Context-aware Learned Mesh-based Simulation via Trajectory-Level Meta-Learning
Authors: Philipp Dahlinger
Niklas Freymuth
Tai Hoang
Tobias Würth
Michael Volpp
Luise Kärger
Gerhard Neumann
Venue: Accepted by TMLR ()

Abstract: Simulating object deformations is a critical challenge across many scientific domains, including robotics, manufacturing, and structural mechanics.
Learned Graph Network Simulators (GNSs) offer a promising alternative to traditional mesh-based physics simulators.
Their speed and inherent differentiability make them particularly well suited for applications that require fast and accurate simulations, such as robotic manipulation or manufacturing optimization.
However, existing learned simulators typically rely on single-step observations, which limits their ability to exploit temporal context.
Without this information, these models fail to infer, e.g., material properties.
Further, they rely on auto-regressive rollouts, which quickly accumulate error for long trajectories.
We instead frame mesh-based simulation as a trajectory-level meta-learning problem.
Using Conditional Neural Processes, our method enables rapid adaptation to new simulation scenarios from limited initial data while capturing their latent simulation properties.
We utilize movement primitives to directly predict fast, stable and accurate simulations from a single model call.
The resulting approach, Movement-primitive Meta-MeshGraphNet (M3GN), provides higher simulation accuracy at a fraction of the runtime cost compared to state-of-the-art GNSs across several tasks.

Published in Transactions on Machine Learning Research (01/2026)
Context-aware Learned Trajectory-Level Meta-Learning
Mesh-based
Simulation
via
Philipp Dahlinger
philipp. Segiigppddhlinngeeiaee edu
Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe
Niklas Freymuth
niklas. fielas.freymathh wwwphnntttkkktt... Skit. edu
Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe
Tai Hoang
taš. toi.hoong Skit. edu
Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe
Tobias Würth
tobias. wuerth @kitt edu
Institute of Vehicle System Technology, Karlsruhe Institude of Technology. Korlsruhe
Michael Volpp
voichee..oolpp@iee bow.llph.debbocc...oooooo bosch. com
Autonomous Learning Robots, Karlsruhe Institute of Technology, Karlsruhe Bosch Center for Artificial Intellignnce
Luise Kärger
kuise.uuerger Okit, edu
Institute of Vehicle System Technology, Korlsruhe Instibude of Technology, Korlsruhe
Gerhard Neumann
gerhard. neumann Gkit edu
Autonomous Learning Robots, Karlsrahe Institute of Technology, Karlsruhe
Reviewed on OpenReview: https: // opt.revieeeeeeeee net/ forum? id= j5uACS2Doh
Abstract
Simulating object deformations is critical challenge across many scientific domains, including robotics, manufacturing, and structural mochanics. Learned Graph Network Simulators (GNSs) offer promising alternative to traditional mesh-based physics simulators. Their speed and inherent differentiability make them particularly well suited for applications that require fast and accurate sinulations, such robotic manipulation manufacturing optimization. However, existing learned simulators typically rely on single-step observations, which limits their ability to exploit temporal context. Without this information, these models fail to infer, e.g., material properties. Further, they rely on auto-regressive rollouts, which quickly accumulate error for long trajectories. We instead frame mesh- simulation as a trajectory-level meta-learning problem. Using Conditional Neural Processes, our method enables rapid adaptation to simulation scenarios from limited initial data while capturing their latent simulation properties. We utilize movement primitives to directly predict fast, stable and accurate simulations from a single model call. The resulting approach, Movement- primitive Meta-MesIIGRAPILNN (M3GN), provides higher simulation accuracy at a fraction of the runtime cost compared to state-of-the-art GNSs across several tasks.
1 Introduction
The simulation complex physical systems is crucial to a wide variety engineering disciplines, including solid mechanics (Yazid al., 2009; Zienkiewicz Taylor, 2005; Stanova et al., 2015), fluid dynamics (Chung, 1978: Zienkiewicz al., 2013; Connor & Brebbia, 2013), and electromagnotism (Jin, 2015; Polycarpou, 2022; Reddy, 1994). In particular, the simulation of object deformations under external forces finds widespread

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Time
Context Data
Prediction
Shared
Shared Encoder
Encoder
Encoder
Latent Task Description Xe
ProDMP Weights
ProDMP Trajectory Generator
Simulator GNN
Context Aggregation
Figure 1: Movemenn--rimiiiie Mela-MEsHIGAPIING (M3GN) aims to learn accurate simulation dynamics from a few initial observations, enabling it to infer material properties from limited historical data. Given a context set initial system states, node-level latent features are computed for every pair of states using a shared Graph Neural Network (GNN) encoder. These features are then aggregated to form a node-level latent. task description Zy. This description concatenated with the last system state predict Probabiliste Dynamic Movement Primitive (ProDMP) weights, which are used compute per-node trajectories.
application in, e.g., robotic applications (Scheikl et al., 2022; Wang Zhu, 2023; Linkerhägner al., 2023). Mesh-based simulations are appealing for such problems due to the computational efficiency and accuracy of the underlying finite element method (Brenner & Scott, 2008; Reddy, 2019). However, the diversity of the problems to be modeled usually demands task-specific simulators in order to accurately capture the relevant physical quantities (Reddy & Gartling, 2010). Such specialized simulators can be slow and cumbersome to use, especially for large-scale simulations (Paszynski, 2016; Hughes et al., 2005).
Thus, data-driven surrogate models trained on reference simulations have become an appealing alternative (Guo et al., 2016; Da Wang al., 2021; Li al., 2022). Among them, general-purpose Graph Network Simulators (GNSs) have recently become incrcasingly popular (Battaglia et al., 2018; Pfaff et al., 2021; Allen et al., 2022b; 2023; Lippe et al., 2023; Linkerhägner et al., 2023; Yu et al., 2024; Würth et al., 2025). GNSs encode the simulated system as graph of interacting entities whose dynamics are predicted using GNNs (Bronstein et al., 2021). GNS are one to two orders magnitude faster than classical simulators (Pfaff ct al., 2021) while being fully differentiable. These properties make them highly effective for, e.g., inverse design problems (Allen et al., 2022b; Xu et al., 2021), robotics (Shi et al., 2023; Hoang et al., 2025), and other engineering applications that benefit from fast simulation surrogates (Simon, 2021). However, existing ENSS-assed models rarely exploit historical context, limiting their ability to infer material properties long-term system behavior.
Consider, for example, robotic manipulation task, where the robot needs to maintain accurate model of some deformable object with unknown material properties to achieve certain goal (Antonova et al., 2022: Shi et al., 2023; Hoang et al., 2025). Current learned models rely mainly on the current observation, preventing them from using the historical context infer knowledge about the material or other relevant. bchaviors (Linkerhägner et al., 2023). To alleviate this issue, we propose provide a few initial mesh states to learned simulators, allowing the model to observe how the given system behaves before having to simulate subsequent steps. Equipped with this knowledge, the model may infer latent material properties, facilitating for more accurate simulations that precisely extend the previous context. To the best of our knowledge, we are the first to study this specific experimental setup.
This setting naturally fits into a meta-learning framework (Schmidhuber, 1992; Thrun & Pratt, 1998; Vilalta & Drissi, 2005; Hospedales et al., 2022), where the sequence initial states induces the unique task of predicting its subsequent simulated trajectory. Concretely, we employ Conditional Neural Processs (CNPs) (Garnelo et al., 2018a) to aggregate the provided context sets and the dynamics inferred from them into latent.
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descriptor, which is then used to predict the rest of the trajectory. This formulation reveals opportunities to better adapt the standard GNS training setup to meta-learning scenarios, particularly when models must. condition on initial context.
GNSs are typically trained through simple next-step supervision (Battaglia et al., 2018; Pfaff et al., 2021; Allen et al., 2023). During inference, entire trajectories are simulated by iteratively predicting per-node dynamics from an initial system state in an autoregressive manner, only ever considering the previous system state or fixed-size history. This approach is prone to error accumulation over time, decrcasing accuracy for longer time horizons (Brandstetter et al., 2022; Han et al., 2022; Radler et al., 2025).
Since the initial context set influences the full trajectory, we decide to utilize trajectory-level predictions, where our model consumes the context steps and directly outpuls the remaining simulation in a single forward pass. This formulation offers several advantages over alternatives such as recurrent architectures (Hochreiter & Schmidhuber, 1997; Ruiz et al., 2020; Mienye ct al., 2024) or multi-step training (Shi et al., 2023). By predicting the entire trajectory in single forward pass, we improve training stability and memory elficiency by avoiding gradient updates across multiple passes. This single- pass approach also increases inference speed and completely sidesteps the error accumulation common in autoregressive rollouts.
To this end, we employ node-level Probabilistic Dynamic Movement Primitives (ProDMPs) (Schaal, 2006; Paraschos et al., 2013; Li et al., 2023), which allow us to output complete trajectories using only the latent descriptor as inputs. By representing trajectories with basis functions, ProDMPs neither require autoregressive rollouts, which is costly and error-prone for long horizons, nor large memory for constructing temporal convolutions, unlike prior works (Dahlinger et al., 2025; Xu et al., 2024; Cini et al., 2025). These components enable efficient training and rapid adaptation to trajectory-speeifie simulation parameters, such as unknown material properties, without requiring explicit parameter knowledge during training inference.
The resulting method, called Movement- Primittii Meta-MesHGGRPPINN (M3GN), allows the generation of context-dependent simulation trajectorics that accurately infer and integrate unknown system propertics. Figure provides an overview our approach, while Figure 2 shows examples for different tasks. To validate the effectiveness of M3GN, we adapt existing experiment suites (Linkerhägner et al., 2023; Dahlinger et al., 2025) to our trajectory-based meta-learning sctup, and additionally introduce two novel tasks based challenging deformable object simulations with varying object materials. Our results show that our method provides superior simulation accuracy compared to several variants of MESHCRAPPHET (MGN) (Pfaff et al., 2021) and recent trajectory-based learned simulators (Xu et al., 2024; Dahlinger et al., 2025)². Further, M3GN's ProDMPs trajectory reprosentation reduces the number model calls at inference, improving inference runtime by up to 32 times compared to MGN.
In summary, we propose M3GN, novel GNS that (i) extracts a latent descriptor from initial states and uses it to generate the remaining node-level trajectory in single shot, while rapidly adapting to varying material properties; (ii) achieves srate-of-the-art inference speed by coupling CNP-based context aggregation with a physically consistent trajectory formulation via ProDMPs; and (iii) surpasses recent GNSs on challenging deformation benchmarks, yielding superior long- long-horizon accuracy and stability. To support our claims, we introduce a new experimental setup in which the initial part of each trajectory serves as context data for the model.
