PlasticityNet: Learning to Simulate Metal, Sand, and Snow for Optimization Time Integration
Authors: Xuan Li
Yadi Cao
Minchen Li
Yin Yang
Craig Schroeder
Chenfanfu Jiang
Venue: NeurIPS 2022 Accept (2022)
Keywords: neural network
plasticity
optimization-based time integrator

Abstract: In this paper, we propose a neural network-based approach for learning to represent the behavior of plastic solid materials ranging from rubber and metal to sand and snow. Unlike elastic forces such as spring forces, these plastic forces do not result from the positional gradient of any potential energy, imposing great challenges on the stability and flexibility of their simulation. Our method effectively resolves this issue by learning a generalizable plastic energy whose derivative closely matches the analytical behavior of plastic forces. Our method, for the first time, enables the simulation of a wide range of arbitrary elasticity-plasticity combinations using time step-independent, unconditionally stable optimization-based time integrators. We demonstrate the efficacy of our method by learning and producing challenging 2D and 3D effects of metal, sand, and snow with complex dynamics.

PlasticityNet: Learning to Simulate Metal, Sand, and Snow for Optimization Time Integration
Xuan Li Department of Mathematics University of California, Los Angeles xuanli 11 @math. www.ulllle...oo
Yadi Cao
Department of Computer Science University of California, Los Angeles yadicas www..cllaeeeooo edu
Minchen Li Department of Mathematics University of California, Los Angeles minchen@math ucla.
Yin Yang School of Computing
University of Utah yin. yanguutt edu
Craig Schroeder
Chenfanfu Jiang Department of Mathematics University of California, Los Angeles affjiang@math, ucla. edu
Department of Computer Science and Engineering University of California, Riverside craigsθcs. sur.ucre
Abstract
In this paper, we propose a neural network- based approach for learning to represent the behavior of plastic solid materials ranging from rubber and metal sand and snow. Unlike elastic forces such as spring forces, these plastic forces do not result from the positional gradient of any potential energy, imposing great challenges the stability and flexibility of their simulation. Our method effectively resolves this issue by learning generalizable plastic energy whose derivative closely matches the analytical behavior of plastic forces. Our method, for the first time, enables the simulation of wide range of arbitrary plasticity-llasticee time step-independent, unconditionally stable optimization-base basedd time integrators. We demonstrate the efficacy of our method by learning and producing challenging 2D and 3D effects of metal, sand, and snow with complex dynamics.
combinations using
Introduction
Combining machine learning with physical simulations has recently attracted a lot of attention. vast amount of existing research adopts end-to-end approach, where the specific underlying computa- tional physics system is treated as black box [46] 41]. Harnessing the power of neural networks, this research has been successfully applied in computer animation [10], multibody systems [3 6,6 59 [1]. human musculature simulation 20), computational fluid dynamics
[3], and non-linear contin- mechanics [5]. An alternative direction is represented by physics-informed neural networks (PINN) 21). where in its original form, the residual of a partial differential equation directly used the loss function
that the network training is physics-aware learning process. PINN becomes powerful when the design space of the input to the network can be parameterized, which accelerates both the roll-out and the inverse optimization process (51). Another noteworthy category learning the physical modeling where the machine can either help increase the model resolution in coarser grid [24], inject nonlinearity linear model (37], or apply learnable model reduction reduce the system degrees-off of-frredoffff (DOF) for acceleration 47 48)
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

