Learning to Control Free-Form Soft Swimmers
Authors: Changyu Hu
Yanke Qu
Qiuan Yang
Xiaoyu Xiong
Kui Wu
Wei Li
Tao Du
Venue: NeurIPS 2025 poster ()
Keywords: Soft Swimmer Agents
Swimming Simulation Environment
Morphological Generalization
Reinforcement Learning
Primary area: reinforcement_learning

Abstract: Swimming in nature achieves remarkable performance through diverse morphological adaptations and intricate solid-fluid interaction, yet exploring this capability in artificial soft swimmers remains challenging due to the high-dimensional control complexity and the computational cost of resolving hydrodynamic details. Traditional approaches often rely on morphology-dependent heuristics and simplified fluid models, which constrain exploration and preclude advanced strategies like vortex exploitation. To address this, we propose an automated framework that combines a unified, reduced-mode control space with a high-fidelity GPU-accelerated simulator. Our control space naturally captures deformation patterns for diverse morphologies, minimizing manual design, while our simulator efficiently resolves the crucial fluid-structure interactions required for learning. We evaluate our method on a wide range of morphologies, from bio-inspired to unconventional. From this general framework, high-performance swimming patterns emerge that qualitatively reproduce canonical gaits observed in nature without requiring domain-specific priors, where state-of-the-art baselines often fail, particularly on complex topologies like a torus. Our work lays a foundation for future opportunities in automated co-design of soft robots in complex hydrodynamic environments. The code is available at https://github.com/changyu-hu/FreeFlow.

Learning to Control Free-Form Soft Swimmers
Changyu Hu!
Yanke Qu'
Qiuan Yangå.
Xiaoyu Xionga.
Kui Wu
Wei Li
Tao Du
Tsinghua University
Shanghai Qi Zhi Institute
3 LIGHTSPEED
"Shanghai Jiao Tong University
Abstract
Swimming in nature achieves remarkable performance through diverse morpholog- ical adaptations and intricate solid-fluid interaction, yet exploring this capability in artificial soft swimmers remains challenging due to the high-dimensional con- trol complexity and the computational cost of resolving hydrodynamic details. heuristics and simpli-
Traditional approaches often rely norphology-deeendeee fied fluid models, which constrain exploration and preclude advanced strategies like vortex exploitation. To address this, propose automated framework that combines a unified, reduced-mode control space with a high-fidelity GPU- accelerated simulator. Our control space naturally captures deformation patterns for diverse morphologies, minimizing manual design, while our simulator efficiently resolves the crucial fluid- fluid-truccurre interactions required for learning. We evaluate our method on a wide range of morphologies, from bio-inspired to unconven- tional. From this general framework, performmne swimming patterns emerge that qualitatively reproduce canonical gaits observed in nature without requiring domain- domain-specific priors, where state-of-the-ar baselines often fail, particularly on complex topologies like torus. Our work lays a foundation for future opportuni- ties in automated co-design of soft robots in complex hydrodynamic environments. The code is available at sijhttp://// pasttps:/gitteee /ttp//ggttttttt////eeeeeee.....h com/ wwwanyyuahurrreeeeeeeeee
Introduction
Underwater swimming exemplifies nature's ability to generate versatile movement strategies through free-form morphological adaptations from the traveling waves of eel-like swimmers to the jet propulsion of cephalopods (Dickinson al.) [2000] Hinch al. 2012). This diversity in soft-bods organisms demonstrates how complex yet efficient control emerges from the interplay between body deformations and fluid dynamics, offering inspiration for bioinspired roboties and adaptive underwater systems. However, exploring such capabilities in artificial free-form soft swimmers poses two challenges. First, unlike articulated rigid-body robots with standardized joint-torque actuation, soft bodies require high-dimensional control policies to coordinate continuum deformations across arbitrary morphologies, lacking a unified control paradigm. Second, learning these policies demands physically-gre grrouundd simulations that balance computational efficiency with hydrodynamic fidelity-a tradeoff often skewed toward speed existing frameworks. As result, current approaches typically heuristics, e.g., predefined muscle layouts (Min et al., 2219 Ma al. 202I) or voxel-aligned contractions (Bhatia al. (02T). which require fine tuning and restrict exploration of control space. Furthermore, although recent simulation environments (Wang) al. 2023a, Xian et al., 2023) enable data-driven control through simplified fluid models, these approximations omit critical hydrodynamic phenomena like vortex coadding-limitte the discovery of efficient gaits observed in biological swimmers.
resort to surphology-dependeed
To address the limitations of domain-expert, morphology-dependen actuation design, we present unified control framework that automates both deformation space construction and policy learning
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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for free-form soft swimmers. Our approach is grounded in a key biological insight (Zhang et al. 2022): natural swimmers exploit spatially low-frequency deformation modes to interact efficiently with fluids, rather than activating the infinitely many degrees of freedom in their soft body. Inspired by this, we introduce morphology-agnostin reduced modes, which compactly encode dominant deformation patterns across arbitrary morphologies with a few parameters. We first leverage geodesic farthest-point sampling to distribute control points adaptively over the swimmer's body. Coupled with linear blend skinning (LBS), these points define a deformation basis that interpolates coarse motions across the entire body.. We propose a dynamics correction process to further adjust deformations to physically plausible configurations while preserving kinematic intent. This approach also ensures motions are entirely driven by internal forces, avoiding unphysical momentum injection.
To address the nccuracy-efficiene tradeoff of existing simulation environments, we develop a GPU- accelerated simulator tailored for learning swimming strategy, ensuring both hydrodynamic fidelity and computational efficiency required for reinforcement learning (RI... Our simulator integrates the Lattice Boltzmann Method (HOME-LBM by [i al. 2023) for fluid dynamics, for its inherent parallelism and physical plausibility. The soft swimmers are modeled as finite elements in order to express different morphologies freely, integrated with the state-of-the-ar GPU solver (Chen et al.) 2024). We incorporate two-way coupling framework, ensuring that body deformations dynamically interact with fluid a mechanism essential for thrust generation. Our simulator supports training policies 128 x 128 x 512 grid in only few hours, successfully reproducing physically plausible swimming phenomena.
We evaluate our framework on diverse set of 3D soft swimmer morphologies, from bio-inspired fish to unconventional morphologies (Fig. 2p. demonstrating universal applicability. Our method achieves observable movement patterns in the majority of tested models in forward swimming task, achieving a 50% higher success rate in learning effective swimming gaits compared to state-of-the-art baselines (Wang et al.) 2023a), which struggle to produce meaningful motion for the majority of tested morphologies. Furthermore, it learns sophisticated behaviors like vortex exploitation, which simplified fluid models cannot capture. From this general framework, gaits corresponding to canonical biological swimming strategies (e.g.. undulation, oscillation, pulsation) emerge automatically without prior kinematic assumptions, establishing robust pipeline for automated soft swimmer control.
In summary, our work presents the following contributions:
1. We introduce a unified, reduced mode control framework for free-form soft swimmers that automates policy learning for various morphologies. Our approach significantly reduces reliance on human-designed priors while preserving deformation expressiveness.
2. We present a GPU-accelerated simulator optimized for learning coft-body swimming strate- gies, enabling efficient RL training while capturing hydrodynamic phenomena critical to swimming.
3. Our experiments demonstrate -fate-of-the-art swimming performance over prior works across a diverse set of morphologics.
2 Related Work
Unified control for diverse morphologies Many studies on the robot design aim at optimizing general control model for different input structures. Some works are based on articulated rigid bodies due to their simplicity, including /Zhao al. (2020); Gupla al. 2022 Lu al. (0225). However, the restriction of degrees of freedom (DoFs) of rigid bodies inhibits them from encoding high-DoF motions. On the other hand, many works turn to soft body design instead, including [Bhatia et al. (022)) using mass-spring method and Hu et al. (2019): Wang et al. (2023a]b); Spielbbrg et al. (2019) using material point method (MPM). Besides, there are other soft-body works based on reduced
modes (Zhang et al. 2017 Liang et al. 2023 Barbic and James 2005) and finite element method (FEM) Ma et al.) 2021] Du al. 2021 Geilinger et al. 2019 Tan et al., 2012). Leveraging the advantage of efficiency in rigid bodies and flexibility in soft bodies, some studies (Liu et al. 2022 Xu
jet al. 2021 Xu 2019 Wang et al. 2019 Li al. 2024) combine these two representations to form bone-flesh structure. However, these works focus on actuating soft bodies by rigid link, limiting their control to the underlying rigid joints. The task of optimizing a unified control for soft robot with free structure is still worth exploring. Moreover, most work in this field focuses on terrestrial

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robot locomotion, and their extension to swimmers is unobvious and non-trivial due to the high computational costs in fluids simulation and the coupling between fluids and solids.
Robot-learning environments There has recently been an increasing interest in and demand for physics-based learning environments artificial intelligence and robotics research. Most works (Makoviychuk et al. 2021] Xiang et al. 2020; Todorov et al. 2012: Coumans 2015 Graule et al. 2022, et al. 2021 focus on high- gesh-pefformance learning environments for simulating and controlling rigid soft robots alone. Among these works, efforts in building learning environments for fluids are less common due to the computational cost of solving physics-bru based fluids and solid- fluid interactions. Most existing works (Min et al. 2019 Ma et al. clastic swimmers with biomimetic actuators in simplified fluids and learns their swimming skills
2018 Ren et al. 2022) build
with deep RL (Min al., 2019) differentiable simulation (Ma et al., 2021). As these simplified fluid models overlook fluid properties (e.g.. vorticity and incompressibillty) and two-way elastic-
coupling, learning advanced swimming skills like jellyfish pulsation and handling multiple swimmers are intrinsically difficult (Min et al. (019) in these works. There are some works involving the simulation of flow field, including Liu et al. (2022); Wang al. (2023a): Ma et al. (2021): )); Holl and
Thuerey (2024): Xian et al. 2023), but they either do not support fluid-elastic coupling or sufter from sticking artifacts, which limits their capability of modeling diverse and flexible swimming robots.
