veins, which are essential maintain unidirectional blood flow in the circulatory system (Bazigou & Makinen 2013 Enderle & Bronzino, 2012). It involves thin, flexible leaflet interacting with a pulsatile, incompressible fluid under physiological conditions. The strong contact, large deformation, and highly transient behavior make this problem especially challenging for effective simulation (Buxton & Clarke 2006). We learn the transient dynamics through autoregressive simulation, which are mostly explored in GNN-based works (Plaff et al. 2020 Sanchez-Gonzalez al. 2020). Mathematically, we simulate the trajectory as: ú,+1 = Fe Fa(u)). u = (o(ū üs+T Fo We build the venous valve simulation model based on biology-relatee literature and generate dataset where each sample has different valve material properties, flow velocities, and other parameters. The variations designed to reflect wide range of human conditions across ages, genders, and health status. Detailed settings can be found in the Appendix B.2 The simulation time is 1s and 0.01s per interval, with 101 frames in total. Each frame records current geometry, stress, pressure, and velocity.
Table 4: Performance on Venous Valve. We record RMSE-all (L), the average RMSE of the whole rollout trajectory and all samples. Results of other physical quantities are listed Table 20
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
Transolver
0.3262
3055.56
91.83
0.0901
0.3432
4941.86
85.18
MGN
0.5540
4436.56
166.03
0.1362
0,5391
6646.33
158.35
HOOD
0,4647
3616.09
135.67
0.1174
0,4956
6080.94
126.05
AMG
0.4029
3784.96
107.45
0.1199
0.3809
5432.73
103.43
Fisale
0.2794
2658.59
80.23
0.0768
0.2565
4365.29
73.31
Error Kap risale step 65)
Map
(rine stop
(cine step
(cine scop
Transolver (rire step 65)
Error vap of Transolved lver (time step 65)
Error Nip of Transolier (time step 75)
Figure 4: Visualization of ground truth and prediction results. The red circle indicates the distortion of solid shape, where Fisale can effectively handle.
As shown in Table Fisale achieves the best results across all physical quantities, with particularly strong performance at the fluid-solid interface. Figure 4 visualizes predictions two time steps. It is observable that maintaining solid shape consistency becomes increasingly challenging as the rollout progresses. Thanks to the explicit modeling of the interface and the use of unified representation to

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Published as a conference paper at ICLR 2026
capture dynamic interactions, Fisale preserves solid geometry more effectively over long trajectories. This design allows it to handle cross cross-domain information exchange and maintain stability even the later stages of rollout. Moreover, when fluid flows through the narrow valve opening, it generates sharp increases pressure and velocity over short periods, making it hard to accurately predict in that region. Fisale addresses this by PCM, which decomposes this complex process into sequence of substeps. This reduces the difficulty of modeling each physical domain and its associated quantities. In contrast, other models typically adopt monolithic modeling strategy over the entire domain, which often struggles to capture rapid dynamic changes around the interface. This further highlights the advantage of the domain aware design handling complex two-way FSI phenomena.
4.3 FLEXIBLE WING
To better reflect real aerodynamics, we study a flexible wing scenario where the wing deforms under airflow. Unlike rigid-wing assumptions with fixed fluid-solid interfaces, flexible wings involve strong two-way coupling and nonlinear behaviors such geometry-depender loading and large deformations, making prediction more challenging. We treat this as steady-state inference task: given set of problem parameters (like the wind velocity, wing material, geometry and etc.), the model directly predicts the steady-state response. Follow the 2D rigid Airfrans (Bonnet et al. 2022), we build 3D flexible wing dataset for evaluation. Each sample contains more than 35,000 mesh points and varies flight and design parameters (see Appendix B.3), covering diverse flight conditions.
Table Performance on Flexible Wing task. Relative 1.2 is recorded. Second-best performance underlined. (a) the relative L2 error of the prediction results; (b) the visualization of prediction errors.
Solid
Fluid
Interface
Mean (4)
Deformation
Geo-FNO
0.0207
0.0802
0.0564
0.0524
GINO
0.2838
0.5681
0.5715
0.4745
CoDANO
0.0355
0.1930
0.1002
0.1096
LNO
0.0173
0.0264
0.0269
0.0235
wing Stress
Galerkin
0.0396
0.0699
0.0635
0.0577
GNOT
0.0081
0.0558
0.0227
0.0289
ONO
0.1446
0.2362
0.2728
0,2179
Transolver
0.0051
0.0242
0.0164
MGN
0.0096
0.0229
0.0281
0.0202
Surface Pressure
HOOD
0.0088
0.0218
0.0266
0.0191
AMG
0.2507
0.1692
0.2357
0.2185
Fisale
0.0042
0.0155
0.0211
0.0136
Fisale
(a) The error of prediction results.
(b) The visualization prediction errors.
As presented in Table 5 Fisale achieves the best performance across all domains. When dealing with massive mesh points, several baselines seriously degenerate due to complex dynamics. This is because dense fluid points can overwhelm solid-related information, making it difficult capture solid changes. turn, this also disrupts the accurate modeling of fluid evolution. The right part of 3 the prediction errors. For the wing structure, deformation is primarily localized the tip. stress is concentrated at the root, and the applicd wind pressure is distributed over the lateral surfaces. Fach of these exhibits distinct spatial patterns driven by the interaction between the wind and the wing. Accurately capturing these patterns is therefore crucial. By modeling each domain separately, Fisale avoids letting solid information be overwhelmed by massive fluid points. Moreover, explicitly modeling the interface enables effective capture of spatially varying physics across different regions of the wing surface, thereby improving the performance complex, large-scale scale scenarios.
CONCLUSION
We propose Fisale, data-driven framework for solving complex two-way fluid-solid interaction (FSI) problems. By explicitly modeling the solid, fluid and coupling interface as separate components, and leveraging multiscale latent arbitrary Lagrangian- Eulerria (ALE) grids along with partitioned cou- pling modules (PCM), Fisale effectively captures nonlinear, cross-domait dynamics. Experiments on challenging tasks demonstrate the effectiveness of Fisale in solving complex two-way FSI problems.
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ETHICS STATEMENT
Our work only focuses on solving two-way fluid-solid interaction (FST) problems with deep learning method, so there is no potential ethical risk.
ACKNOWLEDGMENTS
This work is supported partly by the National Natural Science Foundation of China (NSFC) 62576013, National Key Research Plan under grant No. 20.224YFC2207404,, the Jiangsu Provincial Key Research and Development Program under Grant BE2022065-1, BE2022065-3, and the Ningxia Domain- Specific Large Model Health Industry R&D No.2024JBGS001.
REPRODUCIBILITY STATEMENT
In the main text, provide rigorous mathematical formulations of the model architecture. The overall pipeline is described Appendix A Hyperparameter settings and implementation details provided in Appendix D and the dataset is described in Appendix B Our code and dataset publicly available in the GitHub repository linked the abstract.
LLM USAGE CLARIFICATION
We declare that the Large Language Models (LLMs) are only used for language polishing and grammar correction during the writing of this manuscript. All research content, conclusions, and data solely produced by the authors, without subjective arguments, conclusions, data generated by LLMs. We take full responsibility for the entire content of this paper.