---

Despite its great success, training a neural network to replace a traditional simulator is not always the preferred choice. This is partially due to the challenges in the trained model's 's generality and portability. For example, trained model on a particle-based deformable body solver (such as the Material Point Method (MPM (19) cannot be directly applied to the mesh-based Finite Element Method (FEM) 49], while in traditional continuum mechanics, the constitutive model that describes the relationship between force and deformation is an independent module from the underlying geometric description or simulation scheme. Indeed, by simply switching the constitutive model and applying minor changes to the existing and general simulation pipeline, a wide range of materials can be simulated in the same framework, ranging from sand 42 23 and metal [39, , to snow and glacier [58).
11 57 35|
Many elastic materials, including those represented by mass- systems 2 and common hypere- lastic solids (50), are usually governed by analytical elastic potential energy functions in terms of the deformation. These models are well fitted to experiments and proven to be simple, accurate, and pre- dictive. Although most of these energy functions are highly nonlinear and non-convex, reformulating the dynamic simulation process numerical optimization problem and solving it using projected Newton and line search can guarantee global convergence to a solution (28). Beyond hyperelasticity, plasticity is much more challenging. The mechanical response of plastic materials imposes extra difficulties in the implementation as it is path-dependant and non-smooth. One common handling of plasticity is the return mapping algorithm, which applies the effects of plastic deformation to the elastic forces. However, this leads to asymmetrical force derivatives, which eliminate the possibility of integrating the plasticity into the energy function in single optimization and complicates the pipeline. In the recent work of Energetic Consistent Inelasticity (ECI) (34), the plasticity is analyt- ically modeled
optimization problem just like simulating pure elastic materials. However, their analytical derivation only works for St. Kireho- (StVK) elasticity with the von-Mises plasticity.
energy functional, and the simulation can be formulated
In this work, we propose Plasticity.ett, a neural network-based approach for learning an energy- based force that locally approximates elastic forces with plasticity models and is compatible with optimization time integrators. PlasticityNet framework supports any combinations of elastic models and plastic models and works with both MPM and FEM discretizations. With optimization time integrators, we demonstrate that our framework can simulate vast types of plasticities, such as metal, sand, and snow, with large time step sizes.
2 Related work
Classic Plasticity Models The classic plastic models utilized the geometry information of the plas- ticity and are available for many applications. In the computer graphics community, researchers have followed mechanical literature on the Drucker-Prager elastoplasticity model 42 91, and developed
particle-based simulations of dry (23 and wet [52] sand. Extending a similar Cam anmClay plasticity
model, snow avalanches [1] 35), glacier calving (58) and food fracturing [5 are also captured with high visual plausibility as well as physical accuracy. For metals and dough-like materials, the von-Mises plasticity model [39] usually adopted, while (56) 1X presented its anisotropic extensions. Still, the implementation of these models in modern, optimization-bi based simulators is cumbersome due to the mon-integrable forces. Recently. (34] proposed an elastoplastic energy of von-Mises plasticity under StVK elasticity for optimization time integrator, which can be viewed as a special-case analytical solution to our framework under the same combination of elasticity and plasticity. But our framework works for arbitrary combinations.
Data-Driven Plasticity Models The machine learning approach has been used to find new plastic models using large sets of measurements and parameters, outperforming many long-standing hand- crafted models. The macro- constitutive relationship is learned from the results of the micro-level simulations 40, 45). A similar approach is applied in [53] to learn anisotropic hyperelasticity, where additional geometrical information is included in the input. PINN can also be applied in plastic model finding from experimental measurements [] 54 25), where the loss includes the stress and Hessian, to infer stress with more accuracy in the implicit simulators. However, there does not exist any prior work, to the best of the author's 's knowledge, that tried to find variational form for arbitrary plasticity model.
2

---

Optimization Time Integration The optimization time integrators have advantages in terms of stability under large deformations and large time step sizes. Many of the nonlinear systems of equations that arise from implicit simulation can be integrated to get equivalent optimization problems, which allow robust optimization techniques to be applied. The MPM simulator in this work is based on [14], which formulated the backward Euler time integration with hyperelastic materials as a minimization problem. [30] and (55) also explored domain decompositions and hiearachical preconditioners to improve robustness and efficiincc. The FEM simulator in this work is based on Incremental Potential Contact (IPC) [29], which proposed a variational form for frictional contacts. Their optimization- sanadd frictional contact framework was also extended to codimensional objects [31], rigid bodies [I] 26]. articulated multibodies [7]. reduced elastic solids [27], embedded interfaces [60], and FEM-MPM coupled domains [33]
3 Background
3.1
Optimization Time Integration
In this section, we briefly introduce the optimization time integration for elastodynamics simulations with the Material Point Method (MPM) and the Finite Element Method (FEM). We refer the readers to (28) and [14] for more details.
FEM discretizes the simulation domain as unstructured meshes (e.g., triangle meshes in 2D), while in MPM, a point cloud composed of material particles is used to discretize the domain. While FEM directly uses the mesh nodes as the simulation of-freeoo (DOF), MPM transfers its particle state to a uniform background grid, whose nodes are used as the DOFs for the integration of forces 19 Robust simulation of elastodynamics can be achieved via implicit time integration, which updates the nodal positions (x) or velocities (V) step by step based on the previous physical states. To step from td to ga+1 t + At with time step size At, with implicit Euler time integration rule, one needs to solve a nonlinear system of equations
M(v*+I - (v" + A
(1)
Here v the velocity DOF formed by concatenating all nodal velocity vectors, similarly concatenated, M is the mass matrix, is the gravitational acceleration vector, and f is the internal force vector. Without plasticity, the internal force on node can be calculated as
P(P
V WWlgg
where q iterates the surrounding elements/particlss of node i in FEM/MPM, V is the initial volume (I + A/ (MPM) or F Vx + ArVv (FEM) the deformation
of the element/particle, gradient, which measures deformation from the undeformed state to the deformed state, and P is the first-Piola Kirchhoff stress, which describes the internal force per unit area within a material. Vway is the gradient of the weight function node evaluated on element/particle center. The weight function for transferring physical quantities between the elements/particls and the mesh/grid nodes. Unlike FEM, the last time step is used in MPM as the reference configuration, and so V Wiig is calculated V
Vwyy
When there exists an energy density function V such that P(F) to solving the following optimization problem
0. '9F:
solving Equation is equivalent
-+ argmin, 2 I|v - (v" gl + Σ v (F,)
(3)
This formulation more favored because with line search methods, convergence to local minimum of Equation 3 can be guaranteed even when simulating challenging cases with stiff materials or large time step sizes. After solving for the velocity y*+1, FEM directly updates mesh nodal positions as x = x" + Aty"+1, while for MPM, the velocity on the grid node is interpolated to particle locations for particle advection. The background grid is reset at the heginning of each time step, which allows MPM to benefit from the conveniences of regular grid and mesh-free formulation at the cost of some accuracy loss due to the transfers between the grid and particles.
3.2 Return Mapping for Plasticity
3