Aquatic animal locomotion Animal swimming has long been an intriguing research topic in biology (Dickinson et al., 20000, Hinch et al.] (2012) and mechanics (Zhang al. 2022 Lauder 2015) Costello et 2001). Previous works have identified several distinctive swimming skills commonly shared by aquatic animals which can be divided into three mainstream underwater swimming skills
undulation, oscillation, pulsation which our pipeline can all automatically discover from their representative swimmers' morphologies.
Swimmer Modeling
Following the body-brain paradigm (Lipson and Pollack 2000), model swimmers through two synergistic components: shape (morphology representation) and controller (deformation policy).
3.1 Shape Modeling
We represent the geometry of free-form soft swimmers using a volumetric mesh M {X, E) defined by its rest-shape vertices X E R and its volumetric element structure E, where d [2, the dimension of space and n the number of vertices. We also define the deformed vertices as x(t) R and the nodal displacements as u(t) x(t) X, where denotes the time. This formulation generalizes across 2D and 3D. While our main results focus on 3D tetrahedral meshes, more 2D results are included in the supplementary materials.
3.2 Controller Modeling
Soft body control can generally be categorized into external and internal approaches. External ap- proaches apply forces directly the body. While simple to implement, they often violate momentum conservation and tend to drag the body toward the target rather than generating propulsion through fluid interaction. Internal approaches generate forces by specifying musele fibers within the soft body, preserving momentum and offering more physically realistic behavior. However, existing methods typically define these muscle fibers either manually by domain experts or through morphology- dependent heuristics. As a result, they suffer from limited control expressiveness, low automation, and insufficient generalization across diverse body designs. We propose novel internal controller that automatically modulates the entire rest shape X of the soft swimmer, ensuring momentum conservation and norphology-agnoste control.
Kinematic displacement field In order to obtain efficient control across varying mesh resolutions and diverse morphologies, we adopt reduced modes defined by linear blend skinning (LBS, Dacobson et al. 2014) as compact, low-dimensional control space. We modify the rest shape by kinematically proposed displacements Ukin constructed through reduced modes derived from geodesic control points.
3

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Deformation
Transation Retation
72
(X)
With LBS only
With dynnmic
M
P2
P2
PL
PL
Figure I: We use deformable square bar to illustrate our reduced mode control space. Two LBS control points, P1 and P2. leveraged to generate motions in this example. (a) Upper left shows the normalized weights distributed on the bar, and w1 and We are the weights p1 and p2 respectively. (b) The deformed green mesh (left) is generated by applying vertex wise weighted combinations of rotations R and translations from control points and p2, with weights wi and w'2. The red region indicates the self-inverted elements. Right figure shows the mesh with dynamic correction. (c) Deformation patterns generated by different distributions of control parameters.
We sample m control points via farthest-poin sampling rest-shape vertices X with geodesic distance, which takes into account the mesh's 's topology and reflects the shortest path in the volume of the mesh. The displacement for each vertex X, is calculated as a weighted sum of transformations from all m control points:
(ummn) = Σ
wiy (X,, (R +t, - X,)
t=1
where R; and t; are learnable rotation and translation modes defined on p: and weight wiy determines the influence of control point on vertex j. This function is defined as radial basis function (RBF) based on the geodesic distance between the vertex Xj and the control point p, (Fig. I a). This formulation assigns higher weights to vertices closer to Pa. producing smooth and spatially localized blend transformations (see ablation studies in supplementary for details). The weights normalized on each vertex. This formulation enables resolution-indes independent control of free-form deformations with only 6m degrees of freedom defined on p; (3m for rotation and 3m for translation).
Dynamic correction While uą provides expressive shape changes, it may introduce inverted elements as the LBS formulation ignores the mesh's volumetric integrity (Fig. left). Inspired by complementary dynamics (Zhang et al. 2020), we compute correction u, by solving perturbation uy from the following energy miniizztiion problem:
2 min { + Usin + Ue X) + 2 k k|/uaal
(2)
where (x, X) is the hyperelastic potential energy of a body in its deformed configuration x relative to rest configuration X (see [ee.. 4.1), and & stiffness coefficient that determines the extent of preserving the original deformation modes. This correction projects the kinematically proposed displacement uę onto the manifold of dynamically feasible configurations (Fig. 1 right), while retaining most of the kinematic deformation modes. The total rest shape deformation Ukin + u consists of both the kinematically proposed displacements and dynamic correction.
Control space properties Compared with previous methods that offer only limited or non- physically plausible actuation, the combined displacement field satisfies three critical requirements: (1) Intrinsic actuation via rest-shape modulation avoids external momentum injection; (2) Generality across arbitrary mesh topologies through geodesic sampling; (3) Compuct dimensionality with 6m parameters (m < n) achieved through LBS control points enabling efficient RI training. As shown Fig. TJ. varying coefficients generates diverse rest-shape changes while maintaining dynamical plausibility.

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Swimming Simulation
Efficient underwater locomotion involves rich, dynamic interaction with the surrounding fluid, manifesting complex flow phenomena such vortex shedding, wake capture and reverse Kármán vortex streets. Accurately capturing these effects requires a simulation framework that balances physical fidelity and computational efficiency. Prior work often relies simplified fluid coupling models, limiting the expressiveness of swimmer dynamics (Min et al. 2019; Ma et al. 2021). We address this limitation by integrating sate--o-the-asl aat--rtt fluid and solid solvers with acU-accelerated, two-way coupling scheme.
4.1 Elastic Simulation
Given discrete volumetric mesh representation of the soft body, deformed nodal positions x is governed by the Cauchy momentum equation. The internal elastic forces are derived from the strain energy potential, which depends on the current shape x and the modulated rest shape X + u:
MX V X + u) fex.s
where M E dud is the mass matrix, V the strain energy, and fex the total external force. We discretize time with standard implicit Euler integration calculate the updated x' from the current position x and velocity v over a time step At, by iteratively minimizing the incremental potential at each step (Gast et al. 2015):
min
xx' y) 2A
M(x' y) V X u),
where y is the inertia term - Atu + Ar M fext. constant computed at the beginning of the time step. We utilizes the state-of-the-arl GPU-accelerated solvers dedicated to elasties (Chen et al. 2024 to improve computational efficiency.
4.2 Fluid Simulation
We consider the lattice Boltzmann method (LBM) as our fluid simulator because it allows for explicit computation of updates. Fluid dynamics can be evolved by mesoscopic distribution function f(v,,xy, xy.t), t), which deseribes the probability of finding a particle at position X, with velocity v at time t. The macroscopic quantities of fluid such density P and velocity can be derived from f. LBM evolves fluid behavior by tracking distribution functions f at discrete lattice nodes on a Cartesian grid. The time integration proceeds through collision-streaming scheme:
fx(x + c, & At) f((x,t) 12((f).
where Sa is the distribution function for lattice direction c; and Ω the collision operator which relaxes the distribution function towards a local thermodynamic equilibrium state. We refer interested readers to Lallemand and Luo (2000) for more details.
The explicit nature of LBM's update rule enables massively parallel computation on Cartesian grids. Each lattice node's 'ss state is updated independently, minimizing synchronization overhead and maximizing GPU utilization. Our simulator adopts high-order moment-ene -encoeed LBM (Li et al. 2023) which achieve higher computational efficiency using less memory while ensuring the accuracy of fluid details.
4.3 Elastic-Fluid Coupling
We adopt a weak two-way coupling strategy that alternately updates the fluid and solid at each time step. Compared with other coupling schemes (e.g. strong coupling), it well balances stability and efficiency the context of learning soft-body swimming controller. The solid influences the fluid through boundary conditions, while the fluid applies pressure forces back onto the solid, which are numerically estimated over interface. address the computational challenges posed by extensive fluid-solid interactions, we further develop fully parallelized intersection detection method that exploits parallelism across both boundary elements and all lattice directions, resulting in significant performance gains. More details are presented in the supplementary materials.
5

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5 Swimming-Skill Learning
Building upon the reduced-mode control space and high-fidelity clastic-fluid coupling, we model the task of acquiring locomotion skills as a reinforcement learning (RL) problem. Our physical simulator serves as the dynamic environment, where the agent must learn deformation policies that exploit hydrodynamic interactions to generate thrust.
Task modeling and policy training. We model the task as a Markov decision process (MDP) with a state space S and an action space A. We adopt the standard multi-layer pereeptron (MLP) network controller to map the state of swimmer to actions applied to its actuators. Our simulator serves as the transition function in this MDP that evolves the current state-action pair (s, a) to the new state s' after one simulation frame. Each task contains a reward function R(s, a, s') and aims maximize its discount accumulation in time (Sutton 2018 2018 مi=o 7 Rg where R. = RR( 84, ag 8(41) stands for the reward collected in the i-th simulation Irame. We train all tasks with the soft actor-critic (SAC) method (Haamooa et al.] 2018), a widely adopted deep reinforcement learning (DRL) method known for its stability, sample efficiency, and ability to handle continuous action spaces effectively.