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Contents
Introduction
1
Preliminaries
3
Method
3.1
Multiscale Latent ALE Grids
3.2
Physical Quantities Encoding and Decoding
3.3
Partitioned Coupling Module
Experiments
4.1
Structure Oscillation
4.2
Venous Valve
4.3
Flexible Wing
10
5
Conclusion
10
Overall Pipeline
18
Dataset
18
B. I Structure Oscillation
19
Venous Valve
20
B.3 Flexible Wing
21
Metries
22
Implementation
23
Standard Deviation
24
Efficiency
25
Ablation Study
27
H
Showcases
29
H. Structure Oscillation
29
H.2 2. Venous Valve
30
H.3
Flexible Wing
31
Attention Pattern
34
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Published as a conference paper at ICLR 2026
A OVERALL PIPELINE
Our overall framework is formulated as Eq and we have described our design in Section incllding multiscale latent ALE grid, physical quantities encoding, decoding and aggregation, and partitioned coupling module. To provide a global view our design, we provide a detailed pseudocode here for clarity and improve the reproductivity.
Algorithm 1: Fisale Pipeline for Tow- Way FSI Problems
Input: Observations of fluid uy, solid u, and interface u, at time
Output: Predicted states ûy, üs üp at t + At
for / = to H do
Initialize latent ALE grid ga via geometry-aware offset and k-NN edge set EXh) (0.h) +- Linear(us)
(0.A)
(0.h) Embed imput features: X, <- Linear(uy). (0.N) x..)) < Linear (u.). x;
for / = to do
for h = to H do
Encode physical quantities onto grid:
Pʃ Encode(x (1-1,h)
ge (1))))))
Ps Encode(x X
(11)))
ga
(()1,k))
Ps Encode(x (1-1,h)
(1-1,A)
(x6
-ga
Partitioned Coupling Module (PCM):
Update solid state: P', P' CrossAAttention((.:: Pb. P Ba)
Update grid: g' LaplacianSmoorthppssp Py- P/+ Ba; E)
Update fluid state: p'y. PK CrossAAtentioo((p): H..P'.. g()
Update interface influence: p" P'. +- SelfAttention('', P'y, Ph, g.)
Decode features:
x
(I,))
< Decode(p"..g.
X (.M)
< (eco
(1,N) Decode (p."' g'.)
X
Aggregate across scales:
(1)
(FN H. 1 (1,b))
( FFN (FN H x=1x (1,b)
(
" FFN(Concat (1,b))
X
if < L then
Chunk fused features back to H pathways:
{x (L.b) ff
Th=1 + Chunk(x (
[x (1.b) H
1= < Chunk x
[xg = - Chunk(x4) xx,
(1,)) H
Output predictions:
ú ú. ú, Linear(x (L) (L)
Xb (1))
X3
B DATASET
We evaluate models in three public and curated datasets, whose information is summarized in Table | Note that these benchmarks involve the following three types of fluid-solid interaction tasks, which are widely explored in studies that focus only on fluid solid:
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Rahman et al., 2024 Li et al. 2025): Given solution sequence {Mo u1...., ur) of a dependenddnnt PDE. the goal is to learn a model Fo that maps the current state the target state:
Single-Step Prediction (Li et al. 2021
Úct Fo(uE)
where A/ spans the next few time steps. During both training and inference, the model always receives the ground truth u, input predict Ut+Ar- This task is fundamental for learning local temporal dynamics and serves as a building block for simulating physical processes.
Autoregressive Simulation (Pfaff al. 2020 Sanchez-Cionzalez et al.
2020 Ma et al.
2024): Similar to single-step prediction, but during inference, the model recursively uses its own previous prediction as input except for the first step:
Úx+1 Fu.),, u F (0(+..)),
..+T Fo(
This method allows for long-term rollout of PDE solutions and is widely used in physics- informed forecasting and control.
Steady-State Inference (Wu et al. 2024 Deng et al. 2024 Li et al. 2023d): Given
set of problem parameters À (e.g., environment conditions, boundary conditions, material properties). the objective learn mapping directly from input parameters to the steady- state solution u of the PDE:
C(u' A) = 0, u = (A)
This parameter-to-solun formulation maps input conditions to the system's equilibrium state and plays central role in engineering and scientific design problems concerned with steady-state behavior.
B. STRUCTURE OSCILLATION
The structure oscillation problem, also known as "FLUSTRUK-A", is well-established benchmark in the field of computational FSI. It models the interaction between an incompressible, viscous fluid and thin, elastic beam attached the rear of rigid cylinder placed in channel. The fluid flow around the cylinder induces unsteady forces on the beam, which, in turn, leads to self-sustained oscillations of the structure. These oscillations are periodic and strongly depend on the material properties of the beam, such as its density, elasticity, and damping (Turkk & Hron 2006). This problem plays crucial role in validating and comparing the performance of FSI solvers due to its nonlinear, coupled nature. Unlike purely fluid or solid benchmarks, PLUSTRUK-A tests solver's ability to accurately capture dynamic feedback between two physical domains. It widely used in academic research and engineering applications, especially in domains where flow-induced vibrations (FIV) are significant, such as aerospace, civil engineering, and biomedical simulations (e.g., modeling blood flow through flexible vessels). The problem is particularly challenging due to the fine balance required between numerical stability and physical fidelity, making it an ideal dataset for developing and henchmarking advanced data-driven physics- based models (Hoffman al. 2012).
The dataset used in our experiments is proposed in CoDA-NO (Rahman et al. 2024). The com- putational domain is two-dimensional channel of length 2.5 and height 0.41, containing fixed circular cylinder of radius 0.05 centered at (0.2, 0.2), and a thin elastic beam attached to the rear of the cylinder with length of 0.35 and thickness of 0.02. The fluid is modeled water with constant density of 1000kg/ m The flow enters the domain through the left boundary with a time-dependent fourth-order polynomial velocity profile that vanishes at the top and bottom walls. The inlet condi- tions vary across 28 predefined configurations, and the peak inlet velocity reaches approximately 4 m/s, enabling diverse and realistic flow conditions for each viscosity setting. The outlet (right boundary) applies pero-pressure condition, and no-slip boundary conditions are enforced on the channel walls, the cylinder, and the elastic beam. To investigate different flow regimes, the dataset includes simulations with four viscosity values: μ E {0.5, 1, 5, 10), resulting in Reynolds numbers approximately ranging from 4000 (for μ = 0.5) to 200 (for μ = 10). For the solid, the density is set to 1000 kg/ m² with Lamé parameters À 4.0 10 and μ 2.0 106. Simulations are run up to a final time Ty- = 10 seconds, using a fixed time step of öt = 0.01, resulting in 1000 time steps per trajectory. Samples share same physical domain and mesh. Each sample contains 1317 mesh points.
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the prediction interval to At 4öt in order to allow the oscillation to evolve sufficiently and evaluate the model's ' ability to predict over longer time horizons. We first conduct training and evaluation data with Reynolds numbers Re E {200, 400, 2000). The data frames randomly split into training, validation, and test sets in a ratio of 8:1:1, resulting in 9561 training samples, 1195 validation samples, and 1196 test samples. We further test the out- of-distribution (OOD) generalization of trained models on a separate set of 498 samples generated with Re 4000. This setup similar in spirit to that used CoDA-NO (Rahman et al. 2024), but with important differences. Specifically, CoDA-NO employs pretraining on the first 700 frames of each trajectory followed by few-shot fine-tuning. pretraining fine-tuning procedures. All models are trained and evaluated under the same conditions, using longer time interval and randomly shuffled samples. This design ensures a fair comparison of in-distribution and OOD performance across models. Additionally, we use different evaluation metric from CoDA-NO. The rationale for this choice is detailed in Appendix C
In experiments,
contrast, our study does not involve any
B.2 VENOUS VALVE
The venous valve problem models the dynamics of valve leaflets within veins, which play critical role ensuring unidirectional blood flow and preventing backflow in the human circulatory system (Bazigou & Makinen 2013 Enderle & Bronzino 2012). The system involves a thin, flexible leaflet that interacts with a pulsatile, incompressible fluid under physiological conditions. The valve opens and closes in response changes in local pressure and flow rate, mimicking the behavior observed in venous circulation. This problem presents several unique challenges. First, the contact between the leaflets during valve closure introduces discontinuities and non-smooth behavior in the fluid- solid interface. Second, the large deformation of the leaflet requires robust modeling of nonlinear elasticity. Third, the highly transient, time denendenn nature of the flow, driven by periodic inlet conditions. demands accurate and stable single-step prediction to effectively simulate the full valve cycle. Due to these complexities, the venous valve problem serves as a stringent benchmark for evaluating FSI solvers, especially those aiming to operate under realistic biomedical conditions. It has important implications biomedical research and healthcare applications, including the study of venous insufficiency, the design of prosthetic valves, and the development of patient-specific simulation tools for diagnosis.