2 Related Work
Graph Network Simulators. Deep neural networks for physical simulations can provide significant speedups over traditional simulators while being fully differentiable (Pfaff et al., 2021; Allen et al., 2022a), making them a natural choice for applications like boseel Reinforcement Learning (Mora et al., 2021) and Inverse Design problems (Baqué et al., 2018; Durasov et al., 2021; Allen et al., 2022a). A popular class of Icarned neural simulators are Graph Network Simulators (GNSs) (Battaglia et al., 2016; Gonzalz et al., 2020). GNSs utilize Message Passing Networks (MPNs), a special type of GNN (Scarselli et al., 2009; Bronstein et al., 2021) that representationally encompasses the function class of many classical solvers (Brandstetter et al.,
1 Coode is provided in the supplement. Here, the reviewers can also find videos showing the qualitative results of M3GN predictions and comparisons to existing baselines.
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Sheet Deformation
Deformable Block
Tissue Manipulation
Falling Teddy Bear
Final M3GN simulation steps for different evaluation tasks. From left to right: sheet deforms under two orthogonal forces, a falling collider deforming a block in 2D, a surgical tool dragging tissue, and a falling teddy bear. All visualizations present the predicted mesh (bluc) alongside a reference wireframc (red) of the ground-truth simulation. M3GN takes the deformable object positions from few initial time steps predict the remaining simulation steps using per-node movement primitives.
Figure 2:
2022). GNS handle physical data by modeling arbitrary entities and their relations as graph. Applications of GNSs include particle-based simulations (Li et al., 2019; Sanchez-Gonzalez et al., 2020; Whilney et al., 2023), atomic force prediction (Hu et al., 2021), and fluid dynamic problems (Brandstetter et al., 2022). These models have additionally been applied the mosh-based prodiction of deformable objects (Pfaff et al., 2021; Weng et al., 2021; Han al., 2022; Fortunaoo et al., 2022: Linkerhägner et al., 2023). Recent extensions handle rigid objects (Allen et al., 2022b; 2023; Lopez-Guevara et. al., 2024) and integrate learned adaptive meshing strategies (Plewa et al., 2005; Freymuth et al., 2023; 2025) into the simulator (Wu et al., 2023).
Existing work that considers unknown material properties in simulations of deformable objects combines the GNS prediction with point cloud information to improve long-term predictions Linkerhägner et al. (2023). This method requires a constant stream of point clouds to ground the simulation in, but can not aggregate this information into a description of the material properties. Additionally, the DEL method (Wang al., 2024) integrates physical priors from the Discrete Element Analysis (DEA) framework with learnable graph kernels, addressing the challenges of simulating 3D particle dynamics from 2D images.
In the context of larger-scale simulations, foundation models are gaining traction in neural simulation tasks, as exemplified by Aurora (Bodnar et al., 2024), large-scale model trained on extensive climate data. While Aurora demonstrates impressive performance on atmospheric predictions, including global air pollution and weather forecasts, requires significantly more data for fine-tuning compared to our approach, which focuses on efficient adaptation with fewer data points. Notably, all previously mentioned GNSs predict system dynamics iteratively from a given state, whereas we directly estimate entire trajectories, improving rollout stability and reducing function calls. Related to approach is the Equivariant Graph Neural Operator (EGNO) (Xu et al., 2024), which also predicts full trajectories using SF(() equivariance to model 3D dynamics and capture spatial and temporal correlations.
All previously discussed approaches rely on supervised learning or fine-tuning large foundation models, whereas we employ meta-learning to enable efficient adaptation to new simulation conditions. Recently, the Meta Neural Graph Operator (MaNGO) (Dahlinger et al., 2025) introduced meta-learning into Graph Network Simulators. Our work differs in several key aspects: MaNGO requires access simulation parameters at least for the training set, while method remains fully parameter-agnoste during both training and inference. Moreover, the meta-learning setup in MaNGO uses different simulation trials with the same material properties as context, whereas we exploit the temporal evolution within single simulation to infer latent. material characteristics. Finally, MaNGO's trajectory representation is less memory-efficient, as it. does not employ a compact formulation such as ProDMPs.
Meta-Learning. Meta-learning (Schmidhuber, 1992; Thrun Pratt, 1998; Vilalta & Drissi, 2005; Hospedales et al., 2022) extracts inductive biases from training set of related tasks in order to increase data efficiency
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unseen tasks drawn from the same task distribution. In contrast to other multi-task learning methods, such as transfer learning (Krizhevsky et al., 2012; Golovin et al., 2017: Zhuang al., 2020), which merely fine-tune or combine standard single-task models, meta-learning makes the multi-task setting explicit in the model architecture (Bengio et al., 1991; Ravi & Larochelle, 2017; Andrychowicz et al., 2016; Volpp et al., 2019; Santoro et al., 2016; Snell et al., 2017). This explicit architecture allows the resulting mela-models to learn how to learn new tasks from a small number of example contexts. popular variant is Mlodl--Agnsste Meta-Learning (MAML) (Finn al., 2017; Grant et al., 2018; Finn et al., 2018; Kim et al., 2018), which employs standard single-task models and formulates multi-task optimization procedure.
Neural Processes (NPs) (Garnelo et al., 2018a;b; Kim et al., 2019; Gordon et al., 2019; Louizos et al., 2019; Volpp et al., 2021; 2023) instead build on a multi-task model architecture (Heskes, 2000; Bakker & Heskes, 2003) but employ standard gradient based optimization algorithms (Kingma & Ba, 2015; Kingma & Welling, 2014; Rezende et al., 2014; Zaheer et al., 2017). Here, we use Conditional Neural Processes (CNPs) (Garnelo et al., 2018a), which aggregate learned features over a variable-sized context set to yield latent task description that our downstream GNS is conditioned on. Compared to regular NPs, CNPs assume a deterministic task description, eliminating the need for a distribution over latent variables. This assumption simplifies and accelerates the training process, as our objective is to predict a single precise simulation trajectory from the context set. While epistemic uncertainty cannot be fully eliminated, the simulation data itself is deterministic and lacks aleatoric noise, making a distributional latent formulation less appealing and inconsistent with the probabilistic assumptions that motivate Neural Processes.
3 Movement-Primitive Meta-MeshGraphNett
In this section, we present the theoretical foundation of the M3GN method, detailing the algorithmic design choices that guided its development.
Graph Network Simulators. Consider graph g (V, E, Xy,Xe) with nodes V, edges E, and associated vector-valued node and edge features Xy and Xg. An MPN (Sanchez-Gonzalez et al., 2020; Pfaff et al., 2021) consists of M message passing steps, which iteratively update the node and edge features based on the graph topology. Each such step is given as
h m++1 = fe" (h." h
hm++1 M (h
h h
E
where h and hed denote embeddings of the system state per node and edge at message passing iteration m, rospectively. & C & are the edges connected to v. Further, + denotes a permutation-invarinn aggregation operation such as the sum, the max, or the mean. The functions f& and f are learned Multilayer Perceptrons (MT.ss). The network's final output are the node-wise learned representations h, := hy that encode local information of the initial node and edge features.
Conventional GNSs encode the state of the simulated system graph, feed it through the MPN, and interpret the per-node outputs as velocities or accelerations (Pfaff et al., 2021). These dynamics are used to forward the simulation in time using, e.g., a forward-Euler integrator "Sanchez-Gonzalee et al., 2020). The graph encodes relative distances and velocities between entities instead of absolute ones, as the resulting equivariance to translation improves generalization next-step Mcan Squared Error (MSE) per node during training, adding carcfully tuned implicit denoising strategies (Pfaff al., 2021; Brandstetter et al., 2022) to stabilize long-term predictions. During inference, they compute trajectories by iteratively predicting and integrating their output in antoregressive fashion. If some simulated objects, like the collider, are known, only the remaining nodes are predicted. Our method instead uses ProDMP to predict a compact representation of a whole trajectory per system node.
Manchez-Gonzale et al., 2020). GNSs usually minimize a
Probabilistic Dynamic Movement Primitives. Movement Primitives (MPs) (Schaal, 2006; Paraschos et al., 2013) allow for compact and smooth trajectory representations y via set of basis functions parameterized by set of weights w. This temporal smoothness is highly beneficial for, e.g., robotic applications (Li et al., 2024; Otto et al., 2022). Recent methods integrate MPs with neural networks to enhance their
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expressive capabilities (Seker et al., 2019; Bahl et al., 2020; Li et al., 2023). Dynamic Movement Primitives (DMPs) (Schaal, 2006) use spring-damper dynamical system governed by parameters a and B. To manipulate the trajectory, an external forcing term f is added, before the system converges to a predefined goal g:
-O== = a (3 - v) Tÿ) f(x), f(x) = =
(1)
Here, T influences execution speed, while f depends on the basis functions &p in forco-space, the weights w and the exponential decaying phase x. Solving this equation typically is computationally intensive, particularly when the gradient dy/de dw is required (Bahl et al., 2020). ProDMPs (Li et al., 2023) instead solve Equation (7)
with pre-computed basis functions + in position-space as
(t) = (ty) + c2)2(1) + (()
The term (t)) + (232(() only depends on the initial conditions [y(4)), :(to]]. ProDMPs thus generate smooth trajectories at an arbitrary temporal resolution from oow-dimeniional weights w. They crucially allow for efficient gradient computation, and can respect different initial conditions such as positions or velocities. We provide extensive mathematical background of ProDMPs in Appendix A.
Meta-Learning and Graph Network Simulators. To enable generalization across tasks with varying properties, we frame GNS as a meta-learning problem. In this setup, each task corresponds to simulation of a deformable object with unknown material properties. The goal is to learn a simulator that can adapt quickly to a specific scenario using a limited amount of context data. Following the notation of Volpp el al. (2021), the meta-dataset D Di.L consists of simulation trajectories D; = {Gi, ....1...11r. Gr.h, where T is the trajectory length. Each simulation step Gt, = (mu,, un,z) represents a graph capturing both the deformable object mesh m4, (describing its position and topology) and an optional rigid collider ut,1- Physical proximity is used to define graph edges that model interactions between the deformable object and the collider. At test time, the first 7 < I simulation frames, G1,ITTs are observed as a context set to predict the remaining trajectory. Following prior work (Pfaff et al., 2021), we assume access to the full collider trajectory during prediction", resulting in the complete contect set:
Dj (9 Grr U (ui,r++,..... U
(2)
To provide clear reference point for discussion, define the anchor time step the final time step T of the context set. The corresponding anchor graph, GI,Te represents the system's 's state at this point and serves as the starting state for trajectory prediction by the GNSs. Notably, the anchor graph G,r alone does not capture the complete system state, as the material properties of the deformable object remain unknown. These properties must be inferred from the prior simulation steps, GI,.TT, to enable accurate simulation predictions. Figure illustrates this setup.