Unified state representation. Designing an effective state representation for soft swimmers poses unique challenges: (1) their nigh-dimensional deformations preclude exhaustive state encoding; (2) morphological diversity demands topology-agnostic observations to avoid case-by-case engineering. To address these, we design morphology-robut state space s defined as
[alccal; Vhecal, Omean, d. I, Clust) )
(6)
which includes the local positions Tlocal and velocities Viecal of a set of sample points on the model, the average velocity of all vertices Umean- the direction d. distance to the target position and the action of the last step Aust for typical smooth term (see Eq. 7p. In our implementation, we take the LBS control points as sample points directly. At each step, we treat the current sample points as point cloud and solve the Procrustes problem (Solomon, 20155 obtain closest rotation and translation from its original pose. The positions, velocities, and target direction are then transformed into this local coordinate frame more effectively capture the local deformation patterns of the soft swimmer and ensure the learned policy is rotation- and translation-invariant by construction.
Leveraging our novel soft-body control representation, we query an action vector E IR at each control step, where mn the number of LBS control points (.2). Each control point has degrees of freedom-3 3 for translation and 3 for rotation- within bounded ranges to ensure plausible motion. The number of control points can be fixed or manually specified, enabling resolution- independent control.
Action.
Reward. Since we use LBS control points and the weights are normalized on each vertex, the mapping from the action space to the deformation space is not injective but exhibits some redundancy. For instance, when all the control points take the same action of rotation and translation, there is no actual actuation applied to the model because of the unchanged rest shape. Therefore, we employ penalty term in the reward to restrict the redundant degrees of freedom in action space. typical smooth term is also added. The reward is defined as
R = Reask + Asmooth + XregPree:
Reask Umean d.
Pomouthh = - ||a Amall / (6m), Preg = - //1la///1mmmm)
It consists of three components: task-specific term Rusk. smoothness term Psmooth with coefficient Asmouth that encourages natural actions, and a regularization term Preg with coefficient Areg that penalizes redundant actions. The task-specific terms of reward Riak is the dot product of velocity and the direction to target. See supplementary materials for details.
Task Setup. Our primary evaluation focuses on the forward swimming task in 3D. To further probe the versatility of our framework, we also introduce three advanced tasks evaluated on a 2D swimmer: target navigation, energy-efficient locomotion, and flow resistance. The detailed setup and results for these tasks are presented in the supplementary materials, demonstrating the framework's 's adaptability to diverse objectives.

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Clownfish
Eel
Octopus
Leaf'
Turtic
Jellyfish
Torus
Eight
Spiral
Trumpet
Tube
Enneper
Figure 2: A collection of swimmer morphologies used in our experiments. The top six are bionic morphologies, while the bottom six are abstract morphologies with unconventional topologies.
Results
6.1 Experimental Setup
Dataset We construct novel collection of 12 soft swimmer morphologies Fig. 2). including 6 bio-inspired and abstract morphologies. The bio-inspired morphologies cover representative swimming mechanisms implemented by aquatic animals after millions years of evolution in nature: eel-like undulation, octopus- like oscillation, and jellyfish-like pulsation. The abstract morphologies are deliberately designed to test the framework's ability unconventional swimming scenarios beyond biological templates- semplates--senarioo scenarios where effective deformation patterns may not exist. These morphologies intentionally lack obvious deformation pathways for propulsion, forcing the controller to discover novel fluid-strune siru-trr interaction strategies through exploration. All meshes are normalized to the same scale and tetrahedralized by iTeeWWld (Hu al. 2020), comprising about 400 to 1, 1,500 vertices and 1,000 to 6,000 finite elements.
Baselines
We evaluate method against three baselines:
Domain-expert controller. Following expert-designed templates (Lin et al. 2019), we implement manually tuned actuators for well-understonn morphologies- -xieell--x muscles for clownfish and circular muscles for jellyfish. We directly transfer clownfish's musele design to the since they are geometrically analogous. For the torus, we apply four segments of tangential-direction muscles following the actuation approach outlined in DifiPD (Du al, 202T) for terrestrial environments.
Clustering-bast based controller. DiffuseBot/SotiZoo (Wang et al. 2023bla) are two state-of-the-ar approaches in soft swimmer control, we adopt their clustering based method one of the SOTA baselines. Following the approach, we segment swimmers into user-defined body regions via K- means clustering on centers of finite elements and then use principal component analysis (PCA) to extract dominant deformation directions for each region to define muscle orientations.
Difserentiable controller. We test SoftZoo's controller design in their open-sourced pipeline (Wang et al. 2023a). For comprehensive comparison, we adapt the framework by freezing morphology and material properties to isolate actuator optimization effects. However, this baseline faces critical technical limitations in our experimental setting: (1) its differentiable MPM simulation becomes numerically unstable beyond 3 seconds of simulated time (compared with 15 seconds in our task), causing gradient explosions that prevent policy convergence; (2) It fails to achieve 128 x 128 x 512 spatial resolution due memory constraints. These two factors prevent the baseline from completing all 12 experiments under our temporal and spatial settings. We therefore exclude it from quantitative comparisons. Qualitative results in the supplementary materials show that learned policies tend to repeat similar stretching patterns with limited diversity.
More details for all baselines provided in our supplementary materials.
6.2 Quantitative Results
As shown Tbl. our method outperforms baseline controllers across most morphologies in the forward-swimming task. Performances evaluated by rewards reflecting swimming distance.

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Table 1: Normalized reward (mean + std over trials) for the forward swimming task. Bold indicates the best performance per morphology; gray entries denote controllers failing to make a visible movement (reward less than 0.3).
Model
Method
Clownfish
Fel
Octopus
Leaf
Turtle
Jellyfish
Domain-expert
11.96 0.10
0.17 0.06
18.85 1: 0.42 0.87 = 0.13
Clustering-bused
-1.12 0.22
13.2 = 1.0
0.05 0.03 0.00
1.78 = 0.07
-118 0.16
Ours
10.34 0.42
6.26 1.18
0.88 1 0.29
8.67 1 0.94
23.43 1 1.14
Torus
Eight
Spiral
Trumpet
Tube
Enneper
Domain-cxpert Clustering- based
-0.31 4: 0.31
-0.07 + 0.01
-0.04 0.01
-0.13 0.00
0.16 + 0.05
1.02 + 0.56
0.20 + 0.03
Ours
15.05 = 1.27
3.99 = 0.43
3,20 0.37
-0.16 + 0.22
3,99 # 1.24
12.33 # 1.02
Domain-expert designed muscle templates FFg.. 3 bottom two rows) perform well on bio-inspired shapes like the clownfish, achieving up to 115% of our method's performance due to their well- understood swimming patterns. However, this advantage quickly deteriorates eel (3%). which is geometrically similar but different in proportion of its parts, revealing high sensitivity to geometric changes. Moreover, muscle templates originally optimized for terrestrial locomotion (e.g., torus) exhibit poor transferability underwater (fails), highlighting the limits of human intuition.
The clustering-based controllers implemented by of the SOTA methods in soft swimmer learning fails to produce effective gaits for out of 12 morphologies and breaks down entirely on abstract shapes, often resulting in unstable oscillations nearly motionless poses. This is because the cluster- ing method restricts deformations to the principal axes precomputed muscle fibers, insufficient to produce extensive fluid interaction necessary for effective swimming.
In contrast, our framework demonstrates robust performance, enabling over 80% of the tested morphologies to achieve forward locomotion. This high success rate, particularly unconventional shapes like the torus and Enneper surface, suggests a broader implication: swimming potential may be a latent property in wider range of geometries than previously assumed. This generality stems from our automated pipeline, which adapts naturally to diverse topologies, and our novel rest-shape deformation strategy, which enables expressive yet effective motions. Consequently, our framework acts not just as a controller, but as computational tool to reveal and realize the swimming aptitude of arbitrary designs, challenging prior assumptions about what constitutes a viable swimmer.
Training Stability To address the stochastic nature of reinforcement learning, we evaluated the training stability of our framework. We trained policies for fish-like swimmer in 2D across six independent runs with different random seeds while keeping all hyperparameters constant. The learning process proved to be highly consistent, with all runs converging to similar high level of performance. The final mean normalized reward was 6.71 with a low standard deviation of 0.77 0.77. The learning curves, which detail in the supplementary materials, show a stable and monotonic increase in reward, confirming that our method is robust and its performance is reproducible.
6.3 Qualitative Analysis
Our method learns effective swimming strategies across diverse morphologies (Fig. 3). producing biologically plausible motions for novel topologies. Several key observations are summarized below. Please refer to our videos in the supplementary materials for their full motions.
The torus-shaped swimmer achieves propulsion through a periodic deformation cycle, dynamically balancing body-fluid momentum exchange (Fig. 3] a. left). First, the torus undergoes controlled self-twisting into an 8-shaped configuration. Then it obtains angular momentum from fluid and starts to rotate, generating vortex-induced forces for forward motion. In contrast, the clustering baseline struggles to make a movement (Fig. a. right).
Torus
Enneper Surface Resembling a saddle-shaped skirt, the swimmer performs rhythmic stretch- ing/relaxation cycles, creating a "dancing" motion that leverages pressure gradients across curved surface (Fig. 3 b. left). This emergent behavior achieves stable locomotion despite the morphology's
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15.05
Noar (Clustoring)
(a)
12.33
Near zero (Clustering)
(b)
(Expert)
23.43
R 18.85 (Expert)
Baaelines
Figure 3: Key frames of some swimmers' motions: (a) torus (b) Enneper surface (c) clownfish (d) jellyfish. Swimmers (a) and (b) are compared with clustering baseline, while swimmers in (c)
and (d) are compared with domain-expert baseline. The structures of fluid field are visualized by extracting the isosurface of y criterion (Hunt al. 1988) of velocity field.
negative Gaussian curvature. In contrast, the clustering baseline make slight deformation and can hardly move (Fig. b. right).