To deeply investigate this problem and evaluate the effectiveness of Fisale, constructed simulation model based on related literature (Lin et al. 2023 Wang et al.) 2025 Tikhomolova et al. 2020). The simulation is implemented using COMSOL Multiphysics (Multihyysis) 1998), widely used finite element solver for coupled multiphysics problems. The domain shape of venous valve (Joda et al. 2016 is illustrated in Figure
blood flow
10mn
inlet
valve
outlet
11.53mm
leave thickness
5m5
Figure The physical domain of venous valve simulation model.
To simulate venous valve dynamics across a range of physiological variations, we define parameter sets for leaflet thickness, inlet blood velocity, and the mechanical properties of the valve tissue. These parameters are varied systematically using a full factorial combination scheme, allowing us to generate data that reflect diverse of biological scenarios corresponding to different genders,
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age groups, and biological states. While parameters and combinations may extend beyond typical values observed healthy individuals, they are intentionally included to explore extreme pathological scenarios. This broader coverage is important for studying disease-related valve dysfunction as well as for informing the design and testing of prosthetic valves. The specific parameter settings used in the simulations summarized in T
Table Parameter settings venous valve simulation model.
Leaflet Thickness Inlet Blood Velocity
Valve Material
(mm)
(m/s)
C1 (MPa)
C2 (MPa)
range(0.5.1.0.0.)) range(0...0.6.o.I)
(0.01.0000000110155..00.} {0.00100000.0010.022.00}}
The leaflet is modeled as a hyperelastic Mooney-Rivlin material (Boulanger & Hayes 2001). which is widely used for soft biological tissues. The material behavior governed by two coefficients, C1 and C2, representing the elastic response under deformation. The leaflet thickness is varied from 0.5mm 1.0mm increments 0. 0.1mm, while the inlet blood velocity ranges from 0.1m// to 0.6m// with the same step size. The outlet applies zero-por pero-pressure condition, and no-slip boundary conditions are enforced on the vessel walls. The blood modeled as an incompressible Newtonian fluid, with density 1050kg/m² and dynamic viscosity 0.0035/a s. We formulate it as transient FSI problem. A time-dependent inlet velocity function sin n/ is applied, modeling periodic blood flow cycle with a period second. Simulations are performed with a time step of 0.01s, resulting in 101 frames per trajectory. Each frame records multiple physical quantities, including current geometry, stress, pressure, and velocity. Based on the full factorial combination of parameter settings, we generate total of 900 simulation trajectories. Among them, 720 trajectories used for training, 90 for validation, and 90 for testing. Each sample contains 1693 mesh points on average.
B.3 FLEXIBLE WING
Wind
Wing
Wind
Figure 6: The physical domain of flexible wing simulation model.
The flexible wing problem practical importance in numerous engineering and biological contexts. In aerospace engineering, understanding the aerodynamics of flexible wings is crucial for designing next-gent gextt-ennration aircraft, drones, and morphing airframes that can adapt to changing flight conditions. To better capture realistie aerodynamic behavior, Unlike traditional rigid-wing models that assume a fixed structural geometry and static fluid-solid interface, the flexible wing deform under aerodynamic loading, introducing strong two-way
study FSI problem involving flexible wing.
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coupling between the flow field and the structure. This coupling results in highly nonlinear and geometry-dependere behavior, including large elastic deformations and nontrivial load pressure across the wing surface. In this task, the goal is steady-state inference: given boundary conditions such as inflow velocity and structural material properties, the model aims to predict the equilibrium configuration of the wing and the corresponding steady-state fluid flow around it. This differs fundamentally from time-dependent FSI tasks in that it seeks a converged static solution representing the long-time behavior of the system, rather than predicting transient trajectories.
We follow the conventions (Bonnet et al. 2022 Valencia et al. 2025) to generate the dataset. The simulation domain is shown Figure 6 The free-stream kinematic viscosity is 1.5 x 10 10-5m² /s. The wing sections are 24XX NACA airfoils with constant relative thickness (indicated by XX). The wing has a length of 1.5m and root chord length of 1m. The geometric parameters of the wing vary across samples include the relative thickness, the taper ratio (ratio between the tip and root chords), the sweep angle (angle between the quarter-chord line and line perpendicular to the wing root). Additionally, the free-stream velocity, attack angle and the wing materials are also variations for diversity. These parameters are listed in Table
Table 7: Parameter settings of flexible wing.
Sweep Angle (9)
Free-stream Velocity (m/s)
Attack Angle ()
Taper Ratio
Wing Material
Thickness
Carbon Fibers
(10.12.14.16) {0.5,0.6.0.7}
{10,25,40) {50.75,100} (-12,10.14.18)
Titanium Alloy
Al-Zn-Mg Alloy
Here, three different materials namely Carbon Fibers, Alpha-Beta Titanium Alloy and Al-Zn-Mg Alloy have been utilized based on previous literature (Chakraborty & Ghosh 2022), which are widely
used materials for air- air-wing analysis. The detailed material parameters are listed in Table X
Table 8: Material parameters of wings.
Material Name
Density (kg/m²)
Young's 's Modulus (MPa)
Poisson's 's Ratio
Carbon Fibers
2.30 10
2.30
103
0.210
Alpha-Bete Titanium Alloy
4.43
10 10-6
1.13 10
0.342
Al-Zn-Mg Alloy
2.83
10-
7.20 104
0.327
The dataset samples are generated through full factorial combination of the simulation parameters, resulting in total of 1296 unique configurations. Among them, 1036 samples are used for training, 129 for validation, and 131 for testing. Each sample contains 37441 mesh points on average.
C METRICS
We employ different metrics for specific tasks, adhering to the evaluation approaches in related works.
Single-Step Prediction Steady-State Inference: Relative L2 In line with prior studies on single- step prediction (Li et al. 2021 Rahman et al. 2024 Li et al.. 2025) and steady-state inference (Wul
al. 2024; Deng al. 2024; Li al.] (2023d) tasks, we use the relative L2 to assess performance.
Given the input physical quantities u and the predictions úi the relative L.2 is computed as:
u all
Relative L2
Iu
is worth noting that for the structure oscillation task, we adopt the Relative L2 error as the evaluation metric, instead of the Mean Square Error (MSE) used in CoDA-NO (Rahman al. 2024). This
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task involves predicting multiple physical quantities with different units and orders of magnitude. While CoDA-NO reported MSE results based on normalized data to mitigate the influence of scale differences, this approach may not fully reflect model performance on the original data distribution. In contrast, Relative 1.2 is a more commonly used metric for this task. By computing the Relative 1.2 error for each physical quantity separately and then averaging the results, we effectively account for differences in scale while preserving fidelity to the original, unnormalized data.