Model Architecture. Our model architecture, M3GN, is designed to learn from context data and predict future simulation steps by leveraging combination of graph network simulation and meta-learning techniques. The architecture consists of two parts: the computation of the latent task description from the context dala and the actual graph network simulation of future simulation steps. We base our context processing on the Conditional Neural Process (CNP) (Garnelo et al., 2018a), as it efficiently encodes a latent description over tasks given a set of context observations. Omitting the task index to avoid clutter, CNPs expect a context set {(z), (121.81)..... MI....., (ITO, (eT.)yrr)) }} consisting of inputs x, and corresponding targets ye. We translate our context set De from Equation 2 to this format by using each graph G. as an input, and setting its labels as the node-wise velocities. This approach allows the model to focus dynamics rather than absolute positions, which are more task-specific. Assuming a forward-Fules integration scheme with a time step of 1, we numerically approximate the velocities as the difference between the node positions of two consecutive simulation steps. The input graph æ, represents the simulation state at time step i, including mesh and collider, while yt encodes the change in positions between consecutive time steps.
3 Gr, yt = (G - (G.)
"This assumption commonly satisfied in practical sccuarios. For instance, in robotic planning tasks, the robot generates multiple candidate future end-effector trajectories and requires predictions the resulting deformation select. the most suitable plan.
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Archa
Prediction
Mesh
M3ON
Context
Ocein
Unused
Collider
Anchrox
Predicton
Mesh
MGN
Unused
Provided as Ground Truth
Collider
Time
Context Size
Context Size
Context Size
Figure 3: Left: M3GN and MGN task setup. Both methods predict mesh positions based on the initial mesh at the anchor time step. M3GN utilizes previous mesh positions and the last step of the collider trajectory for latent. task description, whereas MGN disregards past information and integrates the ground truth collider trajectory into its step-based model. Right: Exemplary final simulation steps on the Sheet Deformation task of M3GN given different context set sizes. A larger context size results in more accurate prediction (blue) of the ground truth (red wireframe) simulation.
To account for the known collider trajectory, we add its relative position as an additional node feature. Specifically, we include the position of the collider at the last time step T of the simulation, pos(ur), relative to its current position, pos(u,). Preliminary testing indicated that incorporating the complete future collider trajectory -.o(su) did not improve the results on our tasks. Given a context set De with anchor time stop T this results in T tuples (Ze.3r). A shared GNN encoder he computes node-level latent features
t,, ( y1) E R2T x|Vixd.
(3)
for each context time step with feature dimension dz. We then aggregate over the time domain of the context set to obtain 2, = + 2t, E R/x,,d, using + = max as the aggregation operator. Intuitively, 2, is a representation of the task inferred from the context data De, and encodes material properties, future collider movements, and high-level deformations of the simulation. While we use node-level latent fcatures for M3GN, one could additionally aggregate over the nodes to obtain graph-global lask descriptor We explore this choice and different aggregation functions D X in Section 4.
Zg Rdz
Once the task descriptor has been computed, it serves as the input to the predictive stage, enabling simulation of future trajectories. We concatenate the latent description 2, with the node features of the anchor graph Gre and subsequently use a GNN go to predict per-node ProDMP weights
w. = (GT=, zx) E I xd...
(4)
GNNs provide the flexibility to incorporate arbitrary node features. To better capture short-term dynamics, we augment the input with node velocities from the anchor time step (T"); the inclusion of velocity information
was determined through hyperparameter optimization on per-task basis. The ProDMP trajectory generator (w.) E [ Xdworld transforms the predicted outputs of the simulator GNN into per-node object trajectories over the entire simulation horizon. This approach can be seen as a form of temporal bundling (Brandstetter et al., 2022), requiring a single function call. In comparison, existing GNS train mostly on next-step dynamics and require one call per step during their auto-rogrossive inference schome (Pfaff et al., 2021; Allen et al., 2023). The trajectory-level view further allows us to omit noise injection during training, which MGN requires to generalize from learned next-step predictions to multi-step rollouts during inference. We provide a visualization of our model architecture in Figure 1 and refer to Appendix B for further details.
Meta Training. The goal of meta-learning is to automatically encode inductive biases towards the task distribution extracted from the meta-dataset D into the eask-global parameter O. To this end, we minimize the negative conditional log probability (Garnelo et al., 2018a)
(()) = - - R llo Po possmm....1) [Dj]]
5
Each training batch consists of a task Dj for which we sample a context size T uniformly between Thin and Tmax to ensure that the model learns to handle different context. set sizes. We then compute the latent task
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Figure 4: Comparison of the final simulation step between M3GN (blue) and MGN (orange) all datasets. From (Left) to (Right): (Top) Sheet Deformation and Deformable Block with an anchor time step of 2. (Bottom) Tissue Manipulation with context size of 5, and Falling Teddy Bear and Mized Objects Falling with an anchor time step of 20. M3GN provides much better alignment to the ground truth (red wireframe) simulation on all lasks, except for Tissue Munipulation, where MGN also solves the task well.
descriplor 2n and subsequently the predicted node Irajectories I ( zn)) as described in Equation 3 and Equation 4. The likelihood Po defined to the Gaussian
(o(pos(m..1:r) Dff N (pos(mu,,a))
(6)
Since the training simulations are not affected by noise, not modeling the output variance and set it to 1. Together with taking the mean the nodes and time steps stabilize training, optimizing the Gaussian log likelihood from Equation 6 is equivalent to minimizing the MSE
log (poo((mm)).))) Df) ~
(pos(mu..d... (9 zu)) zo))e...)
T dwoold b,0,
The whole architecture trained end-to-end using the loss C(0) from Equation After the meta-tras cett-training, fix 0, which now encodes inductive biases towards the meta-data D.
4
Experiments
Setup. We model mesh vertices as graph nodes and establish edges according the mesh topology, using additional edges between different objects based on Fuclidean distance. We employ one-hot encoding distinguish deformable objects from colliders, using homogeneous graph representation (Pfaff al., 2021). The edges additionally encode the relative distances between their connected nodes. Compared to existing work (Linkerhägner et al., 2023), we include small initial context sequence covering time steps 1,..., T for each simulation, with T being the anchor time step. Both the context and simulator MPNs use 15 message passing steps. Each message passing step uses separate 1-layer MLPs with a latent dimension of 128 and LeakyReLU activations for its node and edge updates.
We evaluate the Full-roliout MSE, computed the average simulation MSEs following all time steps after the anchor time step, and the Per-timestep MSE, defined as the MSE each individual time step. Both metrics are averaged over all trajectories in the test set. For each experiment, report the mean and bootstrapped confidence intervals (Agarwal et al., 2021) over 8 random seeds. We evaluate both metrics for various context. sizes ranging from 2 to 30 steps. Appendix C provides additional details on our experimental setup.
We validate our method on five different simulation datasets based on three different mesh-based physics simulators. These include Deformable Block (DB) task 2D and a 3D Tissue Manipulation (TM) task (Linkerhägner et al., 2023), generated using Simulation Open Framework Architecture (SOFA) (Faure et al., 2012). In both datasets, the Poisson's 's ratio (Lim, 2015) acts the randomized material property.
Datasets.
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M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO EGNO
Deformable Block
Sheet Deformatio (ID)
Sheet Deformation (OOD)
10.0
Full Rollout MSE
Full Rollout MSE
Full Rollout MSE
100000.0
30000.0
1030.0
10
15.0
1.0
Anchor Time Step
Anchor Time Step
Anchor Time Step
50.0
16-41
Tissue Manipulation
30.0
10-4
Falling Teddy Bear
16-3
Mixed Objects Falling
Full Rollout MSE
48.0
30.0
20.0
Full Rollout MSE
20.0
Rollout MSE
20.0
un
18g
3.0
4.0
2.0
3.0
2.0
1.
Anchor Step
Anchor Time Step
Anchor Time Step
Figure 5: MSE (on a log scale) over full rollouts for different methods across all tasks, including an additional plot for the Sheet Deformation task evaluated on an out-of-distribution (OOD) test. set of material properties Overall, M3GN steadily improves its performance when provided with additional context information. Our method generally outperforms both MGN variants, likely due to the Ma-based trajectory formulation and the latent material property representation, with the exception of the Tissue Manipulation task. Equivariant Graph Neural Operator (EGNO) exhibits instability for later anchor time steps and only performs competitively on the Sheet Deformation (ID) task.
Deformable Block simulates different trapezoids that are deformed by a circular collider with constant velocity and varying size and starting position. Each trajectory consists of mesh with 81 nodes that is deformed over 39 time steps. Tissue Manipulation considers surgical robotics scenario where piece of tissuc is deformed by a gripper. The gripper is attached to fixed object position and moves in a random direction with constant velocity. The mesh comprises 361 nodes and the simulation has 100 steps.
We further consider the Sheet Deformation (SD) datasct (Dahlinger et al., 2025), which simulates how a sheot deforms under two constant forces that are applied at different positions of the sheet plane. As the Young's modulus is varied between sheets, this dataset constitutes a simplified stamp forming process, as common mechanical engineering (Zimmerling al., 2022). The simulations are generated with Abaqus (Smith, 2009), comprising 50 time steps and a plate with 225 nodes. We test two different data splits: The in-distribution (ID) split tests on material properties that the model has seen during training, while the out-of-distributtio (OOD) split evaluates Young's modulus values outside the training domain.
The final two tasks place a randomly rotated deformable object at a specific height and let it fall to and collide with the ground. Falling Teddy Bear (FTB) considers the titular teddy bear as only object, whereas six different objects are considered for Mixed Objects Falling (MOF). Each trajectory assigns a random Poisson's ratio and random Young's modulus to the falling object, thus influencing its deformation upon contact with the floor. Each trajectory in the last two tasks consists of 200 time steps. The object meshes have up to 350 nodes and are shown in Figure 11 in the appendix. For simplicity, we only consider the triangular surface meshes for the experimental setup. All task spaces are normalized to []s, Appendix B.1 details the graph encoding, while Appendix D provides further information on dataset sizes and preprocessing.
Basclines and Ablations. We compare to MGN(Pfaff al., 2021), evaluating its performance both with and without additional material information provided as node feature. Importantly, we never supply this feature to M3GN.
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Figure 6: Per-timestep MSE for different methods across all tasks, showing the temporal evolution of prediction eπror averaged over all test trajectories. For DB, SD (ID), and SD (OOD), the context size is set to 2, whereas TM, FTB, and MOF use a context size of 20. The step-based MGN variants exhibit clear error accumulation over time, whereas M3GN maintains stable accuracy throughout the rollout likely due to its temporally consistent MP-based trajectory formulation.