The controllers produce traveling wave body undulations (Fig. C. left), qualita- tively reproducing the canonical undulatory propulsion common to natural anguilliform (eel-like) and carangiform (fish-like) locomotion (Fig. c. right). Leveraging our realistic LBM fluid simulator, the swimmers effectively harness vortex shedding from the tail for propulsion, closely aligning with biological observations- an effect unattainable in simplified simulation environments (Ma 2021
Eel/Clownfish
Wang et al. 2023a).
Jellyfish For the jellyfish, the learned policy based on pulsation-based propulsion, a canonical swimming mode. However, instead of the synchronized bell contraction common in nature (Fig. 3 d. right), our policy discovers a novel variant that propels fluid through alternating contractions along two mutually orthogonal directions (Fig. 3, d. left). It is worth noting that this strategy achieves higher speed than the domain-expert actuation design described above (Sec. 6.
6.4 Extensions to Energetic Efficiency
Beyond maximizing travel distance, key performance metric for both biological and robotic swimmers is energetic efficiency. To demonstrate that our framework can optimize for such objectives, conducted experiment learn energy energy-efficient gaits. Following established biomechanics literature Verma et al.] (2018), we define energy cost as the total work done by the internal forces deform the swimmer's body. This physically-grounde metric is efficiently computed each simulation step.
We augmented the reward function with energy penalty term: Rar Rusk we E, where E the energy cost and we tunable penalty coefficient. We trained policies for the clownfish morphology with varying we and evaluated the trade-off between distance and efficiency using the Cost of Transport (CoT), defined as total energy consumed per meter traveled.
The results in Tbl. 2
show clear and predictable trade-off. As the energy penalty increases. the learned gaits become more conservative, consuming significantly less energy and achieving better COT. An excessively high penalty (w. 0.05) correctly suppresses movement almost entirely,

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Table 2: CoT (lower is better) and travel distance for different energy penalty weights (we)
Wi
Forward Distance
Energy Cost
CoT
0 (Baseline)
1.68
433.2
258.3 J/m
0.005
1.57 m
211.0
134.3 J/m
0.02
1.06
78.8 J
74.7 J/m (Optimal)
0.05
-0.01 m
0.3 J
N/A
confirming that the policy robustly optimizes the combined objective. This demonstrates the flexibility of our framework to incorporate and optimize for complex, chysically-groundee objectives beyond simple locomotion. The precise formulation for the energy cost is detailed in the supplementary material.
6.5 Ablation Studies
In this section, we conduct series of ablation studies to analyze key components of our method, including the effect of control point count motion, the selection of geodesic distance in LBS process, the choice of LBM for fluid simulation, and momentum conservation enabled by internal actuators. Results show that: (1) geodesic distance proves critical for capturing geometry-aware deformation modes compared to Euler distance; (2) the number of control points may affect the magnitude and the complexity of the motion, depending on the morphology; (3) our LBM fluid solver captures hydrodynamic details for swimming where simplified fluid model fails; (4) we ablate fluid interactions show that our internal actuator does not introduce non non-physical momentum.
Details including figures and videos can be found in our supplementary materials.
Conclusions
This work presents unified framework for learning to control free-form soft swimmers. By constructing morphology-agnostic reduced control spaces through LBS and dynamics correction, our framework automates actuator design across diverse morphologies, bypassing the need for domain- expert manual design while preserving physical consistency. Coupled with a GPU-accelerated simulator resolving vortex-mediated thrust mechanisms, our approach enables the emergence of bio-inspired gaits without domain-exper priors and generalizes to unconventional morphologies where prior methods fail, and demonstrates robust performance range of advanced locomotion tasks.
While framework advances automated control, several open challenges remain. Our focus fixed morphologies paves the way for future work full morphology-contro) co-design, for which our unified controller and high-fidelit simulator provide critical foundation. Another significant hurdle sim-to-real transfer; the policies learned in our idealized environment can serve as a vital baseline for future research aimed bridging the reality gap. Finally, while method learns specialized policies, developing universal controller that generalizes across unseen soft-body geometries in fluid remains challenging open problem, likely requiring breakthroughs in meta-learning.
Acknowledgments and Disclosure of Funding
We would like to thank Dr. Chao Yu for her valuable advice on reinforcement learning training. Tao Du acknowledges the research funding support from Tsinghua University and Shanghai Qi Zhi Institute, and Wei Li benefits from SJTU's startup funds.
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20


Reviews:
[{"review_id": "8JhF4UMPkX", "reviewer": "Reviewer_NiLK", "summary": "This paper introduces a new framework for learning control policies for soft-bodied swimming robots with diverse shapes. Unlike traditional approaches that rely on human-designed muscle patterns or simplified fluid simulations, this method uses a low-dimensional, morphology-agnostic control space inspired by how real animals swim. The authors also develop a fast, GPU-based simulator that accurately models the interaction between the swimmer and the surrounding fluid, which is essential for learning realistic swimming strategies.\n\nThe system automatically generates controllers by sampling over reduced deformation modes, removing the need for expert-designed heuristics. Experiments show that the method achieves significantly better performance than existing baselines—especially on unusual shapes like toruses—and can learn advanced behaviors such as exploiting fluid vortices. The results suggest that this framework could support future work in co-designing both the shape and control of soft robots in complex environments.", "questions": "1. What is the simulation frame rate (FPS) achieved for the soft-body fluid-structure simulation?\n\n2. How many training runs were conducted, and how consistent is the performance across different seeds and across the training.", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n\n1. The paper presents a well-integrated framework for soft-body swimming control that combines a GPU-accelerated simulator, a two-way fluid-structure interaction model, and a unified control representation for diverse morphologies. This integration addresses bottlenecks in simulating and learning soft-body locomotion, offering both physical fidelity and computational efficiency.\n\n2. The proposed method demonstrates robust performance not only on biologically-inspired morphologies but also on abstract and unconventional shapes. This highlights the generality and adaptability of the control approach, which does not rely on human-designed heuristics and can discover effective strategies in previously underexplored morphospace.\n\n3. The paper provides high-quality visualizations of the deformation modes, control points, and learned swimming trajectories, which help clarify the system’s behavior and effectiveness. These visual results significantly aid in understanding the qualitative performance of the method.\n\n# Weaknesses:\n\n1. The core contributions primarily lie in the modeling of the soft-body and its interaction with fluid dynamics, while the learning approach itself follows established reinforcement learning method. As such, the work may be better suited to other venues\n\n2. The evaluation largely relies on task-specific rewards that may include regularization or smoothing terms, which can obscure direct assessment of locomotion quality. Inclusion of more interpretable and standardized metrics—such as forward swimming speed, displacement over time, or time-to-target—would provide clearer insight into the quantitative effectiveness of the learned policies.\n\n3. Given the stochastic nature of reinforcement learning, it is important to report training consistency and stability. However, the manuscript does not specify how many training trials were repeated or whether the final performance is robust across seeds. Including training curves and variance across runs would strengthen the empirical evaluation.", "was_revised": true, "final_justification": "I've read the rebuttal by the authors, and my concerns are reasonably addressed. In this case, I'm keeping my original evaluation as borderline accept.", "rebuttal": "We sincerely thank you for your recognition of our learning framework and visualizations, as well as your constructive feedback regarding the evaluation metrics and training stability analysis. We address each of the weaknesses and questions raised below.\n\n> **[W1]** The core contributions primarily lie in the modeling of the soft-body and its interaction with fluid dynamics, while the learning approach itself follows established reinforcement learning method. As such, the work may be better suited to other venues.\n\nWe thank the reviewer for recognizing our contributions in modeling the soft-body and its interaction with fluid dynamics.\n\nOur work directly addresses the emerging topic of soft robot control in complex physical environments, a subject of growing interest within the learning community (e.g., recent works at NeurIPS [1,2] and ICLR [3]).\n\nWhile our work builds upon established RL algorithms, we believe our contributions are of direct service to the learning community in the following ways:\n\n- **A unified and effective design** for low-dimensional control of general soft objects. This enables the learning community to scale up prior research from simple morphologies to a much broader and more expressive class of soft robots.\n- **A high-fidelity solid-fluid simulator**. This provides the community with a challenging new benchmark for developing and testing RL algorithms on more physically realistic swimming behaviors.\n- **Empirical validation** of RL's feasibility for diverse soft swimmers. This result, demonstrating that RL can learn effective gaits for non-traditional soft objects, is a novel and non-trivial finding in itself, opening up new research questions for the community.\n\nIn summary, we believe our work is a strong fit for the NeurIPS community, and we hope these contributions will foster further research into this challenging and important domain.\n\n---\n\n> **[W2]** The evaluation largely relies on task-specific rewards that may include regularization or smoothing terms, which can obscure direct assessment of locomotion quality. Inclusion of more interpretable and standardized metrics—such as forward swimming speed, displacement over time, or time-to-target—would provide clearer insight into the quantitative effectiveness of the learned policies.