---

With plasticity, objects can undergo both plastic and elastic deformations, and the deformation gradient the current time step can be decomposed as
Plastic Region
Elastic Region
F++1 pp
(4)
based on the finite strain theory. Here F encodes the permanent plastic deformation of the rest shape, and p is the elastic defor- mation that results in effective clastic forces. In theory. F is con- strained within certain elastic regions. Computation-wise, an elastic predictor F
2(F
F
"F"""""""""" can be computed first by assuming F = P F is outside the elastic region, it will be projected back onto the boundary of the region obtain F F =(F (Fig.
Figure 1: An illustration of return mapping.
ure Tp. This projection 2 is called a return mapping. Within this framework, the implicit elastoplastic nodal force can be computed as [34
Σ V TZ(F 4
")
PPP
Wagg
(5)
where rFF) P(F)F" is the Kirchoff stress. The above forces are integrable only if T((())) TT can be represented as the gradient of some energy function:
NV
(2(
OF
Most combinations of elastic constitutive models and plastic return mappings do not satisfy this integrability condition because the Jacobian field of the right-hand side is asymmetrical. Note that directly feeding (F) into an elastic potential does not form a potential energy for the elastoplastic forces defined in [Equation 5 (34] only found one specific combination such that elastoplastic potential energy exists. Thus, it remains challenging to simulate versatile plastic behaviors with optimization time integrators and achieve robust performance.
4
PlasticityNet
We propose PlasticityNet, a neural network-based elastoplastic model that finds a family of local potential energies whose negative gradients approximate the elastoplastic forces within small neighborhood so that plasticity can be conveniently simulated using optimization time integrators. The model architecture is illustrate in Figure Specifically, instead of finding global energy function v (F), we search for an energy (F, Fo), parameterized by Fo. such that
Nv (F.Fo))FFF = - (Z
dv OF
and
F. Fo) R ((((F))F
(7)
OF
To exactly enforee the first equality, we propose the following linear correction:
(F = NNo(F, (VrNNo Fo) - T (Z F)
(8)
Here A + B B, tr(ATB is the matrix inner product. It can be verified that 8 OF
(F
F=F, == =
Then we only need to focus on the approximation part in Equation 7 function for our neural network as
We design the training loss
OVD OF (F, Fo) T(2
(
Er, Er
(9)
During training, F is only sampled near Fy. Please refer Section 5.1 for details.

---

4.1 Hardening of Plasticity
Hardening effects are widely observed in metals and snow. With harden- ing, the clastic region will expand by certain amount whenever F falls in the plastic region. To account for hardening, the return mapping (F,, h) and the energy (F, Fo, h) will depend on an extra hardening state h, which controls the shape of the elastic region. This hardening state is function of F. However, to maintain integrability with respect to F, we approximately update h based on Fo, which is assumed to be close to F.
F Fo
R 2
R
Figure 2: An overview of PlasticityNet. It map from R2d to R.
4.2 Optimization Time Integration with PlasticityNet
Fixed-Point Iteration The gradient of our learned elastoplastic potential energy FF, Fo) only approximates the effective stresses locally near Fg- To approach the accurate solution of Equatton ] with elastoplastic forces, we apply a fixed-point iteration Fo to let it converge to F*+1 Specifically, solve sequence of optimization problems
tnt1 argminn b|lv - (v" g&(())l + Σ
Fy Mg), for j = 0, 0,,,,,... (10)
treating the concatenated deformation gradients Fg and hardening states h constants, which are only updated before each optimization FjJ F(y"+I.)) and h h(Fo). At convergence, will obtain the true solution of [gquation In practice, few number fixed-point iterations can already generate high-quality results.
Stability Regularizer We augment our learned potential with an extra quadratic regularizer to stabilize the simulation especially when the material is stiff or the time step size is large:
ve (F, Fo) = .N. Fo) (VRANo Fo) T(2(F)))
F+ F Foll. (11)
Here z is the shear modulus of the material that V is learning. Note that this extra term is added after the model trained instead of during the training. This extra term does not change the gradient Fo, so it will not change the fixed point of Procedure TO Please Section 5.3 for comparison between simulations with and without this regularizer.
4.3 Learning Volume- Pruuuee-reserving Return Mapping
J62
Elastic Region
The return mapping Z required by PlasticityNet can be either given analytically or learned. Note that with different combinations of many practical clasticity and plasticity models, the return mapping may not have a closed-form solution, and the projection can only be performed by solving a nonlinear system of equations.
El
Here we provide a simple approach to learn volume-preserving return mapping, which ensures that del = det(F). For isotropic materials, the projection can be performed in the diag- onal space, i.c., with F = U Diag(2)) V being the singular value decomposition of F; the projection is only needed for Σ. In the diagonal space, a volume-preservee
Figure 3: Volume-preserving projection.
path is a straight line in the Hencky strain (defined e log( Σ)) space, which is perpen- dicular to the diagonal line. The direction of the projection path 1. The volume prenervin projection in the Hencky strain can be unified by c - 5 for some 8y, with (E(E) exp(H) and Z(F) = UDiag (2 An illustration shown in Figure 3]
The elastic region is usually represented by implicit function y () < 0. We can use neural network to predict 5% where the training leverages the differentiability of the implicit representation for the elastic region boundary. The volume-preserving path usually has two intersections with the
5