MGN generates the next mesh state by iteratively predicting the velocities for the current simulation step. It is trained to minimize the 1-step MSE over node velocities, incorporating Gaussian input noise during training (Brandstetter et al., 2022). This noise serves to mitigate error accumulation and stabilize auto- regressive rollouts during inference. To ensure a fair comparison, we adopt the same hyperparameters as our method and optimize the imput noise level per task.
We also explore the effect of incorporating historical information, specifically previous velocities, as node features for both MGN and M3GN. For MGN, using both the current and previous velocities improves performance significantly on many tasks. For M3GN, including only the current velocity yiclds similar validation
benefits. Figure 12 in the Appendix presents the results of preliminary luning experiments split. Additionally, Table summarizes the specific history configurations and other relevant hyperparameter for each method and task. As a comparison to a graph-basod meta-learning method, we evaluate M3GN against the Meta Neural Graph Operator (MaNGO) (Dahlinger et al., 2025). To adapt MaNGO to our trajectory-based meta-learning setup, we modify its architecture accordingly. Specifically, we remove the MaNGO encoder, which aggregates information across multiple trajectories. Instead, we provide the context data as the initial position in the umprocessed input trajectory of the MaNGO decoder and repeat the anchor-timestey data until the end of the trajectory. MaNGO then processes this sequence using spatial message passing and temporal convolution to produce its predicted trajectory. We discard the predicted portion corresponding to the context time steps and use the remaining time steps as the MaNGO prediction.
As an additional baseline, we compare our approach to the Equivariant Graph Neural Operator (EGNO) (Xu et al., 2024), which employs equivariant messagc-passing layers and predicts the remaining simulation steps in single pass, closely aligning with our setup. However, training EGNO proved unstable with 15 message- passing steps, and the best results were achieved using only 5 steps. We hypothesize that this instability may stem from the longer prediction horizon of up to 200 steps in our experiments, as the bascline was originally evaluated on tasks with a much shorter prediction horizon of only 8 steps. Further details on the implementation of these baselines can be found in Appendix C.
We further study different design choices of M3GN on Sheet Deformation and Deformable Block. To investigate the effect of the meta-learning approach, we train an MGN (MP) variant that uses ProDMPs predictions, but omits context aggregation and thus has latent task description z Similarly, compare to
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Figure 7: MSE on a log scale over full rollouts for the Sheet Deformation (Left) and Deformable Block (Right) tasks for different meta-learning and MP variants. Using a ProDMP representation for MGN improves performance. CNPs and NPs with next-step prodiction do not improve over standard MGN. A NP instead of an CNP architecture for M3GN slightly reduces performance.
M3GN (Step-based), which performs next-step prediction of the dynamics instead of predicting ProDMP parameters, but otherwise follows the CNP training scheme to learn a latent task description. Finally, we compare the deterministic CNP approach to both MP and step-based probabilistic Neural Process (NP) approaches. Here, we get diagonal Gaussian distributions as the outputs of the context MPN, which we aggregate using Bayesian context aggregation (Volpp et al., 2021). We further investigate node aggregation of the latent task description is beneficial by applying maximum aggregation of the node features before the context aggregation. While standard CNPs require permutation- sermutation-invariant invariant context aggregation, has temporal structure. We thus experiment with small transformer model with 4 transformer blocks, 4 attention heads, temporal encoding and latent dimension of 32 as aggregator. The transformer takes the sequence of outputs of the context MPN and predicts the aggregated node-level task description Zv.
context
Main Results. Figure visualizes exemplary final simulation steps for M3GN and MGN for all tasks. M3GN aggregates context information to condition node-level ProDMP representations of the simulated trajectory. This approach leads to accurate simulations, providing much better alignment to the ground truth simulation than the step-based MGN on all tasks. Appendix E.4 shows visualizations of full simulation rollouts for all tasks and methods.3
Figure 5 provides the full-rollout MSE for M3GN, MGN, and MGN (with Material Information) across tasks. Adding material information improves performance only on the Deformable Block and Tissue Manipulation datasets. In contrast, on the Falling Teddy Bear and Sheet Deformation (OOD) tasks, MGN (with Material Information) tends to overfit, likely resulting in decreased performance when material information was provided. In general, using a later anchor time step improves performance across all methods, presumably due to a shorter prediction horizon. The main exception is EGNO Sheet Deformation (OOD), which becomes unstable for later anchor lime steps.
M3GN surpasses all baselines Deformable Block. For Sheet Deformation, M3GN and EGNO significantly outperform the other basclincs across context sizcs. Furthermore, M3GN generalizes well to unseen matorial properties the OOD setup, whereas, e.g., MGN and MaNGO fail to properly extrapolate. For Tissue Manipulation, the step- baselines slightly outerperform M3GN in terms of MSE. Yet, M3GN and MaNGO provide plausible and visually consistent simulations, with the step-based basclines primarily capturing finer details more accurately. In contrast, EGNO performs significantly worse in this setting, achieving an MSE that is an order of magnitude higher than that of M3GN.
On Falling Teddy Bear and Mized Objects Falling, the step-based MGN without material information performs well on the former but, along with the other step-based method, fails to provide accurate long ter-term simulations on the latter. For Mized Objects Falling, the predicted trajectories of both MGN variants qualitatively deviate from the ground truth. Both MGN and MGN (with Material Information) exhibit object drift or misalignment, e.g., between colliding bodies. In contrast, for M3GN, the ProDMP's temporally consistent movements
Videos of these simulations are the supplementary material.
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M3GN (Ours, No Node Agg., Max Context Agg.)
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Figure 8: MSE log scale over full rollouts for the Sheet Deformation (Left) and Deformable Block (Right) tasks for different context and node aggregation methods. The node-level maximum context aggregation of M3GN performs best for Sheet Deformation, while all methods work roughly equally well on the Deformable Block lask.
combined with context aggregation produce stable and coherent simulations, substantially improving over the step-based baselines. Examples of those behaviors are shown at the bottom of Figure 4. MaNGO achieves exceptional good results on Falling Teddy Bear, but it is surpassed by M3GN on Mixed Objects Falling. Interestingly, trajectory-based methods do not benefit. from later anchor steps on these tasks. We suspect the increase in MSE a consequence how the metric is averaged: early timesteps are trivial (the object is simply falling), so including them lowers the mean error. When anchoring later, these casy steps are removed, and the MSE is computed only over the harder, post-inpact. segment, leading to higher average error.
Additional Experiments. Figure 6 shows representative Per- timestep MSE results, while Appendix E provides the complete set of evaluations for different context sizes. These figures highlight the error accumulation common in step-based learned simulatiors. In
M3GN (Ours)
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contrast, M3GN maintains stable accuracy throughout the rollout. Sheet Deformation (ID), we observe a temporary increase in error for M3GN around time steps to This behavior seems to stem from rapid change in node velocities, which ProDMP representation smooths over. While additional basis functions would increase the ProDMP's capacity and thus likely mitigate this issue, we deliberately avoided excessive hyperparamcter tuning to ensure a fair comparison and comparable optimization budgets across methods.
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Next, Figure 7 finds that both MP representations and a meta- learning objective are crucial for accurate simulations. Interestingly, using NPs with ProDMPs only slightly degrades performance com- pared to M3GN, suggesting that the NP's latent distribution does not benefit our GNS setup. Figure 8 additionally compares different methods to aggregate the context set. The node-level maximum con- text aggregation works best for Sheet Deformation. For Deformable Block, there is no significant difference between aggregations, causing to use the comparatively simple node-level maximum context aggregation for all experiments.
Figure 9: Runtime comparison on four tasks between the learned methods and the different ground truth simulators. Note the log scale on the y-axis. The complete runtime and memory usage evaluation in Appendix, Table 3.
Figure 19 in the Appendix presents two additional ablation studics that further characterize the behavior of M3GN. We first investigate the stability of M3GN under noisy context observations and compare it against MGN as well as a variant of M3GN that does not receive any context data. While M3GN remains robust under moderate noise levels, we obscrve that the highest noise setting lcads to worse performance than the no-context variant, which we attribute overfitting to uninformative and misleading context signals. We further evaluate generalization across materials using dataset composed of five distinct material parameter
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settings by comparing it to specialized basclines. Five separate MGN models are trained, each specialized to single material and evaluated on unseen initial conditions with the same material properties, serving as an upper bound for single-material approaches. In contrast, a single M3GN model is trained on the combined dataset and must infer material propertics from the context set. For small context sizes, M3GN achieves performance comparable to the specialized MGN models, while for larger context sizes M3GN slightly outperforms MGN, demonstrating in total its ability to effectively leverage context information and generalize across materials within a unified model.
We additionally present visualization of the latent space for M3GN in Figure 20 in the Appendix, showing that simulations with similar material propertics are clustered together. This latent structure highlights the model's ability to differentiate between material behaviors while preserving the relationships between similar properties.
Figure compares the inference speed of M3GN that of the stop-based MGN. M3GN's ProDMP trajectory representation significantly decreases the amount of required model calls for the simulation, since, after encoding, single GNN forward pass can is used to compute the full trajectory. Additionally, the context set is easily encoded in parallel, resulting in relatively minor cost for the context computation and aggregation for all tasks. While MGN does not perform any context processing, it requires one forward pass per timestep, making its rollout inherently sequential. Together, these properties result in an inference-time speedup of up to a factor of 32 for M3GN compared to MGN (for the Tissue Manipulation task), and up to a factor of 700 compared to the ground-truth simulators (for the Sheet Deformation task). A complete comparison of execution time and memory consumption during inference is provided in Table in the Appendix. Improving over existing learned simulators by more than one, and over classical simulators by more than two orders of magnitude enables rapid development for downstream engineering applications, such manufacturing optimization.
5 Conclusion
We introduce Movement-Primitte Meta-MessHGRAPANNY (M3GN), a novel Graph Network Simulator that combines movement primitives with trajectory-level meta-learning for efficient and accurate long-term predictions in physical simulations. Our method dynamically adapls to available context information during inference, enabling accurate prediction of deformations under unknown object properties. Additionally, it effectively mitigates error accumulation while reducing the number of required learned simulator function calls. To validate the effectiveness of M3GN, we propose two new deformation prediction tasks with uncertain material properties. Results on these tasks and existing datasets demonstrate that method consistently outperforms two strong Graph Network Simulator baselines, even when the baselines are provided with oracle information about the material propertics.