\n\nThank you for the question. We would like to clarify that the normalized reward reported in the tables in our manuscript has excluded all regularization terms, purely reflecting the forward swimming distance achieved within the time limit. For example, the data in Table 1(Section 6.2, Quantitative Results), when divided by 5 (action signal frequency in our experiment), gives the actual movement distance (in meters). Regarding the forward swimming speed metric, as the forward swimming task fixes the time horizon, the speed is in direct proportion to the reported swimming distance (i.e., dividing it by the fixed time horizon). From table1, It can be observed that our optimized swimming strategies have an average speed ranging from 0.1 to 0.3 m/s (the body length of the swimmers is around 1 meter).\n\nFor the turning task in our supplementary material, as well as for the two new tasks we added in the rebuttal (energy-efficient swimming and station-keeping), we will add the standardized performance metrics you suggested to the revised manuscript.\n\n---\n\n> **[W3]** Given the stochastic nature of reinforcement learning, it is important to report training consistency and stability. However, the manuscript does not specify how many training trials were repeated or whether the final performance is robust across seeds. Including training curves and variance across runs would strengthen the empirical evaluation.\n**[Q2]** How many training runs were conducted, and how consistent is the performance across different seeds and across the training.\n\nWe fully agree that training stability should be validated given RL's stochasticity. To address this, we conducted additional experiments on the 2D fish morphology. We trained forward swimming policies across 6 random seeds with identical hyperparameters. \n\nWe analyzed the intermediate rewards collected at 5%, 15%, 25%, 50% and 100% of training episodes. The results are reported below in the table.\n\n|Training Progress|Mean Smoothed Reward|Std. Dev.|\n|---|---|---|\n5%|-0.0047          |0.0742\n15%|4.6269|0.6278\n25%|6.1198|0.9111\n50%|6.7665|0.8122\n100%|6.7135|0.7733\n\nFrom this data, we draw three key conclusions:\n\n- **Consistent Final Performance**: The low standard deviation in the final (100%) reward demonstrates that all runs reliably converge to policies of similar high quality (as a reference, the final average speed is about 0.09 m/s, while the standard deviation is only 0.01m/s).\n- **Stable Learning Trajectory**: The consistently low standard deviation at every milestone (25%, 50%, 75%) confirms the learning process itself is stable and reproducible.\n- **Effective Progress**: The monotonic increase in the mean reward across all milestones shows this stable process is also effective at making consistent progress.\n\nCollectively, these quantitative results demonstrate that our training framework is robust, and its performance is consistent and reproducible across different random seeds. We would be happy to provide more evaluations if requested.\n\n---\n\n> **[Q1]** What is the simulation frame rate (FPS) achieved for the soft-body fluid-structure simulation?\n\nThank you for your question regarding the computational cost.\n\nOur system achieves approximately 30 FPS for the 3D scenes presented in the paper, using a single GPU. To provide a concrete example, the clownfish in the forward swimming task (see Fig. 2), which uses a 128x128x512 fluid grid, a fish model comprising 3,543 finite elements, and a time step of 0.005s, runs at an average of 29.9 FPS.\n\nFurther details on the simulation parameters can be found in Section 3 of the supplementary material.\n\nWe are truly grateful for your insightful comments and constructive feedback. We hope that the additional details and experimental results we provided have effectively addressed your concerns. Please feel free to let us know if there are any remaining concerns. We look forward to hearing your thoughts on our rebuttal.\n\n---\n\nReferences:\n\n[1] Bhatia, J., Jackson, H., Tian, Y., Xu, J., & Matusik, W. (2021). Evolution gym: A large-scale benchmark for evolving soft robots. Advances in Neural Information Processing Systems, 34, 2201-2214.\n\n[2] Wang, T. H. J., Zheng, J., Ma, P., Du, Y., Kim, B., Spielberg, A., ... & Rus, D. (2023). Diffusebot: Breeding soft robots with physics-augmented generative diffusion models. Advances in Neural Information Processing Systems, 36, 44398-44423.\n\n[3] Wang, T. H., Ma, P., Spielberg, A. E., Xian, Z., Zhang, H., Tenenbaum, J. B., ... & Gan, C. SoftZoo: A Soft Robot Co-design Benchmark For Locomotion In Diverse Environments. In The Eleventh International Conference on Learning Representations."}, {"review_id": "I7aK4PIyiy", "reviewer": "Reviewer_Z1aA", "summary": "This paper proposes a novel, automated framework for learning control policies for soft-bodied swimmers with arbitrary morphologies. It introduces a unified, reduced-mode control space using geodesic sampling and linear blend skinning to enable morphology-agnostic actuation. A GPU-accelerated simulator based on the Lattice Boltzmann Method is developed to efficiently capture fluid-structure interactions at high fidelity. The method is evaluated across a diverse set of morphologies, including biologically inspired and abstract forms, and demonstrates improved performance over state-of-the-art (SOTA) baselines in terms of swimming effectiveness.", "questions": "- You state that your method discovers swimming strategies “similar to those found in nature.” How do you measure this similarity? Have you considered comparing swimming trajectories or gait metrics with biological data?\n\n- Does your method account for or optimize the energetic efficiency of swimming? If not, could your framework be extended to incorporate energy-based objectives ?\n\n- Have you considered deploying these controllers on physical soft robots? What challenges do you anticipate in transferring them from simulation to the real world?", "limitations": "yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "In general, I found the paper well written and an interesting contribution. In particular:\n\nStrengths:\n- The proposed reduced-mode controller enables efficient and expressive actuation across diverse geometries, reducing the need for expert-designed actuators.\n- The use of LBM for fluid dynamics and a custom GPU-accelerated simulator ensures physically realistic interactions critical for learning effective swimming strategies. The simulator is a good contribution to the research community.\n- The method generalizes to unconventional morphologies where baseline methods fail, showing the strength of a data-driven, morphology-agnostic approach.\n\nWeaknesses:\n- The framework does not seem to optimize or evaluate energy usage. While action magnitudes are regularized, there is no explicit metric  to assess or minimize energy consumption. \n- Claims about the similarity between learned behaviors and evolved biological swimming strategies are based on qualitative visual resemblance only. No quantitative biomechanical comparisons or gait analyses are provided. \n- Despite referencing \"co-design,\" the work only addresses control learning on fixed shapes, and does not co-optimize morphology and control jointly.\n\nIn general, especially the abstract oversells the results in my opinion. For example: \"We showcase how our automated approach discovers high-performance swimming patterns similar to those found in nature over years of evolution.\" Without a more detailed analysis, this claim does not seem to substantiated. Additionally, energy efficiency is a crucial aspect of biological organisms, and doesn't seem considered here.\n\nAdditionally,  the authors write that\"Our work unlocks future opportunities in automated co-design of morphology and control of soft robots in complex hydrodynamic environments.\" Why does this work in particular unlock this possibility and not existing works on soft robots?", "was_revised": true, "final_justification": "The authors addressed my concerns and I will keep my score of 4 (boderline accept). For a full accept and increased impact, it would have needed to included the co-design of bodies and morphologies.", "rebuttal": "We sincerely thank the reviewer for the insightful and constructive feedback on our manuscript. We found the comments regarding energy efficiency, quantitative biological comparison, the scope of co-design, and sim-to-real transfer to be particularly valuable. These suggestions have helped us to significantly strengthen the evaluation and framing of our work.\n\n> **[W1]** The framework does not seem to optimize or evaluate energy usage. While action magnitudes are regularized, there is no explicit metric to assess or minimize energy consumption. **[Q2]** Does your method account for or optimize the energetic efficiency of swimming? If not, could your framework be extended to incorporate energy-based objectives ?\n\nWe thank the reviewer for raising this point. Energy efficiency is a key performance metric for swimming, and we prepared **a new experiment** to optimize the energetic efficiency of soft swimmers in our framework. We follow established methodologies from the biomechanics literature [1] and adopt a physically-grounded measure of energy consumption. Our framework is designed such that this **energy cost** (see table below) can be directly and efficiently calculated at each simulation step. This allows us to incorporate it as a penalty term within the reward function, enabling the training of policies that explicitly optimize for energy efficiency.\n\nTo demonstrate that our framework is capable of learning **energy-efficient gaits**, we have trained policies with different penalty coefficients (w_e) for this energy cost on the 3D clownfish model, which we selected as a representative case study. We believe the findings from this model are indicative of the framework's general capabilities across all our tested morphologies. The results from this experiment are summarized below:\n\n|w_e|Forward Distance|Energy Cost|Cost of Transport (CoT)\n|---|---|---|---|\n0 |1.68 m |433.2 J |258.3 J/m\n0.005| 1.57 m |211.0 J| 134.3 J/m\n0.02 |1.06 m| 78.8 J | **74.7 J/m (Optimal)**\n0.05| -0.01 m |0.3 J|-\n\nTo analyze gait efficiency, we adopted the widely-accepted Cost of Transport (CoT) metric, which normalizes energy consumption by distance traveled (lower is better). Our results demonstrate a clear trade-off between distance and efficiency. As the penalty increases ($0\\to 0.005\\to 0.02$), the trained policy demonstrates better **energetic efficiency of swimming**, while an excessive penalty (0.05) correctly produces a near-stationary policy. This validates that our framework robustly optimizes for energy-based objectives in a logical manner, and we believe these results fully address the reviewer's concerns.\n\n---\n\n> **[W2]** Claims about the similarity between learned behaviors and evolved biological swimming strategies are based on qualitative visual resemblance only. No quantitative biomechanical comparisons or gait analyses are provided. **[Q1]** You state that your method discovers swimming strategies “similar to those found in nature.” How do you measure this similarity? Have you considered comparing swimming trajectories or gait metrics with biological data?\n\nThank you for your question.\n\nOur comparison to biological swimming is indeed primarily qualitative. A direct quantitative analysis presents non-trivial challenges, particularly in acquiring precise biological kinematic data and in bridging the significant sim-to-real gap in matching animal physiology.