---

Sand
Snow
Metal
56
580
1900
1900
2033
1000
10.0
1940
X030
Figure 4: Training losses of our 2D models.
elastic region boundary. To eliminate this ambiguity, we clamp the output of the neural network with a maximum |ll|.. We define our neural-network-basse return mapping on the diagonal space as:
Texple
870 TeT
y y Σ
0,
87 Σ = min min 2
(12)
Σ
S 0.
The training loss function for single E is defined as
((E:0) - (z) max (2) - -|211,00, 0},
y M
(13)
M
Here, the first term is to pull the points outside the elastic region back onto the boundary. The second term is to avoid these points to be always projected onto the diagonal due to the clamping in 87g. To account for hardening, we only need to let the dy network accept extra hardening state variable h: (E, h) = min(WNN h), ||ε]]]. The learned return mapping is then ready to be used by our Neat
5 Experiments
We show examples to demonstrate the capability of our PlasticityNet in learning versatile plasticity models. Our physical simulators implemented using C++, and we applied PyTTorch to learn the potential energies, which are then loaded into our simulators with TorchScript. All our potential energies are trained as multilayer perceptrons using the Adam optimizer [22 on a single Nvidia RTX 3090 GPU. Please see Appendix A.1 for more training details. All ground-truth data are generated using standard explicit time integration with analytical plasticity returning mapping under small time step sizes for stability. With our PlasticityNet, we can robustly simulate elastoplastic behaviors with much larger time step sizes using optimization time integrators.
5.1 Training
The training of PlasticityNet only requires the return mapping (either given analytically or pre-trained) for the plasticity model and the Kirchhoff stress for the underlying elasticity model. There no need for extra labeled data. At each epoch, we will sample new batch of (F, Fo. h). The sampling of deformation gradients is based on its singular value decomposition F = U Diag V) with U, V being two rotation matrices, To sample F and Fo so that their singular values are close to each other, we set Fo = R Diag(e RR2 and F R3 Diag(e++5e)RR, where is randomly sampled vector, de a random perturbation, and R 's are randomly sampled rotation matrices. The hardening state sampled uniformly from appropriate range depending the plasticity model. Please Appendix A.2 for definitions of hardening states and their range selections. this work, we uniformly sample from [1,, 1)dh de from 1-0.1, 0.1]d for sand plasticity and metal plasticity, and [-0.2, 0.2) for the snow plasticity. The training loss curves of our 2D models are shown Figure
5.2 Testing on 2D Simulations
In this section, explicit time integrators are used to generate the ground-truth data for the validation of the optimization time integrators with PlasticityNet on multiple experiments. The quantitative comparisons are plotted in Figure 5 We additionally include the computational costs in Table I We

---

Snow
Mctal
101
0.00
244
My Mm
0.13
040
Sand
638
0.08 03
04
00
(a)
(b)
(c)
Figure 5: (a) (b) The intersection-oovr--niioo (IOU) 17 measure between the ground truth and our
results. The IOUs computed using the mass distributions the MPM grid. (c) The average FEM nodal position difference. Note that the bounding box of the 2D metal frame is 0.1m x 0.18m.
Table I: Computational costs of 2D experiments.
Ours
Explicit
Experiment
Time step (s)
s/frame
Time step (8)
s/frame
Sand
le:3
12.58
le-5
6.20
Snow
le:3
35,56
le-5
6.78
Von-Mises Metal
le-2
1.08
le-5
5.39
Neohookean Metal
le-2
1.03
le-5
7.88
MPM-FEM Coupling
le-3
38.90
le-6
184.58
remark that the main objective of our work is not surpass the performance of the existing simulation of every constitutive model, but to provide methodology that enables the usage of implicit plasticity in optimization time integration framework.
Sand Plasticity We start by learning the elastoplas- tic model of dry sand (Figure 6). The model com- bination is St. Venant-Kirchhoff (StVK) elasticity, and the closed-form Drucker Prager plasticity return mapping [23] (See Appendix A.2.1). In this example, we simulate a column of sand falling onto the ground under gravity with MPM. Our method generates vi- sually identical results compared the ground truth, both with the same time step size and 100x larger time step size. The quantitative comparison between our results and the ground truth is shown in Figure Sa Note that there no hardening mechanism in this plasticity model,
Ours Tede)
Figure 6: Sand plasticity.
PlasticityNet does not need the hardening state in its input.
Snow Plasticity with Hardening Snow an elastoplastic material that can become stiffer under compression. Essentially, this is the effect of hard- ening where its elastic region get expanded. The variation in the stiffness across the snow body makes it easily fracture. Here simulate snowball hitting the ground in the MPM simulator (Figure 7p. We use Neo-Hookean elasticity with the closed-form non- associative Cam-Clay plasticity return mapping [15] (See Appendix A.2.2). Our method generates similar results compared to the ground truth when using the same time step size. The quantitative comparison of our results and the ground truth is shown in Figure 5b much larger time step sizes. However, more numerical damping artifacts are introduced as the time step size increases, which results in slightly different behaviors compared to the ground truth.
Ground 1e-5s)
1e:5s)
Ous (dr 1e:3s)
Figure Snow plasticity with hardening-
Our framework remains stable even under
7