Autoregressive Simulation: RMSE Consistent with works (Pfaff al., 2020 Sanchez-Gonzalez ct al. 2020 Ma et al. 2024) focused on autoregressive simulation tasks, we use Root Mean Square Error (RMSE) as the evaluation metric. Given the input physical quantities u and the predictions ú,
RMSE is calculated as:
RMSE
Σ u ú.11
During training, we compute the loss using normalized data, which is used for backpropagation and parameter updates, In the evaluation and test phases, we report the RMSE for each physical quantity within each domain based on the original data.
D IMPLEMENTATION
As shown in Table 9 Fisale and all baseline models are trained and tested using the same training strategy. We utilize relative 1.2 as loss function for single-step prediction and steady-state inference tasks, and RMSE for autoregressive simulation task. For different physical domains, we add each loss with equal weights. To ensure fair comparisons, we first approximate the parameter count of all baselines to match that of Fisale and then adjust their parameters to minimize overfitting and achieve better performance. All experiments are conducted on a single RTX 3090 GPU (24GB memory) and repeated three times. We provide the parameter count of each model in Table I0 for reference.
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 the number of samples in 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.
Training Configuration
Model Configuration
Datasets
Epochs
LR
Batch Level (L)
Pathway (H) Grid Shape (M) Channels (D)
Structure
[16, 16]
100
1x 10-3
50
2
2
[64, 64]
Oscillation
[8, 8]
Venous
[16, 16] [8, 8]
100
1 10-3
50
2
2
[64, 64]
Valve
Flexible
5, 5, 5]
100 x 10-4
2
[96, 128)
Wing
4, 4, 4]
We implement baselines based on official and popular implementations. For autoregressive simulation task, we uniformly add noise with mean of 0 and a variance of 0.001 to improve the error accumulation control during rollout. Since several neural operators and transformer-basee baselines are primarily designed for fluid scenarios with Eulerian settings, they are not naturally suited for handling scenarios with Lagrangian views, such as two-way FSI problems. We preprocess the data to adapt it for use with these baselines. Specifically, we first align the feature dimensions of different domains using padding, then concatenate along the length dimension to combine all domains into a single large sequence of points. For models that struggle handle the Lagrangian setting (Rahman jet al. 20044) Li et al.) 2023ct Wu et al., 2023), we map each point to a regular grid, transforming the data into Fulerian representation. Specifically, for point sequence of size N D, discretize
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Table 10: Parameter count of baseline models and Fisale.
Structure Oscillation
Venous Valve
Flexible Wing
Geo-FNO
1.60 M
1.60 M
5.22 M
GINO
1.72
5.17
LSM
2.23 M
2.23 M
/
CoDANO
1.83 M
1.83 M
5.01
LNO
1.62 M
5.34 M
Galerkin
1.74 M
1.74 M
3.18 M
GNOT
1.64 M
1.64 M
4.64 M
ONO
1.63 M
5.07 M
Transolver
1.55 M
1.55 M
4.61
MGN
1,82 M
1.82 M
4,86 M
HOOD
1.79
1.79 M
5.31
AMG
1.34 M
1.34 M
4.71 M
Fisale
1.54 M
1.54
4.88 M
cubic space [sr, y, E []1, into a regular grid, with the number of discretization points along each axis set to N]. We then concatenate each point's coordinates and physical quantities with the corresponding grid points based on their matching order. Excess grid cells are padded align dimensions. For other baseline models, we utilize geometry transformation functions provided in their implementations. Additionally, the fixed boundary condition is maintained by directly setting the displacement of the corresponding region as zero like the operation in MGN Pfaff et al. (2020).
Efficiency report GPU memory usage based on measurements from the operating system, which include memory pre-allocated by PyTorch's 'ss caching allocator. Although not all of this memory is actively used at every moment, remains unavailable to other processes once reserved. Therefore, this reporting strategy offers conservative yet practical estimate of the actual hardware demands during training. better reflects the real-world resource constraints typically encountered in deployment scenarios. To ensure fairness, all models evaluated using PyTorch's default memory management on a single RTX 3090 GPU.
E STANDARD DEVIATION
Table 11: Standard deviations Structure Ocsillation experiment.
Solid (x101 (x10-3)
Fluid (x10-3)
Interface ( (x10-*
Geo-FNO
+0.01
+0.05
+0.01
GINO
+0.50
#2.02
+0.78
LSM
+0.01
+0.72
+0.02
CoDANO
#0.07
+0.05
10.07
LNO
+0.03
+0.29
#0.12
Galerkin
+0.08
+0.03
+0.02
GNOT
+0.02
+0.05
+0.06
ONO
#0.24
10.92
10.12
Transolver
+0.07
+0.05
+0.08
MGN
+0.04
+0.03
+0.03
HOOD
+0.05
+0.03
10.04
AMG
+0.01
+0.01
+0.01
Fisale
+0.01
+0.04
+0.01
We repeat all the experiments three times and provide standard deviations here in Table II 12 and 13 It is worth noting that autoregressive simulation involves rollout process based sequential
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predictions, where error accumulation is inevitable. When the rollout trajectory is long, such accumulated errors can become unstable and difficult to control. In addition, our venous valve experiment poses greater challenge to result stability due to the presence of multiple target physical quantities across different physical domains. As a result, this task exhibits higher variance compared the other tasks. Moreover, ensure fairness and maintain the integrity of each method, adopt the initialization schemes provided in the official and popular implementations. rather than applying unified initialization across all baselines. As result, the scale and variance of network parameters vary across models, which may lead to differences in performance variance. As shown in Table [2] although Fisale does not always exhibit the lowest variance across all physical quantities, its overall performance remains consistently stable. This indicates that Fisale not only achieves superior performance, but also maintains robustness in long-time rollout, which is an essential property for reliable simulation in complex FSI scenarios.
Table 12: Standard deviations on Venous Valve experiment.
Solid
Fluid
Interface
Geometry (x10-2)
Stress (x109)
Pressure (x109)
Velocity (x)
Geometry (x10-2)
Stress (x109)
Pressure (x109)
(x10-3)
Geo-FNO
+3.88
1319.62
+12.61
+1.6
#4.19
488.77
#11.81
LSM
+4.26
+2216.0
+13.08
+2.1
#4.56
+286.48
+11.13
CoDANO
+6.67
+552.81
+18.19
+2.87
=6.30
+603.29
+16.16
Galerkin
+0.71
#79.86
#4.24
#0.86
=1.02
1107.30
#2.62
GNOT
+1.66
+258.56
+8.69
11.0
=1.89
2236.67
#3.87
Transolver
+2.18
+176.98
#3.37
#0.81
=2.5
1184.31
#4.23
MGN
16.28
+417.58
+15.00
#2.11
=5.46
#448.81
19.54
HOOD
+4.50
+359.47
+13.77
#1.48
33.5
#351.29
#7.29
AMG
+55.8
+332.23
+8.47
+1.37
=3.18
+313.47
7736
Fisale
+2.01
1189.26
+3.05
+0.92
#192
+223.60
#3.58
Table 13: Standard deviations Flexible Wing experiment. Some baselines are not included due to severe degradation, where standard deviations are too unstable to provide meaningful reference.
Solid (x10-2)
Fluid (x10-
Interface (x10-2)
Geo-FNO CoDANO
#0.01
#0.04
#0.24
+0.18
+0.15
+0.27
LNO
+0.02
+0.02
+00.4
Galerkin
+0.09
+0.19
+0.25
GNOT
+0.02
+0.04
+0.01
Transolver
+0.01
+0.04
+0.03
MGN
+0.01
+0.02
+0.01
HOOD
+0.01
+0.02
+0.02
Fisale
+0.01
+0.01
+0.02
F EFFICIENCY
Efficiency is key concern in practical applications, especially in large-scale scenarios involving massive mesh points. Therefore, we provide a dedicated discussion on the computational efficiency of Fisale here.