\n\nWe would like to add that our qualitative similarity extends beyond visual resemblance to a more scientific, **taxonomy correspondence**. Previous biomechanical research has identified several canonical swimming modes in aquatic animals, including **undulation** (eel), **pulsation** (jellyfish) and **oscillation** (paramecium) propulsion [2]. We have confirmed that our fluid-solid simulation can reproduce these three mechanisms.\n\nOur unified framework, without any expert knowledge as input, spontaneously exhibits multiple biological swimming modes across diverse swimmer morphologies during training. The emergence of such diverse behaviors has a biological root and is a non-trivial result, which we largely attribute to the physical fidelity of our simulation. In contrast, prior works in soft swimmer learning have often focused on or exhibited undulation swimming mode only (e.g., DiffAqua [3], which focuses on anguilliform motion). This demonstrates a significant degree of generality in our approach.\n\nWe will carefully revise our wording in the manuscript to emphasize that our findings have a **qualitative connection** to these canonical biological swimming modes, and clarify our similarities with them to better position our work.\n\n---\n\n> **[W3]** Despite referencing \"co-design,\" the work only addresses control learning on fixed shapes, and does not co-optimize morphology and control jointly.\n\nThank you for your comment. Co-designing soft robots is an important emerging topic, yet it faces challenges in scaling to high-fidelity 3D simulations and in developing unified design representations. While existing works have made valuable progress, they often operate within certain constraints, such as simplified 2D design space (e.g., EvoGym [4]) or challenges in efficiently discovering controllers for complex 3D environments (e.g., SoftZoo [5]). Although our work focuses on controller optimization only, the framework we have developed shows a promising sign of tackling both of these challenges, providing a unified control representation within a high-fidelity 3D environment. We will tone down the co-design claim in our manuscript and revise its text accordingly to better clarify the position of our work.\n\n---\n\n> **[Q3]** Have you considered deploying these controllers on physical soft robots? What challenges do you anticipate in transferring them from simulation to the real world?\n\nWe thank the reviewer for this question. Sim-to-real transfer for soft robots in fluid is a big challenge for the field, requiring a multi-disciplinary effort that extends beyond the scope of a single work like ours. The path to physical deployment of soft robotic swimmers involves at least three significant hurdles, which are shared challenges across the soft robotics community. First, **modeling** requires non-trivial and probably labor-intensive system identification of physical parameters like fluid viscosity, material stiffness and damping, to narrow the sim-to-real gap. Next, **manufacturing** introduces issues of consistency and fabrication imperfections, meaning any physical prototype will have variations not present in the idealized digital model. Finally, **deployment** must contend with real-world complexities such as sensor noise, state estimation errors, and actuation latency. Addressing these challenges constitutes a significant research topic in its own right. Our current work focuses on a critical prerequisite: establishing a robust framework to discover what an optimal policy looks like under idealized conditions. We believe this provides a vital baseline and a clear target for future research dedicated to addressing the well-established challenges of sim-to-real transfer.\n\nWe are truly grateful for your insightful comments and constructive feedback. We hope that the additional details and experimental results we provided have effectively addressed your concerns. Please feel free to let us know if there are any remaining concerns. We look forward to hearing your thoughts on our rebuttal.\n\n---\n\nReferences:\n\n[1]. Verma, S., Novati, G., & Koumoutsakos, P. (2018). Efficient collective swimming by harnessing vortices through deep reinforcement learning. Proceedings of the National Academy of Sciences, 115(23), 5849-5854.\n\n[2]. Zhang, D., Zhang, J. D., & Huang, W. X. (2022). Physical models and vortex dynamics of swimming and flying: A review. Acta Mechanica, 233(4), 1249-1288.\n\n[3]. Ma, P., Du, T., Zhang, J. Z., Wu, K., Spielberg, A., Katzschmann, R. K., & Matusik, W. (2021). Diffaqua: A differentiable computational design pipeline for soft underwater swimmers with shape interpolation. ACM Transactions on Graphics (TOG), 40(4), 1-14.\n\n[4]. Bhatia, J., Jackson, H., Tian, Y., Xu, J., & Matusik, W. (2021). Evolution gym: A large-scale benchmark for evolving soft robots. Advances in Neural Information Processing Systems, 34, 2201-2214.\n\n[5]. Wang, T. H., Ma, P., Spielberg, A. E., Xian, Z., Zhang, H., Tenenbaum, J. B., ... & Gan, C. SoftZoo: A Soft Robot Co-design Benchmark For Locomotion In Diverse Environments. In The Eleventh International Conference on Learning Representations."}, {"review_id": "ZON583WB7H", "reviewer": "Reviewer_xudV", "summary": "This paper proposes an automated framework featuring a unified reduced-mode control space that naturally captures typical deformation patterns observed across diverse and unconventional swimmers, and a GPU-accelerated simulator to efficiently resolve fluid-structure interactions, specifically tailored for learning soft-body swimming controllers. The proposed method is evaluated on 12 soft-body morphologies and benchmarked against domain-expert designed and state of the art (clustering-based and differentiable) control approaches. The proposed approach performs similarly to the state of the art for the biology based swimming morphologies and outperforms the state of the art for abstract swimmers.", "questions": "Currently, each swimmer is trained from scratch, it is unclear whether the trained policy can be generalized or transfer to unseen morphologies. How can this be addressed?\nPlease clarify the computational cost and training speed of the simulator and training method.\nPlease provide more details of each reward functions and how reward weights were chosen. Could a poorly tuned reward limit the emergence of complex locomotion strategies?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nThe paper introduces a framework that unities soft-body control with reduced deformation modes and implemented with a GPU-accelerated simulator. The method is evaluated on 12 diverse morphologies, six intuitive biology-inspired swimmers and six abstract shapes. The evaluation experiments are well-designed and sufficient to support the paper's claims.\n\nThe paper is well-written and easy to follow. The figures and videos are helpful for understanding this work.\n\nThe framework could be used for co-design problems for both morphology and control or for studying locomotion when the analytical modelling is challenging.\n\nThe integration of a custom LBM-based GPU simulator to resolve dynamics in RL is novel. Furthermore, the combination of geodesic-based-LBS and dynamic correction for internal actuation is novel as far as I understand.\n\nThe evaluation of the approach is rigorous and thorough. The approach outperforms the state of the art for unconventional shapes and performs similarly to the other approaches for the biologically-inspired shapes.\n\nThe approach is shown to generalize to a set of arbitrary morphogenesis, something that prior work struggles with.\n\nWeakness:\nThere is no sim-to-real discussion in this paper or interpretation on why the controllers work. \nThe paper hints the implementation of automated morphology-control co-design, but it does not implement joint optimization. \nThe framework does not yet support general controllers that transfer across unseen shapes, each policy is trained per-shape from scratch. Also, the performance of the controllers under different conditions (eg disturbances) would be another interesting evaluation \nThere is no discussion on the complexity and/or computational cost of the approach, or comparison with state of the art across these lines. This is a big challenge for the field as far as I understand.", "was_revised": true, "final_justification": "Good contribution.", "rebuttal": "We thank the Reviewer for the positive evaluation and for providing detailed, constructive feedback on our work. We are encouraged by the reviewer's assessment and have found the suggestions regarding generalization, robustness, computational cost, and reward design to be very helpful for improving our manuscript.\n\n**[W1] Interpretation on Why the Controllers Work**\n\nOn a physical level, the controllers work by discovering and executing time-varying body deformations that generate effective fluid-structure interactions for propulsion. Our analysis shows that the reinforcement learning agent, through trial and error, learns to create kinematic patterns that correspond to well-understood principles of thrust generation in fluids:\n\n- For our **undulatory swimmers (eel and fish models)**, the learned policies generate traveling waves along the body that shed vortices in a manner consistent with a **reverse von Kármán vortex street**. This specific wake signature is a known indicator of net thrust production [1].\n- For our **jellyfish model**, the controller learns to produce rhythmic bell contractions that create and shed a **starting vortex ring** with each pulse. This generates an impulsive jet of fluid, propelling the body forward, which is the canonical mechanism [2] for pulsatile swimmers.\n\nTherefore, the interpretation is that our framework enables the agent to learn the fundamental physics of fluid-based locomotion from scratch, successfully identifying gaits that manipulate pressure fields and vortex dynamics to its advantage.\n\n---\n\n**[W1] Discussion on Sim-to-Real Transfer**\n\nWe thank the reviewer for this question. Sim-to-real transfer for soft robots in fluid is a big challenge for the field, requiring a multi-disciplinary effort that extends beyond the scope of a single work like ours. Bridging the sim-to-real gap is a well-known challenge for the field, involving significant hurdles in system identification, manufacturing inconsistencies, and sensor/actuator dynamics. Our work focuses on the critical prerequisite of establishing a robust simulation framework to discover optimal policies under idealized conditions, providing a vital baseline for future work dedicated to this challenge.\n\n---\n\n**[W2] Lack of Joint Optimization**\n\nThank you for your comment. Co-designing soft robots is an important emerging topic, yet it faces challenges in scaling to high-fidelity 3D simulations and in developing unified design representations. While existing works have made valuable progress, they often operate within certain constraints, such as simplified 2D design space (e.g., EvoGym [3]) or challenges in efficiently discovering controllers for complex 3D environments (e.g., SoftZoo [4]). Although our work focuses on controller optimization only, the framework we have developed shows a promising sign of tackling both of these challenges, providing a unified control representation within a high-fidelity 3D environment. We will tone down the co-design claim in our manuscript and revise its text accordingly to better clarify the position of our work.