Limitations and Future Work. Our method may struggle to capture rapid, high-frequency dynamics, as the smooth ProDMP formulation can lead to temporary errors when node velocities change abruptly. We also plan to integrate online re-planning of trajectories, predicting trajectory sogments with every model forward pass. This process may enhance coordination between simulated nodes across segments while maintaining the benefits of compact multi-step trajectory representation. In addition, the M3GN framework is not limited to deformation tasks; it can be extended to new settings such as fluid simulations other non-mesh-based enviromments, provided that an appropriate graph structure can be constructed from the underlying data.
Broader Impact Statement Our proposed Graph Network Simulator can positively impact various fields relying on computational modeling and simulation by significantly reducing computational cost compared to traditional simulators while providing accurate simulations. However, efficient and accurate simulation of physical systems also comes with potential negative impacts, such as, e.g., the development of advanced weapon models.
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A
Mathematical formulations of Movement Primitives
We provide an overview of the probabilistic dynamic movement primitives (ProDMP) formulations utilized in this paper, starting with the foundational methods: Dynamic Movement Primitives (DMPs) and Probabilistic Movement Primitives (ProMPs).
A.1 DMPs
Schaal (2006) introduced Dynamic Movement Primitives (DMPs), which integrate a forcing term into a dynamical system to generate smooth trajectories from given initial conditions", such as a robot's position and velocity at a particular time. A DMP trajectory is governed by a second-order linear ordinary differential
equation (ODE) as follows:
~ - v) -T)) f(x), f(x) Epi(x)e x E((())
(7)
where y = y(),, n = dy/ddt and n == ddyd/dd ddt denote the position, velocity, and acceleration of the system at a parameters, is the goal attractor, and T
specific time t, respectively. Constants a and / are is a time constant modulating the speed of trajectory execution.
The functions (a)) represent the basis functions for the forcing term, as shown in Fig. 10a, while the phase variable xxt) E [0, 1] captures the execution progress. The trajectory's shape is determined by the weight parameters E t for = ,...., N and the goal term g. The trajectory [ve]==o:r is typically computed by numerically integrating the dynamical system from the start to the endpoint. However, this numerical process is computationally expensive (Bahl et al., 2020; Li el al., 2023), as its cost scales with the trajectory length and the resolution of the numerical integration.
A.2 ProMPs
Paraschos et al. (2013) introduced the Probabilistic Movement Primitives (ProMPs) framework for modeling trajectory distributions, effectively capturing both temporal and inter-dimensional correlations. Unlike DMPs, which rely on a forcing term, ProMPs directly model the desired trajectory and its distribution using linear basis function representation. Given weight vector w a weight. vector distribution p(w) NNw Ew), the corresponding trajectory trajectory distribution is computed as follows:
Compute Trajectory:
[ull=-or = TT
(8)
Compute Distribution: M(ve)e=0::: My, Ey) = - TT
(9)
Here, the matrix & contains the basis functions for each time step E 7), shown in Fig. 10a. The trajectory shape is determined by the weight parameters w; E w through matrix-vector multiplication. Despite their simplicity and computational efficiency, ProMPs lack an intrinsic dynamic system, limiting their ability to specify a given initial condition for trajectory or predict smooth transitions between two ProMP trajectories with differing parameter vectors.
A.3 ProDMPs
Solving the ODE underlying DMPs Li et al. (2023) observed that the governing equation of DMPs, described in Eq. (7), admits an analytical solution. We re-express the original ODE from Eq. (7) and its homogeneous counterpart in standard ODE forms as follows:
aB
f(x)
Non-homo. ODE:
- 72 y
7 29= F(r,9), F(,
(10)
72
aB
Homo. ODE:
i - シサ 72 ==
(11)
'In mathematics, an initial condition refers the value of function derivatives at starting point, which can specified at any time, necessarily at. 0.
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2.0
1.0
DMPs ProMPs basis
PrDMPP Goal Basis
0.6
1.5
0.8
ProDMP weights
0.6
0.4
1.0
0.4
0.5
0.2
0.0
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Normalized Time
Normalized time
Normalized Time
(a) Basis functions of DMPs / ProMPs
(b) Positional basis functions of ProDMPs' weights
(c) Positional basis function of ProDMPs goal
Figure 10: Illustaaionn of basis functions used in MP methods. (a) Normalized radial basis functions used in DMPs in Eq-(7) and ProMPs in Eq-( respectively. (b) Positional basis functions of ProDMPs' weights and (c) ProDMPs' goal g in (17). In ProDMPs, g is concatenated with the weights vector and treated as one dimension of the resulting vector wy. Both weights and goal basis functions are computed from solving the DMPs' underlying ODE, following the procedure from (q. to (16)
The solution to this ODE is essentially the position trajectory, and its time derivative yields the velocity trajectory. They are formulated through several time-dependent function as:
w + C181 +
y [y2P2
V1P1
V292
91
(12)
9
w
1 1i2P 1/P1
d242 01
+ +ath + c2g2-.
(13)
99
Here, the learnable parameters [w, g]" which control the shape of the trajectory, are separable from the remaining time-dependent functions /1, 12. PL, P2, qi Q2. These functions computed by solving the ODE in Eq. (10), (11):
(t) exp (- -))
a 82(1) texp
t)
(14)
2r
(())) 72 /'- ('exp
2T
(('))
Pa(() 72
exp 2r
xx( x((')
(15)
(1(()
( ( 2T
1-
exp
2T
1,
(2(1) 2r
exp
2T
(16)
Here, the function 31, 3/2 are the complementary solutions to the homogeneous ODE presented Eq.(11), with in, d2 their time dcrivatives respectively.
It's worth noting that P1 and P2 cannot be derived analytically due to the complexity of the forcing basis terms Pa. Consequently, these terms must be computed numerically. However, isolating the learnable parameters, namely w and g, enables the reuse of other time-dependent functions across all generated trajectorics.
ProDMPs identify these reusable terms as the position and velocity basis functions, denoted by +bt) and (t), respectively. Fig. 10b and Fig. 10c illustrate the resulting position basis functions for the weights w and the goal 9. respectively. These functions are pro-computed offline and treated constants during online learning. When w and g are combined into concatenated vector, represented as wy: the position and velocity trajectories can be expressed in manner similar to that used by ProMPs:
Position: v(t) ot)) + (t)) + c2nn(t,
(17)
Velocity: i(t) Tt) + (t) + (t).
(18)
In the main paper, for simplicity and notation convenience, we use w instead of wy to describe the parameters and goal of ProDMPs.
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Trajectory's Initial Condition The coefficients 01 and 02 are solutions to the initial value problem defined by Eqs. (1)).. Assuming the trajectory starts at time 4, with position yb and velocity yb, we denote the values of the complementary functions and their derivatives at the condition time to as yı, W2,, 31, and v2,, Similarly, the values of the position and velocity basis functions at to are denoted as Py and o respectively. Using these notations, 61 and C2 are compuled as:
92, yo - 92, 8h Big Ung Dog 81b
M2, #1 U2, L Wig
91, V2, 92, Vih
(19)
C2
to U1, Mh. B1 912, 22,
Un, #I L Wig
91, 92, 916
Set Goal Convergence Relative to Initial Condition The goal attractor 9 in the ProDMPs framework represents an asymptotic convergence point for the dynamical system as t + oc, typically defined as an absolute coordinate. However, the goal term can also be modcled relative to the initial position ys- In this approach, the relative goal grel predicted, and its absolute counterpart computed as Jabs = grel + ybb This approach is particularly useful for predicting the goal in the coordinate system relative to a node's starting position. Since we aim to achieve translatiou-equiiariant approach (where absolute node positions are encoded as relative edge features between nodes), predicting relative goal positions aligns well with this design principle.
B Architecture and Method Details
This section offers detailed insights into our methodology and the architectural decisions guiding our approach.
B.1 Graph Encodings
In processing the imitial graph Gr,re, we create edges between the mesh and the collider based on a radius graph. Specifically, we connect mesh and collider nodes for Deformable Block and Tissue Manipulation if their euclidean distance is smaller than 0.3. In the Tissue Manipulation task, the collider given as a single node which is connected to the tip of the tissue. It marks the grasping point of a gripper. Tn the Sheet Deformation task, we add an additional node feature the nodes which get directly influenced by the external force. Therefore, no collider is used in this task. For the Falling Teddy Bear and the Mixed Objects Falling task we implicitly model the ground as collider by adding the current. position of every node to its node features Manchez-Gonzalee et al., 2018). This quantity gets updated for the step-baser methods.
B.2 ProDMP Details
Initialiaation of ProDMPs necessitates node velocities for the anchor time step TC, We employ a linear approximation, leveraging data from the previous time step T 1.
Similar to the relative encoding node positions in the MPN, we employ a technique in ProDMP to derive relative trajectories. Initially, we integrate a relative goal position as part of the node weights w,. Utilizing this approach, trajectories commence from the origin and traverse towards their respective relative goals. Subsequently, adjust all positions by the initial position. This strategy fosters model generalization across various nodes.
The parameter 7, as described in Equation 7, is learned globally across all tasks using a compact MLP. The model's final layer employs a scaled sigmoid function for parameter estimation.
C Experimental Protocol
In order to promote reproducibility, we provide details of our experimental methodology. Table presents the hyperparameters used in our experiments. For a comprehensive description of the creation of all datasets, please refer to Appendix D.
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The training took place of 48 hours, except for the Sheet Deformation task, where the computation budget was set to 24 hours. Consequently, the number of epochs varied, as the batching differed significantly between the trajectory-based method M3GN and the step-based MGN. We adapted the batchsize of the step-based methods in order to use the GPU memory efficiently. M3GN is always trained on one full trajectory per batch. Here, the whole context. processed in parallel and the remaining trajectory predicted and compared to the ground truth.
NVIDIA A100 GPU, with each method given the same computation budget
We conducted multi-staged grid-based hyperparameter search for the learning rate, input noise, and other hyperparameters the latent task description dimension. In general, we optimized all methods all tasks separately, however, we noticed that over different tasks and methods some parameters had the same best configuration. We did not use the lest data for this, but tuned all hyperparameters on a separate validation split. This split was also used to determine the best epoch checkpoint to mitigate any overfitting effects.
In the end, all methods worked well with a learning rate of 5.0 x 10-4 except in the Sheet Deformation task. Here, our hyperparameter optimization indicated that the trajectory based methods benefit from a smaller learning rate of 1.0 x 10-5. For MGN, experimented with different imput noise scales. Notably, for the Deformable Block and the Sheet Deformation task, smaller noise scale improved performance significantly. In the falling objects tasks, we also explored second-order predictions, such as node accelerations, instead of velocity predictions. Following the approach in Pfaff et al. (2021), we adjusted the labels accordingly and conducted preliminary evaluations. However, since direct velocity predictions yielded superior results, we opted for them as our final approach, as presented in the main paper.