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In the computation pipeline of Fisale, the major modules include ALE initialization, physical quantity encoding and decoding, attention-base baaiidnnbbaeddd Partitioned Coupling Module (PCM), and FFN- based aggregation.
For ALE initialization, the most expensive step computing offiett, which involves caleu- lating distances between N observed physical points and M ALE grid points, resulting in complexity of O() After constructing the ALE grid, we apply k-Nearest Neighbors (kNN) to build edge connections, which incurs complexity of (M.
In the encoding stage, attention-like weights and spatial mappings are computed, both with complexity MMDM)) Similarly, the decoding stage also involves mapping from grids to physical points, again costing OMMND)
During the PCM evolution, we adopt linear attention, leading to a complexity of (MM
The FFN-based aggregation module uses stacked linear layers over N physical points, contributing complexity of O(N
We have ignored constant factors such as level L. pathway H. problem dimension C' and scalar coefficients. The total computational complexity is:
O(NMD + M ND + MD?)
In this expression, M (the number of ALE grid points) and D (the feature channels) are user-defined hyperparameters, while N (the number of physical observation points) is determined by the task and dataset scale. Typically, both M & N and D < N. For instance, in the Flexible Wing task, N 10$, while in the other two tasks, N > 103. In contrast, M and are both on the order 102. Therefore,
approximate the overall computational complexity of Fisale growing linearly with the problem size N, making scalable to large-scale FSI scenarios. We further conduct an experiment to demonstrate the efficiency of Fisale.
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 single epoch. Additionally, the recorded runtime includes the time spent constructing graph edges. As result, models like HOOD (Grigorev al. 2023) and AMG (Li at. 2025), which involve multiple rounds of dynamic edge construction during iteration steps, become particularly time-consuming.
Parameters (M)
Running Time (S)
GPU Memory (GiB)
Mcan Relative L2 (L)
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
ONO
5.07
152.87
7.46
0.2179
Transolver
4.61
245.75
11.09
0.0164
MGN
4.86
506.77
22.50
0,0202
HOOD
5.31
1500
22.80
0.0191
AMG
4.71
> 1500
21.92
0.2185
Fisale
4.88
296.30
3.10
0.0136
As shown in Table [4] Fisale achieves favorable trade-off between efficiency and predictive perfor- mance. Benefiting from its parallel module design, Fisale maintains a relatively wide architecture under similar parameter scale, which alleviates the memory burden associated with gradient storage during backpropagation. Moreover, by decomposing the physical domain into fluid, solid, and interface components, Fisale effectively splits large matrices into smaller submatrices. This reduces
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the memory footprint of intermediate computations. Together, these design choices lead signifi- cantly lower GPÚ memory usage compared to other attention- based, GNN-based models and most neural operators. Although Fisale involves kNN operations and multiple attention passes within each module, which limit its speed advantage, its runtime remains within a practically acceptable range. In summary, Fisale delivers high prediction accuracy with minimal memory overhead and without significant computational time increase, making it a good candidate for large-scale, real-world FSI applications.
G ABLATION STUDY
Beyond the main results, we conduct a series of ablation studies to comprehensively evaluate the design. All ablation study experiments are conducted on the Flexible Wing task, which is challenging long-range steady-state inference problem and contains more than 35,000 mesh points per instance.
Explicit interface component In our design, recognizing the importance of the coupling interface, we explicitly model it as separate component on par with the solid and fluid. Here, we explore the necessity of this operation. Specifically, since the interface inherently shares properties (solid stress, fluid pressure, and velocity) from both the fluid and solid domains, we cannot directly merge it into one of them. Hence, we duplicate the interface coordinates: one copy is concatenated with the physical attributes of the solid and included as part of the solid input, while the other is concatenated with the attributes of the fluid and treated as part of the fluid input. For the output, we average the coordinates from the each attribute. The network architecture remains unchanged except for removing the projection and deprojection of the interface component. The result is shown in Table 15 We can observe that removing the interface component leads to notable degraded performance. Since the behavior of the fluid-solid interface differs from both the solid and the fluid, merging it with either domain not only reduces the accuracy of the interface itself, but also adversely affects the evolution of the fluid and solid domains. This confirms that modeling the interface as an independent component contributes positively to the predictions.
branches the final position of the interface and calculate the loss
Table 15; Ablation study of the explicit interface component.
Solid
Fluid
Interface
Mean Relative L2 (L) Decrease
Fisale w/o explicit interface
0.0042
0.0155
0.0211
0.0136
0.0061
0.0212
0.0251
0.0175
28.68%
Ordering of the PCM In fact, the Partitioned Coupling Algorithm is flexible framework, and the specific ordering used in the our design is one representative choice rather than a fixed requirement. As we described in Section 2 a common and intuitive coupling loop (Placzek et al., (2009) proceeds as: the fluid exerts pressure on the solid, causing deformation; this alters the geometry and updates the grid positions; the updated grid affects the fluid state, which is then updated; finally, the updated fluid and solid states jointly determine the interface dynamics. Given this cycle, is reasonable to begin with either the solid the fluid, as both influence the evolution of the system. Different update sequences can be also physically justified. To further explore the order influence, we systematically test alternative update orders by permuting the four components. As shown in Table [6] we observe that: regardless of the specific update order, the model achieves comparable performance across all permutations. This indicates that the PCM is inherently a flexible and robust framework. Since Fisale adopts stacked architecture, the update order of components does not need to remain fixed within each layer. Instead, the interactions among fluid, solid, grid, and interface can be iteratively adjusted through vertical information flow across layers. This iterative propagation helps compensate for local order choices, allowing the model to refine cross-domain interactions over multiple stages. Therefore, while adopt a physically reasonable ordering in implementation, the model's performance remains stable under other plausible orderings, further validating the flexibility of PCM.
Replace PCM with simpler attention module We conduct ablation experiment in which replace the entire staee attention cascade with a simpler module including an attention layer followed by an FFN. We concatenate four components (solid, fluid, interface, grid) input like the
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Table 16: Ablation study of the update ordering within PCM.
Ordering
Solid
Fluid
Interface
Mean Relative L2 (1)
fluid-grid-solfe gordd-oolidinntefface
0.0043
0.0155
0.0206
0.0135
solid-fluid-intenn
0.0042
0.0155
0.0206
0.0134
0.0041
0.0157
0.0216
0,0138
grid-interface-solle
seeerfacc-sooid-ffuie ffuid
0.0041
0.0155
0.0212
0.0136
0.0044
0.0161
0.0213
0.0139
0.0042
0.0155
0.0211
0.0136
representation format in learning-basel fluid field and keep comparable model parameters by modify the latent dimension. As shown the Table 17 the replacement leads to drop in performance. This degradation is observed consistently across all regions. This indicates that cross-domain attentions in our design play a crucial role modeling the intricate interactions among components. Although the simplified variant is more compact in structure, it lacks the ability to disentangle and coordinate domain-specific interactions, which are essential in FSI systems.
Table 17: Ablation study of replacing PCM with a simpler attention module.