---

Metal Plasticity with Hardening Metal is another com- mon plastic material with hardening. In this example, we train PlasticityNet to learn metal plasticity with the StVK elasticity and the closed-form von-Mises plasticity return mapping [39]. (See Appendix A.2.3) We simulate metal frame compressed by a rigid plate in the FEM simulator (Figure 8). where the Incremental Potential Contact (IPC) 299 is used to handle the frictional contact between the solids. When run the explicit time integration gen- erate ground truth, we have to decrease Young's 's modulus to enable using large enough time step sizes that the simulation can be finished in practical time. Our method with the original setting generates visually identical sults using a much large time step size. The quantitative comparison of our result and the ground truth is shown in Figure 5c
Ground truili (E - 107 Pa, di le-58)
Ouns (E - le7 Pa, 1o:56)
1e:2s)
Figure 8: Metal plasticity with harden-
ing.
Metal Plasticity Return Mapping Here we show an example simulated using PlasticityNet with a learned von- Mises plasticity return mapping. The underlying elasticity neo-Hookean, instead of the StVK model in the last ex- ample (See Appendix A.2.4). Note that for leo-Hookean material, there is no closed-form solution available for the von-Mises return mapping. In this case, a nonlinear optimization problem will need to he solved to perform the return mapping for every element/particle in every time step, which could severely slow down the standard explicit time integration. Using the same parameters as the metal compression experiments above, we show that PlasticityNet with learned plasticity return mapping under neo-Hookean elasticity can generate qualitatively similar results (Figure 9) to those from PlasticityNet with closed-form return mapping under the SiVK elasticity.
Figure 9: Learned metal plasticity return mapping with neo-Hookean elasticity.
MPM-FEM Coupling PlasticityNet enables the sim- ulation of plastic materials the MPM-FEM coupling framework BFEMP (33), where only pure elasticity supported. When simulating with explicit BFEMP, the time step size required by stability is the minimum be- tween MPM step size upperbound and FEM step size upperbound. Here we show example where stiff FEM clastic body falls onto MPM sand (Fgurre 10), where the implicit BFEMP can use time step size 1000x larger than the explicit BFEMP and achieves an approximately 5x speedup wall-clock time. We also remark that when the time step size is small (as is required to keep the explicit time integration stable in this case). MPM suffers from excessive numerical damping due the significant amount of particle-gric transfers. This known issue of explicit MPM simulations.
Fxplisit (dr - le-6s)
Churs (di 16-36)
Two-war way coupling between FEM clasticity and MPM sand plasticity.
Figure 10:
Different Energy Representations Here we include some different energy representations we investigated (Figure 11), whose inaccurate results motivated us to develop our final representation Equation 8, These experiments are all conducted on the 2D sand column collapse example. The first straightforward idea is find globally defined neural energy function (FF (F that solves Equation 6, where 0 is the parameter of the neural network. Note that it is theoretically unachievable to train a global potential energy function because the right hand side of Equution 6lis not integrable in the plastic region. But it is still worth trying to explore approximation by minimizing ((0) Ey
(F) (F) F However, the experiment shows that this formulation makes 8 3
the sand column behave like an elastic body. also noteworthy that the sand column cannot even maintain the rest shape at the first frame: it erroneously shrinks suddenly and jumps off the ground Additional insight is provided by realizing that linear correction necessary to exactly vanish stress when the deformation gradient is the identity; so experiment with (F) (F) V (I).

---

(FF)
V(F) Oe(F) V.9e())
Figure 11: Ablation studies on different energy representations.
3D Sand
3D Snow
3D Metal
Figure 13: 3D simulations with sand plasticity, snow palsticity and metal plasticity.
This formulation unfortunately also leads to an insufficient capture of plasticity, giving an elastic and visually distinct incorrect result. These observations motivate us to investigate a family of potential energies to solve Equation 6 locally.
first use (F, Fo) = (F, Fo) VrV. (I, Fo) and train with the loss function in Equation 9) The simulation captures certain plastic behaviors when the deformation small, but the result quickly deviates from the ground truth when the deformation becomes larger. Finally, we come up with Equation 8 to achieve the nice results in Figure 6
5.3 Ablation Studies
Stability Regularizer As ablation study for the stabil- ity regularizer in Equation II, we compare the simulations with and without the regularizer on two 2D examples (Fig
Regsbaiied
ure 12). Without the regularizer, the metal frame can not even stay in its original rest configuration after the first time step. In the sand example, particles in the highlighted regions tend to separate from the sand column in a non- physical manner. These demonstrate that our regularizer significantly improves the stability of the simulation.
Figure 12: The regularizer significantly improves the stability of the simulation.
5.4 Testing on 3D Simulations
Extending PlasticityNet to support 3D simulation is straightforward. We only need to increase the dimension of the inputs to the PlasticityNet. To improve expressiveness of the network, we also increase the dimension of hidden variables. Here we demonstrate the 3D versions of our 2D examples with similar physical parameters in Figure 13 3D sand plasticity, 3D snow plasticity, and 3D metal plasticity. The 3D metal is simulated with At 10-", and for sand and snow, we use At 10 to satisfy the CFL condition g MPM, preventing the particles from traveling farther than the grid cell spacing in single time step.
6 Conclusion
proposed PlasticityNet, neural network- setwork-based elastoplastic model learning framework that agnostic to spatial discretizations. PlasticityNet represents the clastoplastic forces the positional