C.1 EGNO Training
For the Equivariant Graph Neural Operator (FGNO) method, we used the original code from Xu et al. (2024) for the model implementation. Since EGNO can only predict for fixed next horizon, we cut the remaining prediction when using later anchor time step. This is done during training and evaluation.
C.2 MGN Training
We mainly follow Pfaff et al. (2021) for the training of the MGN baseline. The only difference is the incorporation of current and historic velocity node features. Pfaff et al. (2021) consider this in their experiments but they show in their experiment suite that it does not improve the results and can lead to overfitting. This is different to our results. For us, on all tasks except Mixed Objects Falling, adding the current and historic velocities of nodes improves the results. We follow the Gaussian random walk noise injection for the velocity features from Banchez-Gonzalee al. (2020).
D Datasets and Preprocessing information
In this section, we give detailed information about the datasets we used. We report a general overview of all datasets in Table 2. Here each dataset is abbreviated for brevily as the following:
Table 2 lists in detail the datasets used in the paper. Each dataset is abbreviated for brevity and explained as follows:
SD.: Sheet Deformation
DB: Deformable Block
TM.: Tissue Manipulation
FTB.: Falling Teddy Bear
MOF.: Mixed Objects Falling
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Table 1: Table listing the hyperparameters and configurations of the experiments
Parameter
Value
Node feature dimension
128
Latent task description dimension
64
Decoder hidden dimension
128
Message passing blocks
15
Message passing blocks (EGNO)
5
GNN Aggregation function
Mean
GNN Activation function
Leaky ReLU
M3GN Context Aggregation method
Max Aggr.
M3GN Latent Node Aggregation method
No Aggr.
Learning rale
5.0 10-4
Learning rate (Sheet Def. MP-basod methods)
1.0 10
Number of ProDMP basis functions
30
ProDMP
learned
range: [0.3,3.0]
ProDMP Relative start position
True
MGN input mesh noise
0.01
MGN imput mesh noise (Def. Block)
0.001
MGN input mesh noise (Sheet Def.)
0.0001
MGN history length (all tasks except Mized Objects Foll)
2
MGN history length (Mired Objects Foll)
0
M3GN history length (all tasks except Tiss. Man, and Sheet Def. (OOD))
I
M3GN history length (Tiss. Man. and Sheet Def. (OOD)) Aonimum/Maximin Train Context Size (Sheet Deformation) Minimum MaaimmmuMMaxmmuu Maximum Train Context Size (Deformable Block)
0
2 / 15
2 / 15
Minimum/ Maxxmmm Train Context Size (Tissue Manépulation)
2 / 40
Minimum Maxximmm Train Context Size (Falling Teddy Bear)
10 50
Minimum/ Maximum Train Context Size (Mixed Objects Fall)
10 / 50
Threshold to create collider-mest edge
0.3
Table 2:
Table listing the datasets and their configurations
Name
Train/Val/Test Splits
Number of steps
Number of Nodes
Collider interaction
SD.
630/135/133
50
225
External Force
DB.
675/ 135/ /135
39
81
Rigid Collider
TM.
600/ 12/0/1200111 20220
100
361
Grasping point
FTB.
700/15 000//5///10 1555100 1550
200
304
Boundary Condition
MOF.
1800/360/300
200
up to 350
Boundary Condition
D.1
Sheet deformation
We select 9 different Young's modulus ranging between 10 and 1000 from a very deformable to an almost stiff sheet. Then, per material, we compute 100 sinulations using Abaqus where the positions of the two acting forces are randomized. The boundary nodes of the sheet are kept in place. From every material configuration, we take simulations for training and 15 simulations for validation and testing respectively. For the nut-of-distribution task, we trained using Young's modulus values ranging from 60 to 500 and tested using values of 10, 30, 750, and 1000.
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Figure 11: Six objects usod in the Mized Objects Falling task. From left right: Teddy Bear, Bunny, Gummy Bear, Gummy Worm 1, Gummy Worm 2, and Traflic Cone.
D.2 Deformable Block
The original task was introduced in Linkerhägner et al. (2023), generated using SOFA (Faure et al., 2012). It uses different Poisson's s ratios and different trapezoidal meshes. We increase the difficulty of this dataset by introducing more complex initial starting conditions. This done by selecting random Poisson's 's ratio, simulating for 11 steps, and then switching to another Poisson's ratio. Then, the simulation continues for steps. The first steps are then discarded and step 12 then the initial step for the datasel (and referred step throughout the paper).
D.3 Tissue Manipulation
We use the original task introduced in Linkerhägner et al. (2023) without alterations. This dataset was also gencrated using SOFA (Faure et al., 2012).
D.4 Falling Teddy Bear
Each trajectory of the dataset was created by choosing an angle from 360% for the first time step. To vary the material properties, we randomly select combinations of Young's modulus and Poisson's 's ratio from 1000 uniformly spaced values of the Young's modulus in [1 105, 1 108) and 100 uniformly spaced values of the Poisson's ratio in [0.0, 0.499]. This dataset is generated using NVIDIA Isaac Sim (NVIDIA, 2022a), which utilizes PhysX 5.0 (NVIDIA, 2022b) to simulate letrahedral meshes based on initial CAD models.
D.5 Mixed Objects Falling
The simulation the setup from Falling Teddy Bear. In addition to the Teddy, include other objects to encourage diversity. In total, there are six different objects presented in the dataset. We report image of their high-resolution meshes in Figure 11. From every object, we use 300 simulations for the training split and 60 sinulations for the test and validation split respectively.
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M3GN (no history)
M3GN (history)
MGN (no bistory)
MGN (history)
Planar Bending (OOD)
1m-4
Deformable Plate
Planar Bending (ID)
100.0
Full Rollout MSE
Full Rollout MSE
Full Rollout MisouttSSE MSE
10.0
10.0
1.
3
15
Anchor Time Step
Anchor Time Step
Anchor Time Step
10:55
Tissue Manipulation
10-4
Falling Teddy Bear
1m=3
Mixed Objects Falling
30.0
Rollout MSE
10.6
Full Rollout MSE
hollout MSE
8.0
6.0
7.0
6.0
3.0
4.0
4.0
3.6
20 Anchor Time Step
Anchor Time Step
Anchor Time Step
Figure 12: Log-scale MSE over full rollouts on the validation split for M3GN and MGN comparing history
features. The better performing hyperparameter configuration was chosen for the final evaluation on the test dataset.
M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
0.5327
2.0010
0r5 Time
0.0000
Enor
0.332
0000
10000
0.0000
0.8303
1.0000
0.0000
Time S20.00
Time Stops
Thmre Stops
(a) Context Size 2
(b) Context Size
(c) Context Size 10
Figure 13: Per-timestep MSE for the Deformable Block task.
E Additional Results
E.1
Hyperparameter Optimization
We observed that the history inclusion of previous velocities has a big impact on the result the simulation, depending on the task. To obtain optimal performance, we did an hyperparameter optimization on the validation split comparing history features. The results for M3GN andMGN are given in Figure 12.
E.2
MSE over time
To gain better insights into the rollout stability of the model predictions, report the Mcan Squared Error (MSE) over timesteps in Figure 13, 14, 15, 16, 17, and Figure 18. Overall, our model M3GN demonstrates great robustness against error accumulation, benefiting from the inherent trajectory representation provided by the ProDMP method. MGN works well on the Tissue Manipulation task, but fails incorporate the correct context information on other tasks. EGNO performs general worse except on the Sheet Deformation tasks.
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M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
46.5
Deformation
Shee: Deformation (DO)
Sheet Deformation (ID)
1.0
Eyoor coeer Time
Time Steps
Time Steps
Time Steps
(a) Context Size 2
(b) Context Size
(c) Context Size 10
Figure 14: Per-timestep MSE for the Sheet Deformation (TD) task.
M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
Sheet Deformation (000)
6,200156
0.000220
8.000135
09913:30
8,200100
00321
890375
0.20015
8.000056
Time Steps
Time Stops
Time Stopa
(a) Context Size 2
(b) Context Size
(c) Context Size 10
Figure 15: Per-timestep MSE for the Sheet Deformation (OOD) task.
E.3
Additional Ablations
Figure illustrates two complementary evaluations of M3GN. On the left, we analyze the stability M3GN when provided with noisy context data and compare its behavior to MGN as well a variant of M3GN that operates without any context observations. This experiment highlights the robustness of M3GN to imperfect contextual information. On the right, consider dataset composed of five distinct material parameter settings. We train five separate MGN models, cach specialized to a single material, and evaluate them on unseen initial conditions but with their assigned material properties. These models are therefore specialized to a single material and serve as an upper bound for single-material material approaches. In contrast, we train one M3GN model on the union of all material splits, where the model must infer the underlying material properties directly from the context set, demonstrating its ability to generalize across materials within unified model.
E.4 Visualizations
In Figure 20, include latent space visualization of the Sheet Deformation task, where simulations with 9 different Young's Modulus values are clustered according to their material properties. The t-SNE projection of the 64-diimnsiionll latent node vectors demonstrates clear clustering, indicating that the model effectively captures and differentiates material characteristics based on learned task representations.
We further provide additional visualizations for M3GN, MGN and MGN (with Material Information) for exemplary simulations of all tasks. Each visualization shows the same simulated trajectory for different time steps (columns) and different methods (rows).
Figure 21 shows a simulation of the Sheet Deformation task for context set size of 5.
Figure 22 visualizes the Deformable Block task for context set size of 6.
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M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
Trssue Monipulazion
Tissue Maripulation
Tissue Horipulztion
0.0005
1.0000
0.0004
Enol
0.000
1.000
20
102
Time
Time
(a) Context Size 3
(b) Context Size
(c) Context Size 20
Figure 16: Per-timestep MSE for the Tissue Menipulation task.
M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
Falling Teddy Bear
Felling Teday Bewr
Palling Teddyy
5.030
Tite
Time
0.00
2,022
0.200
0.00
0.000
1.000
0.000
100
125
150
175
200
25
60
35.
120
125
150
175
200
25
50
75
103
125
155
135
205
Stops
Time Stops
Staps
(a) Context Size 10
(b) Context Size 20
(c) Context Size 30
Figure 17: Per-timestep MSE for the Falling Teddy Bear task.
Figure 23 shows a Tissue Manipulation visualization for context set size of 6.
Figure 24 provides an examplary Teddy Bear Falling for context set size of 20.
Figure 25 and Figure 26 show two different simulated Mixed Objects Falling for a context set size of 20.
Across tasks, M3GN provides accurate simulations, whereas MGN, especially when not provided the additional material information as oracle knowledge, sometimes fails to respect the material properties or predicts a drift in the solution for later time stops.
E.5 Runtime, memory and parameter comparison.