Params (M)
Solid
Fluid
Interface
Mean Relative 1.2 (1) Decrease
Fisale Simpler Module
4.88
0.0042
0,0155
0,0211
0.0136
4.96
0.0045
0.0178
0.0223
0.0149
9.56%
Multiscale Latent ALE Grids To evaluate the effectiveness of multiscale design within Fisale, we perform two scaling experiments. First, we explore the effect of mesh resolution M. To isolate this factor, we use single pathway and gradually increase M to investigate how it influences model performance. Except for the M and channel number, other settings are kept. The channel is set 128 for each experiment. As shown in Table 18 increasing the resolution level M in the single-pathway setting leads to slight improvements performance. However, overall, the performance remains largely consistent across different values of M. This trend suggests that, in single-resolution configuration, increasing resolution alone offers limited benefit. Each fixed-resolution mesh captures only specific scale of geometric and physical patterns, and lacks the flexibilityy simultancously represent both coarse global structures and fine local details. Moreover, even when the parameter count increases match that of main model, the accuracy still lags behind the multi-resolution two-pathway design. This indicates that the exact value of is not a dominant critical factor in determining the model's learning capability.
Table 18: Scaling results of mesh resolution M with single pathway.
1/3
Params (M)
Solid
Fluid
Interface
Mean Relative L2 (L)
2.48
0.0047
0.0171
0.0226
0.0148
2.63
0,0047
0.0169
0.0224
0.0147
3.66
0.0044
0.0171
0.0221
0.0145
8
5.11
0.0045
0.0172
0.0223
0.0147
7.81
0.0046
0.0167
0.0219
0.0144
10
12.50
0.0049
0.0172
0.0227
0.0149
Second, to further examine whether multi-resolution pathways indeed capture richer and more complementary information, we conduct an additional ablation study on the high-resolution pathway H. The setting and results are shown in Table 19 and Figure 7 respectively.
Through the results we find that increasing the number of resolutions H in the high-resolution pathway leads clear performance gains. Notably, moving from H to H 2 results in
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Table 19: Scaling settings of pathways H.
H
M1/3
Channels
Params (M)
[5]
[128]
2.63
2
[4, 5]
[128, 96]
4.88
[4, 6]
[128, 96, 64]
6.99
4
[3, 5, 6]
[128, 128, 96, 64]
11.99
[3, 4, 5, 6,
[128, 128, 96, 64, 64]
15.69
Error Trends of Solid. Fluid, and Interface w.r.t. H
Scaling Results of Pathways
0.0225
0.01475
0.0200
0.01450
0.0175
Pelebive Erer
0.0150
6.02400
0.0325
0.01375
Improvenent
Sold Flidd
0.0100
Interface
0.01325
0.0050
c.01300
Number Resolutions
Number Resclutions
(a) Error trends of solid, fluid, and interface.
(b) Mean error and relative improvement.
Figure 7: Scaling results of pathways H.
significant improvement in overall accuracy demonstrating that the incorporation of multi-resolution features is highly beneficial for FSI tasks. This supports the idea that coupling information across multiple spatial scales is critical to capture complex cross-domain dynamics. As H continues to increase, we observe gradual but diminishing improvements. This trend suggests two important insights:
Multi-resolution aggregation is indeed effective, especially when moving from single-scale to dual- or tri-scale setups.
Beyond a certain point (H 4 for example), the model reaches a saturation limit, where adding more resolutions and parameters yields only marginal gains even degradation. This due to the limited data capacity or the task-specific resolution requirements already being fulfilled.
These findings highlight both the value and the limitations of increasing resolution diversity within the architecture. Our final selection of hyperparameters strikes a practical balance between performance and efficiency.
H SHOWCASES
H.1 STRUCTURE OSCILLATION
Figures X and present qualitative comparisons and corresponding error maps on the Structure Oscillation benchmark regarding the solid displacement, fluid z-velocity, and fluid pressure. This two-way FSI problem is characterized by strong mutual influence between the fluid and the solid: pressure from the fluid deforms the solid, while the resulting structural motion feeds back to alter the surrounding flow. Accurate prediction in such scenarios requires simultaneous fidelity in both the fluid and solid domains. As shown in Figures 8 and 9 several baseline models suffer from severe artifacts, including discontinuities in flow velocity and distorted solid shapes, which break the physical coupling and lead unstable predictions. In contrast, Fisale maintains global coherence across the
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Ground Truth
Map
0C16
0.0010
0.14
AMG
Rud 12
0.0025
AMC
nlddd Emorr Melterry)
0.610
0,0000
sales Enee
0.006
0.004
GooRNO
GooPND
0.0025
0.1
0.0000
MGN
0.0
Figure 8: Showcase comparison of structure oscillation task. The solid deformation and fluid velocity are shown.
Error Map
0,016
0.014
0,0014
0,012
AMG
0.0012
AMG
Emoi (Daple (ovoleemeeee
0,010
Fud PRSSAAFO
Uk0
0,0008
UNO
Plad Zrrr
8,000
6.0006
GeoFNO
CeoFNO
0,002
0,0002
0,0000
Figure 9: Showcase comparison of structure oscillation task. The solid deformation and fluid pressure shown.
entire domain: the solid structure remains intact and physically plausible, and the surrounding flow field is smooth and consistent.
This consistency stems from Fisale's architectural design: the explicit modeling the coupling interface ensures localized continuity across domains; the multiscale latent ALE representation provides a flexible and unified embedding for cross-domain geometries; and the stacked partitioned coupling modules enable progressive and iterative updates across the fluid-solid system. Together, these components allow Fisale to preserve the integrity of the coupling dynamics, leading to more reliable predictions highly nonlinear, strongly coupled FSI regimes.
H.2 VENOUS VALVE
The Table 20 contains the results of physical quantities that not included in main experimental table. Figure 10 and Figure present qualitative comparisons and corresponding error maps for the venous valve deformation, stress, blood flow velocity, and pressure. In the early to mid stages of the rollout (before time step 55), Transolver able to maintain the overall valve shape. However, the fluid field exhibits more significant emor accumulation, with uneven distribution
valve task, focusing
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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.
Fluid
Interface
Geometry
Velocity (y)
Velocity (z)
Velocity (v)
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
Transolver
0.2867
0.0183
0.0641
0.0132
MGN
0,3851
0.0211
0.1018
0.0193
HOOD
0.3216
0.0204
0.0706
0.0152
AMG
0,3159
0.0193
0.0846
0.0161
Fisale
0.2337
0.0129
0.0542
0.0119
and particularly large errors near the valve opening. As the rollout progresses (after time step 65), the stability of Transolver degrades further, and noticeable geometric distortions appear in the valve structure. This can be attributed to its homogeneous modeling across the entire domain, which limits its ability to distinguish between fluid and solid regions and to capture their dynamic interactions at the interface. In contrast, Fisale explicitly models fluid, solid, and the coupling interface separate components, assigning equal attention to each. This enables the model to better track interface dynamics and structural changes. As shown in Figure and Figure TT, Fisale maintains valve shape stability even over long rollout trajectories and produces more accurate predictions for both fluid and solid fields, especially around the opening during peak states.
H.3 FLEXIBLE WING
presents qualitative comparisons and corresponding enror maps the Flexible Wing dataset, focusing on wing deformation, stress, and surface pressure. The figure reveals clear spatial patterns across these physical quantities: deformation is concentrated at the tip of the wing, where the structural flexibility highest; stress peaks at the root of the wing, where the wing connects the fuselage; and pressure distributed primarily along the wind-facing side of the wing. Such region-specific distribution behaviors introduce challenges for accurate prediction.
Figure 12
The figure also highlights the difficulty faced by models that apply a homogeneous modeling strategy across the entire domain. These models struggle to effectively capture the region-speciff dynamics of each physical quantity. Moreover, in this dataset, the number of mesh points in the fluid domain significantly exceeds that in the solid domain, making it easy for solid-related information to be overwhelmed. Some baselines exhibit severe degradation, while others fail to distinguish fluid, solid, and surface regions, which hinders their ability to extract useful signals from dense mesh representations. This further demonstrates the effectiveness of Fisale's domain-aware design in capturing cross-domain dynamics and maintaining robustness under such challenging conditions.