---

gradients of learned potential energies, so that optimization time integration could be applied to achieve robust and efficient simulation at large time step sizes. We demonstrated that low-level components in traditional physical simulation frameworks can be substituted with neural networks to obtain desired numerical properties that benefit the computation. Notably, this also avoids tedious analytical derivations or expensive nonlinear root-findings without significantly sacrificing the accuracy. We believe our work can inspire more research that applies machine learning to physical simulation in the bottom-up style, maintaining fundamental physical properties and applicability to general scenarios.
Limitations and Future Work There are several limitations of our framework. (1) We cannot guarantee our ixed-point iteration will converge for arbitrary scenes. It is theoretically valuable to explore under what conditions the fixed-point can converge and what loss functions can accelerate (2) Although the regularizer added during the simulation improves the stability of the simulation without changing the solution at convergence, it may introduce some artificial viscosity because the regularized energy penalizes deformations away from Fg. Running more fixed-point iterations can alleviate this issue. It will also be interesting to explore adaptive weighting mechanisms for the regularizer, or convert this soft regularizer into a hard constraint. (3) We do not consider the Hessian of the learned plastic energy in our training. Since we use second-order methods to perform optimization time integration, the properties of the Hessian matrices may have an impact the convergence of the optimization method. Although the Jacobian matrices of the target gradients are asymmetric, it may be helpful if the Hessian of our learned elastoplastic energy can approximate them that the stiffness of the material can be more accurately resolved. (4) Principled physical assumptions of the learned potential energies by PlasticityNet, such as lower-boundee boundedness and convexity, are not enforced. It is interesting to explore whether enforcing these energy properties would positively influence the convergence of the optimizations and fixed-point iterations. (5) A trained PlasticityNNt can be directly re-scaled to accommodate different Young's 's modulus, but it needs to be re-trained for materials with different Poisson's 's ratio or plasticity parameters. It is an important future work to let our model more easily generalize to different parameters. For example, these parameters can become extra inputs to the neural network. The generalized energy can also be integrated into differentiable simulators
the convergence.
16 43
solve many inverse problems 38 36 32].
Acknowledgments and Disclosure of Funding
We would like to thank Pingying Chen for narrating the supplemental video. We would also like to thank the anonymous reviewers for their valuable comments. This work has been supported in part by NSF CAREER 21 2153851, CCF-2153863, ECCS-2023780, IIS-201 1471, I1S-2016414, I1S-2006570. DOE ORNL contract 4000171342.
References
[1] F. As' ad, P. Avery, and C. Farhat. A mncohaeds--nfformedd artificial neural network approach in data-driven constitutive modeling. In AIAA Scitech 2022 Forum, page 0100, 2022.
[2] A. W. Bargteil, T. Shinar, and P. G. Kry. An introduction to physics-based animation. In SIGGRAPH Asia 2020 Courses, pages 1-57, 2020.
[3] Battaglia, R. Pascanu, M. Lai, D. Jimenez Rezende, al. Interaction networks for learning about objects, relations and physics. Advances in neural information processing systems, 29, 2016.
[4] F. D. A. Belbute-Peres. T. Economon, and Z. Kolter. Combining differentiable pde solvers and graph neural networks for fluid flow prediction. In International Conference on Machine Learning, pages 2402-2411. PMLR, 2020.
[5] F. E. Bock, R. C. Aydin, C. J. Cyron, N. Huber, S. R. Kalidindi, and B. Klusemann. A review of the application of machine learning and data mining approaches in continuum materials mechanies. Frontiers in Materials, page 110, 2019.
[6] M. B. Chang, T. Ullman, A. Torralba, and J. B. Tenenbaum. A compositional object-based approach to learning physical dynamics. arXiv preprint arXiv:1612.00344 2016.
10