In Table 3, we provide the runtime, memory, and parameter comparison of all methods on the four benchmark task during inference. We split up the total inference time into context encoding and simulation time for M3GN.
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M3GN (Ours)
MGN
MGN (with Material Information)
MaNGO
EGNO
Mixed Objects Falling
Nixed Oblects Falling
Mixed Oblects Pailing
0.00
0.00
0.00
Time
lime Staps
Tiree Staps
(a) Context Size 10
(b) Context Size
(c) Context Size 30
Figure 18: Per-timestep MSE for the Mixed Objects Falling task.
M3GN (Ours)
MGN
M3GN context 0.0005
M3GN context noise = 0.001
M3GN context noise =
0.01
M3GN no context data
Deformalee Block v2 (Per-Materia MGN models)
Deformable Block
10.0 1
10.0
1e-4
Full Rollout MSE
Full Rollout MSE
10 15
Anchor Time Step
Anchor Time Step
Stability analysis of M3GN using noisy context data, compared with MGN and variant of M3GN that does not observe any context data. Right: Comparison of M3GN and MGN on special dataset. Using five different material parameters, we train five MGN models, each on single material, and evaluate them on unseen initial conditions while assuming the material is known from the training set. As comparison, train single M3GN model on the combined dataset of all five materials, where the model estimates the material properties from the context set.
Figure 19: MSE on a log scale for two ablations on the Deformable Block Dataset. Left:
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1000
750
500
300
150
100
t-SNE Latent Space Visualization
Young's Modulus
Figure 20: This figure shows a latent space visualization of the Sheet Deformation task for trajectories with 9 different Young's Modulus values, using a context size of 10. Each dot represents a 64-dimensional latent node vector projected to 2D using the t-SNE algorithm (van der Maaten & Hinton, 2008). Dots of the same color correspond latent. node descriptions for the same lask, each simulated with unique Young's Modulus. The visualization reveals distinct clustering in the latent space, with similar material properties grouped closer together, highlighting the relationship between material characteristics and the learned task representations. To improve clarity, points corresponding to nodes on the plate's 's edge were excluded, as their constant boundary condition resulted in unvarying latent descriptions.
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Sheet Deformation
1=1
1=6
t=
t= 16
1=
[ 26
36
46
t=1
t= 6
t= 11
t= 16
t= 21
t = 26
t = 36
t = 46
1=1
t= 11
t= 16
1= 21
1 = 26
= 36
46
t=1
t= 6
t=
t= 16
t= 21
t = 26
t 36
t = 46
t=1
t= t= 11
t= 16
t 21
== 26
== 36
t = 46
Figure 21: Simulation over time of an exemplary test trajectory from the Sheet Deformation task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size is set 5. All visualizations show the colored predicted mesh a collider floor, and wireframe (red) of the ground-truth simulation. M3GN can accurately prodict the correct material properties, resulting in highly accurate simulation.
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Deformable Block
1= 6
t t=1 11
1 t= 16
1 1= 21
{ 26
t= 31
36
t=1
t= 6
t= 11
t= 16
t t= 21
t = 26
t = 31
t = 36
1=1
1= 6
t= 11
i t= 16
t= 21
1 = 26
I = 31
i 36
t= 6
t= 11
i t= 16
t t= 21
t = 26
t = 31
t = 36
t=1
t= 6
t= 11
t= 16
t= 21
== 26
t t= 31
t= 36
Figure 22: Simulation over time of an exemplary test trajectory from the Deformable Block task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size is set 5. All visualizations show the colored predicted mesh a collider or floor, and a wireframe (red) of the ground-truth simulation. M3GN can accurately prodict the correct material properties, resulting in highly accurate simulation.
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Tissue Manipulation
t=1
t t= 11
t == 21
t = 31
t 41
t = 61
t t= 81
= 101
61
81
101
==1
t t= 11
t= 21
t t= 31
t 41
t 61
t= 81
= 101
t=1
t 11
t = 21
t = 31
f = 41
f = 61
= 81
101
t= 1=
31
1= 1= 1=
101
Figure 23: Simulation over time of an exemplary test trajectory from the Tissue Manipulation task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size is set to 5. All visualizations show the colored predicted mesh, a collider or floor, and a wireframe (red) of the ground-truth simulation. All methods can solve the task, however MGN is drifting a tiny bit to the left over time.
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Falling Teddy Bear
t1
t= 41
t= 61
t= 81
=001
121
t= 141
t=
21
61
t= 81
t= 101
t= 121
141
t= 181
t= 21
t=
t= 61
t=
101
t= 121
141
t= 181
21
t= 41
t= 61
t= 81
t -011
t= 121
t= 141
t= 181
21
1=
t=
t= 81
100
t= 121
141
181
Figure 24: Simulation over time of an exemplary test trajectory from the Falling Teddy Bear task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size set to 20. All visualizations show the colored predicted mesh, a collider or floor, and a wireframe (red) of the ground-truth simulation. M3GN significantly outperforms both step-based baselines.
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Mixed Objects Fall (Bunny)
t= 21
t= 41
t= 61
t= 81
t= 101
t= 121
t= 141
181
t=
t=
t=
==
t= -01
121
t= 141
181
t= 21
t= 41
t= 61
t= 81
101
t == 121
t= 141
= 181
21
t= 41
t= 61
1= 81
t= 101
t= 121
1= 141
1= 181
1. 1= 21
1 t= 41
t= 61
t= 81
= 101
t= 121
t= = 141
t= 181
Figure 25: Simulation over time of a bunny from the Mixed Objects Fall task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size is set to 20. All visualizations show the colored predicted mesh collider or floor, and wireframe (red) of the ground-trui truth simulation. M3GN significantly outperforms both step-basse based baselines.
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Mixed Objects Fall (Traffic Cone)
= 21
t= 41
t= 61
t= 81
101
t= 121
141
181
t = 21
t=
t=
t=
101
121
t= 141
t= 181
t= 21
t= 41
t= 61
t= 81
t= 101
t= 121
t= 141
181
101
121
141
181
1. - 21
1 -= 41
-= 61
1= 81
101
t= 121
= 141
L 181
Figure 26: Simulation over time of traffic cone from the Mixed Objects Fall task by M3GN (blue), EGNO (purple), MaNGO (red), MGN (orange), and MGN with material information (green). The context set size is set to 20. All visualizations show the colored predicted mesh, collider or floor, and a wireframe (red) of the ground-truth simulation. M3GN significantly outperforms both step-based baselines. The simulation generated by MGN is severly affected by drift.
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Table 3: Runtime, memory, and parameter comparison of all methods on the four benchmark tasks.
Task
Metric
M3GN (Ours)
MGN
EGNO
Mango
Context Encoding Time [s]
0.0062
0.0000
0.0000
0.0000
Simulation Time [s]
0.0132
0.2673
0.0134
0.0628
Deformable Block
Total Inference Time
0,0194
0.2673
0.0134
0.0628
GPU Memory Usage [MB]
68.33
26.94
121.43
126.10
Parameters
2 244 847
259 394
1200
2990 082
Context Encoding Time [s]
0.0066
0.0000
0.0000
0.0000
Simulation Time [s]
0.0132
0.3177
0.0242
0.1247
Sheet Deformation
Total Inference Time
0.0198
0.3177
0.0242
0.1247
GPU Memory Usage [MB]
170.27
29.12
210.69
324.03
# Parameters
2 546 206
259 779
999 258
2990 211
Context Encoding Time [s]
0.0083
0.0000
0.0000
0.0000
Simulation Time [s]
0.0135
0.7531
0.0802
0.4705
Tissue Manipulation
Total Inference Time
0.0218
0.7531
0.0802
0.4705
GPU Memory Usage [MB]
569.70
45.40
759.84
748.71
# Parameters
2 547 614
259 907
1000 794
2 2990 723
Context Encoding Time [s]
0.0354
0.0000
0.0000
0.0000
Simulation Time [s]
0.0143
1.1961
0.1347
0.7442
Mixed Objects Falling
Total Inference Time is]
0.0497
1.1961
0.1347
0.7442
GPU Memory Usage [MB]
1340.79
42.56
1406.92
1405.34
# Parameters
2 950
1 259 651
999 130
2990 083
38


Reviews:
[{"review_id": "TkzEQk0kA9", "reviewer": "Reviewer_ifmq", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper describes a novel problem formulation of deformable object simulation as meta-learning and provides a model architecture and training methodology for that problem. They show that their method (M3GN) is able to leverage the additional context effectively to produce more accurate simulations than a baseline method that is directly given material properties about the object to be simulated.\n\nSpecifically, they make the following contributions:\n\n* Propose a Graph Network Simulation algorithm that 1) uses the initial context to learn latent descriptors; and 2) outputs the weights of Probabilistic Dynamic Movement Primitives (ProDMPs) for each node. \n* Evaluate this method against several baselines and demonstrate effective performance.\n\nOne important comment is that the authors don't explicitly claim that the problem definition is a contribution; however, I think it needs to be (or, at least, the introduction of a novel problem setting needs to be more heavily emphasized). Otherwise, it is not clear that their comparisons are to baselines that are not designed to generalize across different deformable object parameters. \n\n## Strengths\n* The problem formulation seems realistic and allows for simulators to give improved performance without needing to solve the real-to-sim-to-real problem. \n* The use of ProDMPs to model trajectories is interesting/novel and seems to potentially improve the quality of simulation over long time horizons. It also seems well-suited to efficient optimization.\n\n## Weaknesses\n* The paper isn't very clear (to my reading) that the baselines are being applied to a task they aren't really designed for. This is a new problem and the authors are attempting to adapt the baselines to a new problem setting. This is a reasonable choice but needs to be made much clearer to readers as outperforming the baselines isn't as impressive in this context. It would be good to see the authors compare M3GN's performance with that of MGN (or other baselines) when trained on a single deformable object. This would allow evaluation of how well the meta-learning performs in comparison to a skyline method that stacks the deck in favor of MGN. \n\n* The paper doesn't explicitly compare against Dahlinger et al. 2025, which seems like the closest method from related work and the primary other method to apply meta-learning to a similar problem. The discussion in related work simply states that their approach relies on different information (a distinct trial vs a prefix of a trajectory) and is less memory efficient (but the actual comparison for memory costs is not made in the paper). At a minimum, it would be good to more clearly describe the experimental rationale and the pro/cons of the different baselines to make it clear how each baseline adapts to the meta-learning problem.\n\n* The evaluation confounds the benefits that come from two novel changes to the MGN approach: the learned latent parameters based on the context and the ProDMP parameterization of the trajectory. It would be nice to see comparisons that separate out the benefit from each of these additions to the baseline. For example, does MGN with ProDMP output parameterization see substantial performance on its own? If this ablation does not make sense, then the authors should clearly explain why this experiment isn't appropriate.\n\n* The paper is technically dense, and this makes it hard to identify the key points. (I personally had to spend a lot of time going back and forth and it took several passes through the paper before I felt like I understood the main ideas.) For example, the introduction is almost 3 pages long and first discusses GNS broadly, then discusses error accumulation over time, then introduces an example task that is really about the availability of initial context, then jointly introduces meta-learning and trajectory-level predictions, then describes ProDMPs, and then describes the method. Better clarity in the writing would help substantially. Consider, e.g., a structure that explains 1) why GNS is useful; 2) that it needs to train from scratch for a new material; 3) why auto-regressive prediction is problematic, how other methods have tried to fix the problem and the associated drawbacks for memory... etc; 4) how M3GN fixes both of these problems by meta-learning and outputting ProDMP parameters.