\n\n---\n\n**[W3, Q1] General Controllers across Unseen Shapes** \n\nWe thank the reviewer for raising this important point. While significant progress has been made in learning a Universal Controller for different rigid robot shapes in terrestrial locomotion tasks (e.g., MetaMorph [5]), transferring this capability to soft-bodied agents in fluid environments introduces substantial new difficulties due to the infinite-dimensional nature of soft-body deformations and complex fluid interactions.\n\nTackling this infinite-dimensional challenge requires a **compact, low-dimensional representation** that can jointly encode a soft agent's morphology and its actuation across diverse robot shapes. We believe this is the key to solving the generalization problem. Our work offers a promising approach for the actuation component of this representation. The proposed automated, universal method for defining control points can be applied to general soft-body mesh. This could be combined with established techniques for morphology representation (e.g., graph neural networks or point clouds) to form a complete, universal input for a generalized control policy.\n\nTherefore, while our current work focuses on per-shape training, we see it as providing a promising component needed to develop universal controllers across various shapes in the future.\n\n---\n\n**[W4]** Performance under Disturbance\n\nThank you for suggesting this study. To address this, we have conducted new experiments to evaluate the robustness of our learned controllers. \n\nWe tested robustness in a variant of the 2D trajectory tracking task (see Section 4.3 in supplementary materials for details): a straight-line moving target point guided the fish to a fixed goal while random lateral flow disturbances (width ≈ 0.3 body lengths) were applied. We define the following success rate to measure the swimmer's performance under disturbance and higher success rate indicates the policy is more capable of resisting disturbance: a trial is considered successful if the agent manages to reach the target point (defined as being within a distance of 0.2 units) within the episode's time limit after being disturbed. We conducted **50 independent trials** with randomized disturbance positions to ensure a robust evaluation. Across 7 disturbance velocities  reported in the table below, success rates remained high at 96% for disturbance velocity as large as 0.067 (c.f. the fish's swimming speed ≈ 0.05) and 68% at 0.1. These results indicate our trained controller's robustness under moderate disturbance. \n\n|Disturbance Velocity| 0.0 |0.033|0.067|0.1|0.133|0.167|0.2|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|Success Rate|100% |100%|96%|68%|42%|14%|4%|\n\nThe current validation was performed on a 2D navigation policy trained in a quiescent fluid environment (see Section 4.3 in supplementary material). We expect that incorporating disturbances directly into the training curriculum, for instance through domain randomization, would likely lead to even greater policy robustness.\n\n---\n\n**[W5 Q2] Computational Cost and Training Speed**\n\nOur simulator achieves frame rates of approximately 70 FPS for 2D morphologies and 30 FPS for 3D morphologies. Given our specified time step and episode duration (detailed in Supplementary Section 3), a single episode completes in approximately 85 seconds for 2D cases and 100 seconds for 3D cases. For comparison, at an equivalent fluid resolution, the DiffMPM-based simulator used in SoftZoo[4], one of the SOTA works on soft swimmers, requires approximately 350 seconds to complete a forward simulation of the same duration for its 3D scenes.\nAs for the training time, a complete training run for one agent, consisting of 75000 environment steps (equivalent to 3 million simulation steps), typically takes 3-4 hours with parallel environments on 8 GPUs.\n\n---\n\n**[Q3] Reward Function and Weights**\n\nEach reward functions has a clear physical motivation: \n- **The smoothness penalty** is the difference of action vectors between adjacent frames, encouraging more continuous actions, a common technique in robotics to reduce jittery motions.\n- **The action penalty** is the norm of the action vector that prevents excessive deformation.\n\nWe will add more details on them to our supplementary material.\n\nTo determine proper reward weights, we began by tuning the primary task reward to achieve basic locomotion, then incrementally adjusted the regularization weights to refine the emergent gaits—for instance, increasing the smoothness penalty to mitigate jittery movements. This process led us to a set of baseline weights (w_s = -0.02, w_a = -0.01) that produced stable and effective gaits, which we then used consistently across all experiments for fair comparison.\n\n---\n\n**[Q3] Poor Tuned Reward Weights**\n\nYes, a poorly tuned reward can limit the emergent strategy. **Extreme reward weights lead to degenerated, albeit explainable solutions**. An excessively high action penalty, for example, teaches the agent to remain nearly stationary, a trivial locomotion strategy that correctly minimizes the action penalty. Such cases are usually straightforward to identify and fix during initial tuning and confirm that the reward terms function as intended.\n\nWe are truly grateful for your insightful comments and constructive feedback. We hope that the additional details and experimental results we provided have effectively addressed your concerns. Please feel free to let us know if there are any remaining concerns. We look forward to hearing your thoughts on our rebuttal.\n\n---\n\nReferences:\n\n[1]. Triantafyllou, M. S., Triantafyllou, G. S., & Yue, D. K. (2000). Hydrodynamics of fishlike swimming. Annual review of fluid mechanics, 32(1), 33-53.\n\n[2]. Lin, Z., Hess, A., Yu, Z., Cai, S., & Gao, T. (2019). A fluid–structure interaction study of soft robotic swimmer using a fictitious domain/active-strain method. Journal of Computational Physics, 376, 1138-1155.\n\n[3]. Bhatia, J., Jackson, H., Tian, Y., Xu, J., & Matusik, W. (2021). Evolution gym: A large-scale benchmark for evolving soft robots. Advances in Neural Information Processing Systems, 34, 2201-2214.\n\n[4]. Wang, T. H., Ma, P., Spielberg, A. E., Xian, Z., Zhang, H., Tenenbaum, J. B., ... & Gan, C. (2023). SoftZoo: A Soft Robot Co-design Benchmark For Locomotion In Diverse Environments. In The Eleventh International Conference on Learning Representations.\n\n[5]. Gupta, A., Fan, L., Ganguli, S., & Fei-Fei, L. (2022). MetaMorph: Learning Universal Controllers with Transformers. International Conference on Learning Representations."}, {"review_id": "7G6bjK0lyt", "reviewer": "Reviewer_SDpb", "summary": "This paper introduces a unified learning framework for controlling free-form soft-bodied swimmers by combining a reduced-mode internal actuation model with a GPU-accelerated fluid-structure simulator based on the Lattice Boltzmann Method (LBM). The system learns swimming policies through reinforcement learning (SAC) in a morphology-agnostic control space, achieving notably strong performance across a wide range of both bio-inspired and abstract 3D morphologies. Notably, the method does not rely on predefined muscle layouts or hand-crafted actuation heuristics, enabling the emergence of plausible swimming behaviors through physically grounded simulation.", "questions": "1.\tBehavioral Diversity: Can your framework support learning of multiple distinct swimming strategies (e.g., hovering, turning, darting) within a single morphology? Would introducing multi-task learning or a goal-conditioned policy allow this? Demonstrating this capability would significantly strengthen your claims of biological plausibility.\n2.\tGait Transitions: Is it possible to transition smoothly between different learned gaits within the same controller or policy space? If not, do you foresee how your reduced-mode control representation might be extended to support this?\n3.\tBiological Framing: Could you clarify your use of phrases like “rediscovering evolved behaviors”? As it stands, the methodology does not explicitly model evolutionary processes. Framing this as emergent similarity rather than discovery could help avoid misunderstanding.\n4.\tCo-design Potential: You mention future work on joint morphology-control optimization. Do you believe your current control representation is compatible with a differentiable or evolutionary co-design pipeline?\n5.\tAdditional Tasks: Have you considered extending your experiments to include non-trivial objectives such as turning, avoiding obstacles, or energy-efficient locomotion? Such tasks would better demonstrate policy generalization and control flexibility.", "limitations": "While the paper discusses some limitations, such as the absence of joint morphology-control co-design, several important points warrant further consideration. First, the method is restricted to learning a single locomotion strategy per morphology and does not support multi-modal behaviors or context-driven gait transitions—key features of biological locomotion. Second, the task setup is narrowly focused on forward swimming, limiting the evaluation of adaptability or robustness across diverse goals. Third, morphologies are fixed and not co-optimized with control, missing the opportunity to study the interdependence of structure and behavior. Finally, the method is evaluated entirely in simulation without addressing real-world deployment challenges such as actuation delay, sensor noise, or sim-to-real transfer. While the framework is promising, acknowledging these aspects would provide a more balanced view of its current scope and future potential.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n•\tThe integration of geodesic-based sampling, linear blend skinning (LBS), and LBM-based fluid simulation forms a well-designed and efficient pipeline for learning physically plausible control policies in fluid environments.\n•\tThe method is applied successfully to a broad set of morphologies, including highly unconventional topologies where baseline controllers struggle, demonstrating its robustness and generalizability.\n•\tThe learned controllers exhibit swimming behaviors reminiscent of biological strategies (e.g., undulation, pulsation) without requiring any explicit kinematic priors, which is an appealing result.\n•\tThe authors provide thorough ablation studies and quantitative comparisons to validate the design choices.\n\nWeaknesses\n1.\tLimited behavioral diversity and adaptability\nOne of the defining characteristics of biological swimmers is their ability to exhibit multiple distinct locomotion modes (e.g., hovering, darting, gliding) using the same body structure, and to transition between these modes smoothly in response to environmental changes.\nIn contrast, the proposed system is trained on a single task objective (forward swimming) and results in a single learned gait per morphology. There is currently no mechanism to support multi-modal behaviors or gait switching, which limits the biological plausibility and adaptive capacity of the framework.\n2.