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Ground Tuth
PoRt 119
Fradl omi_Paps
Transdwer
Turncker
Grond Tuuh the stpp
Fadde smp
6 Uriyleps
tinegless 33
Ground Tuuh tine stepe -
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Tuncher 3 Inrelless 55
Gound Tuth thne
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Tuncher tireltess
200
460
660
30C
200
306
400
540
Suid Digfacmmn trent
Se Deploeeantiinn
visulization
Map
Figure 10: Showcase comparison of venous valve task. The solid deformation and fluid pressure are shown.
Ground Tuth tire strps
Flate @ tiTe SIESS
Raste tire stEps
Tanseker @ time strps
Tonsceer & time stres 35
prresleps
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Pask 3 tiresltps
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Groand Tuth
Task
Park lireatage 75
Turasher
Turnche
Fasd Webilyy Inooo
Faed Vildoy INc
20000 Color Bar of Result visu liuttii
10000
10000
40000
50000
Color Bar of Error Map
Figure 11: Showcase comparison of venous valve task. The solid stress and fluid velocity shown.
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Ground Truth
Fisale
Transolver
LND
wing Deformation
2355
Error Map
wing Stress
Error Map
Surface Pressure
Error Map
Figure 12: Showcase comparison of venous valve task. The wing deformation, wing stress and surface pressure are shown.
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x10*>
2.00
Attention of updating solid
updating
Attentton
intertace
3.0
1.75
solld
Interiace Fluld
2.0
Interface
1.5
Auu
1.00
0.5
0.75
Interface
Interface
(a.1)
(b.1)
sold
fuidd
Interface
x10
x10
(c.1)
x10
Attention pattern of updating solid
Attention pettern updeting fluid
Attention pettern of updating interfece
sald
Ruid
2010
1.50
3
2.25
Interfcce
intericee
nuid
0.75
intefaace
solid
fuid
Interface
solid
fluid
interface
0.50
(a.2)
(b.2)
interface
(c.2)
Figure 13: Attention pattern last layer PCM modules of Fisale. The top row corresponds to 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.
ATTENTION PATTERN
visualize and analyze the attention patterns each update step of the PCM module to provide intuitive insight into how attention contributes to modeling the two FSI problem and updating the various physical components. It is worth noting that we use linear attention CCoo 2021). where the softmax operation is removed and the computation prioritizes KT V. Although visualize the attention weights as standard attention (Vaswani et al. 2017). analyzing QK? remains meaningful. On the one hand, from the perspective of matrix multiplication, Q(K*V) and (QK")V produce the same result. On the other hand, QK" explicitly characterizes the correlation between the Query and Key vectors. Even without softmax, the magnitude of its entries reflects the strength with which each Query token aggregates information from the Keys. This form of analysis also widely used in existing works (Han et al., 20236 Wu et al. 2024).
cannot directly
Figure 13 illustrates the attention patterns in the last PCM layer of each pathway. The top row corresponds the low-resolution latent ALE grid, and the bottom row corresponds the high- resolution latent ALE grid. Regions with higher activation intensity are highlighted with red boxes, indicating where attention tends to aggregate information for the Query.
From the vertical comparison, we can observe two key phenomena:
Complementary patterns across resolutions. The attention patterns learned on the low- and high-resolution latent ALE grids exhibit clearly complementary behaviors. From the vertical comparison of the highlighted red boxes, we observe that the two resolutions assign different activation strengths to different components. For example, on the low-resolution grid, attention mainly captures the solid-to-solid relationships (Figure a 1)), whereas the high-resolution grid, it shifts toward capturing interactions between the solid and other components (Figure (3.2.))) Similar complementary patterns appear across the other attention layers well. This indicates that the two pathways capture different aspects of the underlying dynamics, further demonstrating the importance of the multi-scu scale design for modeling two-way FSI.
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2. Different learning tendencies for cross-attention and self-attention across resolutions. In the sross-attentionn used for updating the solid and fluid components, the low-resolution grid tends to focus on self-relations (solid-to-solid in Figure 33. and fluid-to- siddtto-fluid in Figure 13 b. 1), aggregating information primarily from the same component. contrast, the high-resolution grid shows the opposite trend, where the solid aggregates information from the fluid and interface (Figure 13 a.2)), and the fluid aggregates information from the solid and interface (Figure b.2). However, for the interface self-attention, observe reversed pattern (Figure c.c.l) 1) and Figure 13 c.2):: although still complementary across resolutions, both resolutions tend assign higher activation inter-region aggregation (e.g.. interface-to-solid or interface-to-fluit oo-liid).. Meanwhile, for self-aggregation, the fluid component shows stronger activation (primarily because the fluid region occupies the majority of the computational domain), whereas the solid and interface components exhibit relatively weaker self self-focused aggregation. This observation further highlights the importance of the multi-scale PCM. Different components require different aggregation priorities when being updated, and the attention patterns learned on different resolution grids emphasize different regions of influence. By combining these complementary behaviors, the multi-scale PCM ensures comprehensive coverage during updates: capturing both self-related information and cross-component interactions. Therefore, it enables more complete and physically meaningful representation of complex wwo-way FSI dynamics.
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Reviews:
[{"review_id": "C2zrtnjDss", "reviewer": "Reviewer_yad9", "summary": "The authors propose a new neural operator framework for handling fluid-solid interaction problems with cross-domain awareness. The proposed architecture Fisale leverages multi-scale latent grids to provide geometry aware embeddings. They further enable problem decomposition to handle nonlinear dependencies.", "questions": "1. It was shown in GNS that the model is able to handle multi-material systems (water-sand, water-jelly) by simply creating material-type embedding and leveraging data-driven training to learn the interaction dynamics. How does that compare against the proposed approach?\n2. Lagrangian systems typically require GNN based processing and models designed to handle Eulerian grid inputs fail to handle inter-particle interactions. This was shown in UPT, Transolver and GIOROM. The problem setting describes a system that is Eulerian or Lagrangian but the results don't seem to include any interaction dynamics datasets despite referencing GNS several times in the paper. \n3. The rationale behind KNN based grid is not fully clear to me. Typically neural operators leverage radius based grids with mean aggregation to enable discretization invariance (GINO, multipole graph kernel network). This is because when the input discretization changes, radius aggregation ensures an entire region is captured. With KNN based grids, the aggregation region changes based on discretization -- how does this enforce discretization convergence? \n4. The approach inherently handles multi-scale inputs. It would be interesting to see if it also handles AMR, as part of future work.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The paper provides good motivation to model ALE systems for fluid-solid interactions. \n2. The authors conduct thorough experiments, with a very detailed appendix section. \n3. The authors have considered SOTA baselines and SOTA problem settings.", "weaknesses": "1. The writing is quite dense in section 3 and it is not clear to me exactly how this proposed architecture compares to \"A Neural Material Point Method for Particle-based Emulation, O Sharabi\"\n2. It seems like grid update is similar to message passing. It seems like section 3.3 is describing typical neural network operations and as such can be moved to the appendix for better readability. \n3. The overall writing can be significantly improved, with many parts being unclear with substantial focus on describing techniques that are well established in ML community (such as update fluid state and update interface influence) which describe the sequence of operations performed within the network, without any intuition for the said operations. Similarly, \"update grid coordinate\" is overly detailed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vkbruuUhQz", "reviewer": "Reviewer_yXgU", "summary": "This work introduces Fisale, a purely data-driven surrogate model for two-way fluid structure interaction between incompressible fluids and deformable solids. The combined solid-fluid + interface state is represented using multiscale latent ALE grids and the dynamic interface is evolved in time using self and cross attention mechanisms in a partitioned interative fashion. Multiscale modeling is handled using parallel processing of latent ALE grids, interspersed with aggregation after every layer, to capture phenomena at various scales. The authors test the effectiveness of their method by predicting future states in three different scenarios: Structure Oscillation, Venous Valve and Flexible Wing, and compare the obtained frames with the numerically simulated ground truth as well as other neural models.", "questions": "1.