\n\n* The paper claims that M3GN is faster than MGN, but doesn't actually present evidence for the speed of the different methods. Similarly, the authors claim that ProDMP as a trajectory parameterization provides memory efficiency over alternative trajectory transformations but don't substantiate this with evidence. \n\n* The related work section currently mixes general background discussion of methods with comparison to/differentiation from related SOTA  methods. This makes it harder to understand what is novel and assess the work.\n\n## Overall Assessment\n\nI think this paper has a lot of potential and I would like to see it accepted. However, as it stands there", "claims_and_evidence_yn": "No", "claims_explanation": "The claims about the overall performance of the method are well substantiated with experiments in comparison to MGN. However, there are some missing pieces of evidence:\n\n* comparison with a relevant meta-learning approach with respect to both prediction performance and cost (time/memory) are not presented and improvement is claimed without substantiating evidence\n* the authors claim that M3GN is 32x faster than MGN but don't directly support this\n\nThe second issue seems like a relatively minor fix. The first one likely requires additional experiments to be run (or a clear description of why the comparison is inappropriate). I understand that MaNGO relies on different context, but I think it is reasonable for the authors to modify MaNGO to fit their setting (or to run experiments that match the configuration for MaNGO).", "audience_yn": "Yes", "audience_explanation": "The problem considered is important and interesting and the results are promising. Overall, I expect that this paper would be interesting a range of TMLR readers who are interested in learned differentiable simulation.", "requested_changes": "* Improve clarity of the presentation that makes it clear which problems are being solved and separates the changes to the problem setup from the description of the solution proposed.\n* Experiments that more clearly isolate the distinction between the benefits of the meta-learning approach presented and the contribution from the trajectory representation of ProDMPs as the output of the network. Ideally, the authors would produce ablations that independently assess the benefit of these changes in comparison to MGN.\n* Comparisons with MaNGO or a more explicit discussion of why the comparison is inappropriate\n* Experiments that explicitly measure the performance gains claimed with respect to speed and memory requirements of the different methods\n* Experiments that compare the performance of M3GN with MGNs trained on a fixed set of mesh dynamics would be nice to gauge the relative performance of M3GN in comparison to a more costly approach of re-training an MGN for each object. This would substantially help to calibrate the results and improvements.\n* Changes to the related work that more clearly distinguish the discussion of background methods that this work builds on from comparing and contrasting this work with alternative problems or methods. Consider, e.g., splitting this section into a background section and a more direct related works section.", "broader_impact_concerns": "None", "additional_comments": "I found the description of ProDMP in the appendix quite helpful --- it might be good to reference that more explicitly in the main body of the text so that readers unfamiliar with ProDMP are aware of the resource.", "extra_scores": {}, "extra_text": {}}, {"review_id": "sune89rv3V", "reviewer": "Reviewer_D5jb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes Movement-Primitive Meta-MeshGraphNet (M3GN), a context-aware, mesh-based learned simulator that leverages trajectory-level meta-learning and Probabilistic Dynamic Movement Primitives (ProDMPs) to improve the accuracy and efficiency of physical deformation simulations. Instead of performing step-wise autoregressive rollouts like conventional Graph Network Simulators (GNS), M3GN predicts full trajectories in a single forward pass conditioned on a few initial context states. This design allows the model to infer latent material properties and to avoid cumulative rollout errors.\n\nStrengths: Innovative integration of meta-learning and movement primitives; strong empirical results with substantial runtime gains.\n\nWeaknesses: The method’s applicability is somewhat narrow, and the model design is relatively complex to reproduce.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The evidence in the paper aligns well with its main claims.\nFirst, the quantitative results across five benchmarks (Deformable Block, Sheet Deformation, Tissue Manipulation, Falling Teddy Bear, and Mixed Objects Falling) consistently show that M3GN outperforms strong baselines such as MeshGraphNet (MGN) and the Equivariant Graph Neural Operator (EGNO) in terms of rollout accuracy and stability.\nSecond, the runtime analysis clearly supports the claim of improved efficiency—M3GN achieves up to 32× faster inference than MGN and 400× faster than traditional simulators due to its trajectory-level prediction and ProDMP formulation.\nFinally, ablation studies (Figures 7–8) confirm that each design choice—meta-learning, ProDMPs, and CNP-based context aggregation—contributes meaningfully to performance.\nTogether, these results provide convincing and well-documented evidence for the paper’s claims of higher accuracy, speed, and robustness in learned mesh-based simulation.", "audience_yn": "Yes", "audience_explanation": "The topic is directly relevant to a portion of TMLR’s audience, particularly those working on physics-informed machine learning, graph-based modeling, and efficient simulation methods.", "requested_changes": "- The method assumes access to future collider information at inference (e.g., inclusion of the final collider position and the note that providing the full future collider trajectory did not help). Please state explicitly what is available in realistic settings, and, if possible, add a variant where only partial/uncertain collider motion is known, with quantitative impact.\n- Training assumes deterministic simulations and fixes $\\sigma_o=1$ (i.e., no output variance). Add a controlled study with synthetic noise or missing context frames to quantify robustness and failure modes.\n- Fig. 9 is compelling; consider adding parameter counts, peak memory, and a breakdown of time spent in context encoding vs. trajectory decoding to substantiate the “single-pass” advantage and the stated 32×/400× speedups.\n- Consider adding a short discussion on how the proposed framework might generalize to non-deformation or non-mesh-based simulations, clarifying the method’s broader relevance.", "broader_impact_concerns": "The paper includes a brief Broader Impact Statement noting potential positive impacts in reducing computational costs for physics simulations and possible misuse in developing advanced weapon models. Overall, the statement is adequate.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "RkPLPnq8j4", "reviewer": "Reviewer_F6am", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work proposes M3GN, a new type of Graph Network Simulator (GNS) that combines trajectory-level meta-learning and probabilistic dynamic movement primitives to solve the problem of simulating deformable objects. Using Conditional Neural Processes (CNP) to enable faster meta-learning, the proposed method resulted in magnitude times faster simulation compared to related works and showed SOTA performance on various simulation tasks. Extensive ablation studies have also been conducted. \n\n**Strength**\n\n- Propose to incorporate the full trajectory, rather than fixed-length steps. The proposed method thus avoids the error accumulation problem in previous step-based methods. \n- Using a meta-learning framework allows the proposed method for faster adaptation to newer tasks, while outputting the full trajectory allows faster inference, resulting in a very fast simulator compared to previous works. \n- Adopting meta-learning to deal with the simulation tasks is novel, especially the concept of regarding each trajectory as a new task is quite interesting\n- I found this work to be well-written and organised, and is easy to follow.\n\n\n**Weaknesses**\n- The proposed method requires some initial observations, which is not required for previous methods. \n- While extensive ablation study is done on the context size/anchor steps side, none of the ablation is done in the hyperparameter side of the proposed method. \n- The authors mentioned another closely related meta-learning-based GNS method named MaNGO. However, the proposed method is not compared with MaNGO in the experiments. The claim of surpassing recent GNSs' performances is a bit weakened.", "claims_and_evidence_yn": "Yes", "claims_explanation": "For the most part. The main contribution in this work is about proposing M3GN, a novel GNS that generate full trajectory in a single time, which can adapt to other material properties. The claim regarding inference speed is fast supported by empirical experiments. The last claimed contribution, regarding the proposed method outperforms recent GNSs on challenging deformation benchmarks, can be further strengthen if a closely related work MaNGO is also compared.", "audience_yn": "Yes", "audience_explanation": "I think this work has potential to be quite impactful. The SOTA result from the perspective of simulator inference speed and inference accuracy and stability over the longer timesteps are quite impressive.", "requested_changes": "**Clarity/Readability**:\n- C1) Very minor but, for figure captions, I would prefer also writing the colours out: for example, M3GN (blue), MGN (orange) to improve readability.\n- C2) There are some inconsistent citations style across the manuscript, for example in “provides superior simulation accuracy compared to several variants of MeshGraphNet (MGN) Pfaff et al. (2021)”, “Pfaff et al.” was not parenthesised. Should be (“Pfaff et al.)” like other references. I’m sure there are more. \n- C3) I might have missed but the actual basis functions used are not described anywhere.\n- C4) Examples of material property can be included for clarity. \n- C5) For Figure 5 (and other figures presenting the results), I’m a bit confused by the scale of y-axis. The caption notes log-scale MSE, but the plots have 1e-6/1e-4 on it. Can the authors confirm that these are indeed correct? I suspect one is written incorrectly, that is, either it’s actually MSE (i.e. the mistake is in the caption) or the mistake is in the plots.\n- C6) For Figures 20-25, the meaning of the purple row (2nd row) is not described. \n\n\n**Suggestions**\n- S1) How is physical proximity used/determined? It’s explained in the manuscript but it’s not used/explained anywhere. \n- S2) Comparison with MaNGO should be included. \n- S3) Intuitively, given the usage of full trajectory of proposed M3GN, one would think that with the increasing anchor steps would naturally increase performance. This is the case for most tasks, however for mixed objects falling and falling teddy bear, increasing the anchor steps too much would actually be a bit harmful. Would be nice to have some discussion about this result. \n- S4) What is the longest timestep the proposed method can be suitable for? Similar to my previous point, to my intuitive understanding, generally longer context would be more beneficial, I’m a bit confused on why one would try to input a shorter context size. Some discussion on the limitation of M3GN if possible regarding context size would be interesting. \n- S5) The authors mentioned that M3GN struggles to capture rapid, high-frequency dynamics when node velocities change abruptly. It would be interesting if there are more empirical results showing this struggle. \n- S6) If I understand correctly, for OOD material of the Sheet Deformation task, the scale of MSE is 10 times more than its ID counterpart, depending on the given context size. I wonder the actual meaning of this; It would be interesting to see the OOD results on other tasks as well if possible.", "broader_impact_concerns": "Concerns are sufficiently addressed in the Broader Impact Statement section in the manuscript.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)