\tAmbitious biological framing not fully supported by methodology\nThe paper frequently refers to the \"rediscovery\" or \"understanding\" of evolved swimming strategies, but it does not model evolutionary or developmental processes. Morphologies are fixed, and control policies are optimized independently for each shape. While the resulting behaviors are certainly interesting and sometimes biologically plausible, these claims may be overstated given the actual scope of the method. A more modest and precise framing would strengthen the overall presentation.\n3.\tLimited novelty in methodology\nThe proposed method combines existing components: reduced-mode actuation via LBS, geodesic sampling, LBM-based simulation, and standard SAC reinforcement learning. These are all well-established techniques. The originality lies more in their practical integration than in introducing fundamentally new algorithms or theoretical insights.\n4.\tSimplified task setup\nAll training scenarios focus on a forward swimming task with a fixed target direction and reward function. There is little exploration of more complex objectives such as turning, obstacle avoidance, or energy-efficient maneuvering. This restricts the evaluation of behavioral richness and policy flexibility, and may limit the method’s applicability to real-world adaptive tasks.", "was_revised": true, "final_justification": "The primary concern I had has been sufficiently clarified in the rebuttal. Accordingly, I have decided to update my evaluation to 'accept", "rebuttal": "We sincerely thank the reviewer for the thorough and constructive feedback on our manuscript. \n\n**[L2] Focused on Forward Swimming**\n\nWe would like to clarify that our framework indeed supports more tasks than just forward swimming:\n\n- **Navigation Task**: Our supplementary material and video already present this task on both our 3D jellyfish and 2D fish morphologies. This task demonstrates the agent's **adaptability** by requiring it to **dynamically adjust** its swimming gait to pursue a series of on-the-fly generated targets, showcasing its ability to handle **diverse goals** within a single episode.\n- We have also conducted two new experiments on **energy-efficient swimming** and **flow resistance**, which we will detail in our subsequent responses.\n\nIn summary, our framework is not limited to forward swimming. These experiments demonstrate its capability to train policies that exhibit adaptability and can achieve diverse goals.\n\n---\n\n**[W1/Q1/L1] Multiple Learned Gaits on Single Morphology**\n\nWe thank the reviewer for this excellent point on behavioral diversity. We believe our **navigation task** already demonstrates the adaptive capability in question. In this task, the learned policy on the 2d fish morphology exhibits **context-driven gait switching**: it naturally adopts a **turning** gait that exhibits **asymmetric** deformation to turn towards a target, then seamlessly transitions to a **darting** gait that exploits **symmetric** deformation for forward propulsion (Figure 5 in supplementary material and fish2d.mp4 in the media attached).\n\nThis adaptive behavior emerges from a single, simple reward function, which shows a promising sign that our framework's capacity to learn multiple strategies and seamless switching without explicit programming for each. \n\nFurthermore, to clarify the \"single task objective\" point, our new **energy-efficient swimming** experiment demonstrates the objective's flexibility. By introducing an energy penalty into the reward, we elicit distinct gaits from the same morphology: one optimized for maximum speed and another for energy efficiency. This shows our framework can produce a spectrum of behaviors by simply adjusting the task objective.\n\n---\n\n**[Q2] Smooth Transitions between Different Gaits**\n\nAs we explained in our response to [W1/Q1/L1] above, our navigation task demonstrates that a single policy can indeed produce smooth, context-driven gait transitions.\n\nWhen the target deviates from the agent's path, the policy generates an **asymmetric gait**: it produces a stronger body bend towards the target and a weaker counter-stroke, creating the necessary yaw torque to change direction. Once aligned, the policy seamlessly transitions back to a **symmetric, rhythmic gait** to maximize forward propulsion.\n\nOur **reduced-mode control representation** is particularly well-suited for this. It parameterizes the holistic deformation of the body, allowing the policy to learn to smoothly modulate just a few control parameters to create either the large-scale asymmetric bends needed for turning or the balanced oscillations for efficient forward swimming. This demonstrates that our representation is sufficiently expressive to support this form of adaptive control.\n\n---\n\n**[W4/Q5] Simplified Task Setup / Additional Tasks**\n\nWe thank the reviewer for the suggestions, which highlight the importance of demonstrating our framework's versatility on more complex tasks. To this end, our experiments now cover three such tasks:\n- **Navigation**: Our navigation task, already present in the supplement, directly addresses the turning task requested in this question. The agent learns to dynamically adjust its gait to follow the target, demonstrating adaptive turning.\n- **Energy-Efficient Locomotion**: We have conducted new experiments where an energy penalty is added to the reward. The agent successfully learns a distinct, smoother gait that minimizes the energy cost, showcasing its ability to optimize for efficiency.\n- **Flow Resistance**: We have added a new task where the agent must maintain its position against a persistent fluid flow, following DiffAqua [1]. This demonstrates the policy's ability to generate continuous corrective forces to maintain stability.\n\nFor a detailed analysis of the energy-efficient locomotion task, we respectfully refer the reviewer to our response to **Reviewer Z1aA [Q2]**, where this is addressed in full.\n\n**Flow Resistance**\n\nWe trained a 2D fish policy to maintain its position against a persistent head-on fluid flow. The policy was rewarded for minimizing its distance from the starting point. The learned policy demonstrated good stability.\n\n|Policy|Average Deviation (m)|\n|---|:---:|\n|**Trained**|0.030|\n|**Zero Action**|0.191|\n|**Random**|0.226|\n\nThis shows the agent learned to generate persistent, continuous counter-thrust to actively resist the environmental disturbance.\n\n---\n\n**[W2/Q3] Biological Framing**\n\nThank you for this valuable feedback. Our biological connection is indeed qualitative: the learned behaviors match canonical swimming modes, such as the **undulation** in our eel/clownfish models and the **pulsation** in our jellyfish model (Fig. 4). Crucially, these distinct modes emerged autonomously from our framework without explicit priors. We agree our use of \"rediscovering evolved behaviors\" was imprecise and, per your suggestion, will revise our text to focus on the emergent similarity of the behaviors rather than “discovery”.\n\n---\n\n**[W3] Limited Novelty in Methodology**\n\n While we employed quite a few established techniques, we believe the merits of our framework also include their novel adaptation and **non-trivial integration** of a system that balances the competing demands of physical fidelity and computational speed required for soft robot learning.\n\nOur methodological novelty lies in combining LBS with complementary dynamics. While LBS is a standard kinematic tool, we explore its application in a physics-based context where it defines a low-dimensional control space, and complementary dynamics ensure the resulting holistic deformations are physically plausible.\n\nFurthermore, the design decisions for assembling a complete system that supports RL for complex swimming behaviors are highly non-trivial. The community has explored several technical combinations, including CFD with A2C [2], differentiable MPM for solids and fluids [3], or simplified fluid models [4]. However, these approaches often present trade-offs that limit their use for the free-form soft swimmer control we demonstrate: full CFD is often too slow for RL, differentiable simulators can struggle with the long-horizon stability required to learn robust controllers, and simplified models lack the physical realism to capture complex hydrodynamics.\n\nTherefore, we believe that identifying and successfully integrating our specific combination of techniques is an original and significant result. It provides the community with a practical and effective platform that balances physical fidelity, computational efficiency, and generality, opening up new possibilities for research in this domain.\n\n---\n\n**[Q4] Co-design Potential**\n\nWe thank the reviewer for this question. We believe our representation is compatible with both co-design paradigms.\n\n**Integration into a Differentiable Pipeline**\n\nThe LBS computation is differentiable. By treating the control points as optimizable variables, our LBS-based kinematic mapping becomes a fully differentiable module. Furthermore, the complementary dynamics solver is also differentiable through standard adjoint methods. Therefore, our entire physics-based control module can be integrated into a gradient-based pipeline.\n\n**Integration into a Gradient-free Pipeline**\n\nOn the other hand, the representation is highly suitable for **gradient-free methods like evolutionary algorithms**. Its **compact and low-dimensional design** introduces minimal computational overhead. This efficiency is a significant benefit for population-based approaches, which require a large number of simulation rollouts to effectively search the design space.\n\n\n---\n\n**[L3/L4] Limitations on Co-design and Sim-to-Real Transfer**\n\nThank you for these valuable comments. We agree that joint co-optimization and sim-to-real transfer are both important and challenging avenues for future research. To provide the more balanced view you suggested, we will expand our conclusion to discuss these topics, framing them as key limitations and exciting directions for our work.\n\nWe are truly grateful for your insightful comments and constructive feedback. We hope that the additional details and experimental results we provided have effectively addressed your concerns. Please feel free to let us know if there are any remaining concerns. We look forward to hearing your thoughts on our rebuttal.\n\n---\n\nReferences:\n\n[1]. Ma, P., Du, T., Zhang, J. Z., ... & Matusik, W. (2021). Diffaqua: A differentiable computational design pipeline for soft underwater swimmers with shape interpolation. ACM Transactions on Graphics (TOG), 40(4), 1-14.\n\n[2]. Zhang, T., Tian, R., Yang, H., ... & Xie, G. (2022). From simulation to reality: A learning framework for fish-like robots to perform control tasks. IEEE Transactions on Robotics, 38(6), 3861-3878.\n\n[3]. Wang, T. H., Ma, P., Spielberg, A. E., ... & Gan, C. (2023). SoftZoo: A Soft Robot Co-design Benchmark For Locomotion In Diverse Environments. In The Eleventh International Conference on Learning Representations.\n\n[4]. Min, S., Won, J., Lee, S., Park, J., & Lee, J. (2019). Softcon: Simulation and control of soft-bodied animals with biomimetic actuators. ACM Transactions on Graphics (TOG), 38(6), 1-12."}]

Author rebuttal:

Meta-review: Most reviewers agree that the paper should be accepted.

The strengths of designing a new approximation for soft body control and faster GPU-accelerated outweigh the limited morphology diversity.
Decision: Accept (poster)