---

[7] Y. Chen, M. Li. L. Lan, H. Su, Y. Yang, and C. Jiang. A unified newton barrier method for multibody dynamics. ACM Trans. Graph. (SIGGRAPH), 41(4), 2022.
[8] R. Courant, K. Friedrichs, and H. Lewy. On the partial difference equations of mathematical physics. IBM journal of Research and Development, 11(2):215-25 1967.
[9] D. C. Drucker. Some implications of work hardening and ideal plasticity. Quarterly of Applied Mathematics, 7(4):4 11-418, 1950.
[10] M.-L. Eckert, K. Um, and N. Thuerey. Scalarflow: a large-scale volumetric data set of scalar transport flows for computer animation and machine learning. ACM Transactions on Graphics (TOG), 38(6):1-16, 2019.
[11] Z.. Ferguson, M. Li, T. Schneider, F. Gil-Ureta, T. Langlois, C. Jiang, D. Zorin, D. M. Kaufman, and D. Panozzo. Intersection-free rigid body dynamics. ACM Transactions on Graphics, 40(4), 2021.
[12] C. Gan, J. Schwartz, Alter. M. Schrimpf. Traer, De Freitas, J. Kubilius, A. Bhandwaldar, N. Haber, M. Sano, et al. Threedworld: A platform for interactive multi-modal physical simulation. arXiv preprint arXiv:2007. 04954, 2020.
[13] Garnier, J. Viquerat, J. Rabault, A. Lareher, A. Kuhnle, and E. Hachem. A review on deep reinforcement learning for fluid mechanies. Computers & Fluids, 225:104973, 2021.
[14] T. F. Gast, C. Schroeder, A. Stomakhin, C. Jiang, and J. M. Teran. Optimization integrator for large time steps. IEEE transactions on visualization and computer graphics, 10100:1103-111: 2015.
[15] J. Gaume, T. Gast, Teran, A. Van Herwijnen, and C. Jiang. Dynamic anticrack propagation in snow. Nature communications, 9(1):1110, 2018.
[16] Y. Hu, L. Anderson, T.-M. Li, Q. Sun, N. Carr, J. Delln- and F. Durand. Difftaichi: Differentiable programming for physical simulation. ICLR, 2020.
[17] Z. Huang, Y. Hu, T. Du, S. Zhou, H. Su, J. B. Tenenbaum, and C. Gan. Plasticinelab: A soft- body manipulation benchmark with differentiable physics. arXiv preprint arXiv:2104.03311, 2021.
[18] C. Jiang, T. Gast, and J. Teran. Anisotropic clastoplasticity for cloth, knit and hair frictional contact. ACM Transactions on Graphics (TOG), 36(4):1-14, 2017.
[19] C. Jiang, C. Schroeder, J. Teran, A. Stomakhin, and A. Selle. The material point method for simulating continuum materials. ACM SIGGRAPH 2016 Courses, pages 1-52. 2016.
[20] Y. Jin, Y. Han, Z. Geng, Teran, and R. Fedkiw. Analytically integratable zero-restlength springs for capturing dynamic modes unrepresented by quasistatic neural networks. arXiv preprint arXiv: 10122, 2022.
[21] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics- informed machine learning. Nature Reviews Physics, 36)):222-440 2021.
[22] D. P. Kingma and Ba. Adam: A method for stochastic optimization. arXiv preprint 1rXiv:1412.6980 2014.
[23] G. Klár, T. Gast, A. Pradhana, C. Fu, C. Schroeder, C. Jiang, and J. Teran. Drucker-prager clastoplasticity for sand animation. ACM Transactions on Graphics (TOG), 35(4):1-12, 2016.
[24] D. Kochkov, J. A. Smith, A. Alieva, Q. Wang, M. P. Brenner, and S. Hoyer. Machine learning- accelerated computational fluid dynamies. Proceedings of the National Academy of Sciences, 1 11(22)), 2021.
[25] A. Koeppe, F. Bamer, M. Selzer, B. Nestler, and B. Markert. Explainable artificial intelligence for mechanies: physics-explaining neural networks for constitutive models. Front. Mater. 8 824958. doi: 10.3389/fmats, 2022.

---

[26] L. Lan, D. M. Kaufman, M. Li, C. Jiang, and Y. Yang. Affine body dynamics; Fast, stable & intersection free simulation of stiff materials. ACM Trans. Graph. (SIGGRAPH), 41(4), 2022.
[27] L. Lan, Y. Yang, D. Kaufman, J. Yao, M. Li, and C. Jiang, Medial ipc: accelerated incremental potential contact with medial elastics. ACM Trans. on Graph., 40(4):1-16, 2021.
[28] M. Li. Robust and Accurate Simulation of Elastodynamics and Contact. PhD thesis, University of Pennsylvania, 2020.
[29] M. Li, Z. Ferguson, Schneider, T. Langlois, D. Zorin, D. Panozzo, C. Jiang, and D. M. Kaufman. Incremental potential contact: Intersection-and inversion-free, large-deformation dynamics. ACM transactions on graphics, 2020.
[30] M. Li, M. Gao, T. Langlois, C. Jiang, and D. M. Kaufman, Decomposed optimization time integrator for large-step elastodynamics. ACM Trans. Gr., 38(4), 2019.
[31] M. Li, D. M. Kaufman, and C. Jiang. Codimensional incremental potential contact. ACM Transactions on Graphics, 40(4), 2021.
[32] S. Li, Z. Huang, T. Du, H. Su, J. B. Tenenbaum, and C. Gan. Contact points discovery for soft-body manipulations with differentiable physics. arXiv preprint arXiv:2205.0288 2022.
[33] X. Li, Y. Fang, M. Li, and C. Jiang. Bfemp: Interpenetration-feee free mpm-fem coupling with barrier contact. Comp. meth. applied mech. eng., 2021.
[34] X. Li, M. Li, and C. Jiang. Energetically consistent inelasticity for optimization time integration. ACM Trans. Gr., 41(4), 2022.
[35] X. Li, B. Sovilla, C. Jiang, and J. Gaume. Three-dimensional and real-scale modeling of flow regimes in dense snow avalanches. Landslides, 18(10):3393-344 2021.
[36] X. Lin, Z. Huang, Y. L., J. B. Tenenbaum, D. Held, and C. Gan. Diffskill: Skill abstraction from differentiable physics for deformable object manipulations with tools. arXiv preprint arXiv:2203. 17275, 2022.
[37] R. Luo, T. Shao, H. Wang, W. Xu, X. Chen, K. Zhou, and Y. Yang. Nnwarp: Neural network- based nonlinear deformation. IEEE transactions on visualization and computer graphics, 26(4):1745-177 1759, 2018.
[38] P. Ma, T. Du, J. B. Tenenbaum, W. Matusik, and C. Gan. Risp: Renderingginvariaat invariant state predictor with differentiable simulation and rendering for cross-domain parameter estimation. arXiv preprint arXiv:2205.00 05578, 2022.
[39] R. v. Mises. Mechanik der festen körper im plastisch-deformablle zustand. Nachrichten von der Gesellschaft der Wissenschaften Göttingen, Mathemaaisch--Plyiikaasss Klasse, 1913:582-592, 1913.
[40] M. Mozaffar, R. Bostanabad, W. Chen, K. Ehmann, J. Cao, and M. Bessa. Deep learning predicts path-dependent plasticity. Proceedings of the National Academy of Sciences, 116(52):26414 26420, 2019.
[41] T. Pfaff, M. Fortunato, A. Sanchez-Gonzaler and W. Battaglia. Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409, 2020.
[42] W. Prager. The theory of plasticity: survey of recent achievements. Proceedings of the Institution of Mechanical Engineers, 169(1):41-77, 1955.
[43] Y. Qiao, J. Liang, V. Koltun, and M. Lin. Differentiable simulation of soft multi-bout systems. Advances in Neural Information Processing Systems, 34:17123-17135, 2021.
[44] M. Raissi, P. Perdikaris, and G. E. Kariiadaiis. Physics-informel neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686-707. 2019,
12