\tThe authors should consider expanding some of the captions for exposition purposes, especially for Figure 2 and Table 3\n\t2.\tIn this work, I believe that even the grid sample positions are inferred from the model, along with the sample values, which is good. But how are the sample positions defined for the initial state. Specifically, how is g defined for the initial state?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 1, "presentation": 3, "contribution": 2, "strengths": "1.\tMultiscale modeling has proved to be an effective way in solving PDEs which involve dynamic boundaries and multiple domains, such as in fluid-structure interaction. Using more samples in interface regions for accuracy while using less samples is domain interiors for efficiency is a well-studied approach. The authors seem to leverage this well in the construction of their architecture.\n\t2.\tIn the experiments provided in the paper, Fisale seems to outperform other baselines, which is a good indication.\n\t3.\tThe experimentation in the main paper, together with the Appendix, is quite extensive, with a great number of relevant baselines.", "weaknesses": "1.\tAlthough the authors claim to leverage the classical numerical formulations like ALE and Partitioned Coupling, it does not seem to be the defining factor here. This is clear from the ablations performed in Appendix G. The increased accuracy in the experiments in the main paper can simply arise from the high representational capacity of the architecture itself, since there are a lot of learnable components. Also, the choice of the dimension of the latent ALE grid (i.e. D) does not seem to be explicitly defined anywhere in the paper. A large value of D can also cause overfitting.  \n\t2.\tThe intuition behind using attention for coupling is not very clear. In my understanding, attention mechanisms are used to model long-range dependencies which arise in text based domains, for example. The interface dynamics in fluid-structure interaction are necessarily short range, as far as both fluid and solid domains are concerned. So, using attention would be wasteful, and might even be numerically unstable. \n\t3.\tThe proposed model does not seem to be generalizable across tasks, which involve different geometry (static and dynamic). There is separate training and dataset for each of the three tasks. In that case the gain in inference time is not so much to offset the training time required for each of the tasks, when compared with a classical numerical simulation. \n\t4.\tThere is no mention of how the initial values of physical quantities q is fed to the model. The only point where it seems to be involved is in defining the initial latent state x^{0,h} = Linear(u). Even that uses a trainable linear layer. Without information about the initial state, no model can accurately predict future states. This supports my earlier observation that the proposed model may just be overfitting the data and not actually learning the dynamics. \n\t5.\t The limitations should be mentioned in the main paper, not in the Appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6JSQNu0oSo", "reviewer": "Reviewer_wVhk", "summary": "The paper proposes to model fluid-solid interaction (FSI) problems by explicitly modeling fluid/solid/coupling interface as three separate components. The approach boils down to:\n\n1) Projecting state data (positions, features) from all three components onto a shared latent grid\n2) Processing data on the grid through a partitioned coupling module (PCM) that sequentially updates solid → grid → fluid → interface\n3) Decoding back to the original points\n\nThe approach achieves superior performance compared to other methods (GNN and Transformer) that treat the FSI problem as one single unified system.", "questions": "I am overall learning towards accept, but answering those question might significantly improve my rating and I also generally improve the paper.\n- Did you try using standard attention instead of linear one? The scale appears manageable to me.\n- Regarding Table 17, performance vs mesh resolution. It appears to me that the performance does not depend on the mesh resolution. That is counter-intuitive to me as I would expect improved performance with increased number of grid points. Do you have any explanation why it does not happen?\n- Which attention is used in Table 16? If linear, would it be possible to update the table with standard attention (flash attention) and compare memory, runtime on top of performance?\n- For the set of problems, how does the method compare against a numerical solver?\n- Authors report Relative L2 instead of MSE for the CoDA-NO dataset. Is it possible for you to provide MSE just for a direct comparison to the CoDA-NO's paper?\n- You claim to be calable to large-scale problems. Do you plan further experiments on large-scale CFD simulations? Would you method be able to handle million point grids?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper is well and clearly written, the introduction is comprehensive, and the problem is demonstrated in an intuitive way with Figure 1.\n- Architecture design is well-motivated:\n  - It suggests a novel approach to handling a gap in prior works, and the problem tackled by the method is practically relevant for many engineering fields.\n  - The approach is inspired by established numerical techniques (ALE and partitioned coupling).\n  - Each component has a physical justification and is grounded in domain knowledge.\n- The method is experimentally validated against multiple baselines and has demonstrated superior performance on multiple tasks. Benchmarks include 2D and 3D problems with different complexity and fidelity.\n- The chosen baselines are state-of-the-art methods in physical modelling. The proposed method consistently outperforms all of them.\n- Ablation studies are thorough and target specific components of the method (e.g. multiscale ablation and having explicit interface).", "weaknesses": "- The paper adopts linear attention which is empirically weaker than standard attention despite the scale of the problems being manageable for flash attention.\n- The method does not scale well with increasing the number of latent points, which, in my opinion, is a limitation, see Questions.\n- Ablation studies indicate that some of the components might not even be necessary:\n  - For example, Table 15 demonstrates that the ordering within PCM doesn't change the performance. Since that is the case, couldn't you simplify the model to:\n    1) update solid via cross attention;\n    2) update fluid via cross attention;\n    3) update interface with self attention;\n    4) update grid?\n\n    Note that 1) and 2) can be potentially done in parallel if solid and fluid concatenated, which would make it even faster. Am I missing anything here? Perhaps PCM converges faster?\n   - In table 16, substituting the PCM module with attention only slightly drops the performance. However, it is not clear to me what attention is used: linear or self-attention. If not latter, then I would expect much better performance from using standard attention which would, in my opinion, question the value of having the PCM module. It would be great if authors could do the ablation measuring runtime, memory and performance. Having a single kernel might also make it faster, hence I am curious about the runtime.\n- The paper does not do an interpretability analysis of the latent ALE grids. While it is not strictly necessary, I do find Transolver slice assignment useful, perhaps the paper would benefit from a similar plot. Besides, studying attention patterns would be a neat addition, similar to how it is done in the EAGLE paper [1].\n\n[1] https://eagle-dataset.github.io/", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oeKbDdC85R", "reviewer": "Reviewer_k2K6", "summary": "This paper introduces Fisale, a data-driven framework for solving complex two-way FSI problems. Inspired by classical numerical methods such as the Arbitrary Lagrangian-Eulerian (ALE) approach and partitioned coupling algorithms, Fisale excels in three reality-related challenging FSI scenarios, covering 2D, 3D and various tasks by leveraging multiscale latent ALE grids and partitioned coupling module.", "questions": "Explain the weaknesses please.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1. Clear structure and good readability.\n2. Innovation in introducing the ALE approach and partitioned coupling module.\n3. Abundant experiment and superior experimental results compared with current methods.\n4. Rich practical application scenarios.", "weaknesses": "1. Lack of consistency in the experimental results: For example, as shown in Figures 8 and 9, the fluid fitting effect of LNO is significantly worse than MGN, but the results in Table 1 are the opposite.\n2. Inadequate experiment on complex scenarios: Each experiment in the paper involves relatively low complexity of surfaces and does not prove whether the model has a good fitting ability for more complex surfaces, such as Shape-Net Car (mentioned in AMG).\n3. Insufficient proof of generalization ability: The experiment cannot fully demonstrate that the model has learned the physical laws in the fluid dynamics scenario rather than the fitting data in a few scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)