---

[45] D. Reimann, K. Nidadavolu, H. ul Hassan, N. Vajragupta, T. Glasmachers, P. Junker, and A. Hartmaier. Modeling macroscopic material behavior with machine learning algorithms trained by micromechanical simulations. Frontiers in Materials, 6:181, 2019.
[46] A. Sanchez-Gonzalez, J. Godwin, T. Pfaff, R. Ying, J. Leskovee, and P. Battaglia. Learning to simulate complex physics with graph networks. In International Conference on Machine Learning, pages 8459-8468. PMLR, 2020.
[47] S. Shen, T. Shao, K. Zhou, C. Jiang, Luo, and Y. Yang. Hod-net: High-order differentiable deep neural networks and applications. 2022.
[48] S. Shen, Yin, T. Shao, H. Wang, C. Jiang, L. Lan, and K. Zhou. High-order differentiable autoencoder for nonlinear model reduction. arXiv preprint arXiv:2102.11022 2021.
[49] E. Sifakis and J. Barbic. Fem simulation of 3d deformable solids: practitioner's guide to theory, discretization and model reduction. In Acm siggraph 2012 courses, pages 1-50. 2012.
[50] A. Stomakhin, R. Howes, C. Schroeder, and J. M. Teran. Energetically consistent invertible elasticity. In Proceedings of the 11th ACM SIGGRAPH/EEuogrpphis conference on Computer Animation, pages 25 32, 2012.
[51] L. Sun, H. Gao, S. Pan, and J.-X. Wang. Surrogate modeling for fluid flows based on physics- constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 361:1 12732, 2020.
[52] A. P. Tampubolon, T. Gast, G. Klár, C. Fu, J. Teran, C. Jiang, and K. Museth. Multi-species simulation of porous sand and water mixtures. ACM Transactions on Graphics (TOG), 36(4):1- 11, 2017.
[53] N. N. Vlassis, R. Ma. and W. Sun. Geometric deep learning for computational mechanics part i: Anisotropic hyperelasticity. Computer Methods in Applied Mechanics and Engineering. 371:113299, 2020.
[54] N. N. Vlassis and W. Sun. Sobolev training of thermodynamic-infomeed neural networks for interpretable elasto-plastin paaastiilltttitty models with level set hardening. Computer Methods in Applied Mechanics and Engineering. 377:1 13695, 2021.
[55] X. Wang, M. Li, Y. Fang, Zhang, M. Gao, Tang, D. Kaufman, and C. Jiang. Hierarchi- cal optimization time integration for cfi-rate mpm stepping. ACM Trans. on Graph., 3963):1-16. 2020.
[56] J. Wolper, Y. Chen, M. Li, Y. Fang, Z. Qu, Lu, M. Cheng, and C. Jiang. Anisompm: Animating anisotropic damage mechanics: Supplemental document, ACM Trans. Graph, 39(4). 2020.
[57] Wolper, Fang, Li, Lu, Gao, and C. Jiang. Cd-mpm: continuum damage material point methods for dynamic fracture animation. ACM Transactions Graphics (TOG), 38(4)11 15, 2019.
[58] J. Wolper, M. Gao, M. P. Lüthi, V. Heller, A. Vieli, C. Jiang, and J. Gaume. A glacier- ocean interaction model for tsunami genesis due iceberg calving. Communications Earth & Environment, 2(1): 2021.
[59] S. Yang, X. He, and B. Zhu. Learning physical constraints with neural projections. Advances in Neural Information Processing Systems, 33:5178-5189, 2020.
[60] Y. Zhao, J. Choo, Y. Jiang, M. Li, C. Jiang, and K. Soga. A barrier method for frictional contact on embedded interfaces. Computer Methods in Applied Mechanics and Engineering, 393: 1 14820, 2022.
13

---

Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits. hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [N/A]
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [N/A]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're rou're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
14


Reviews:
[{"review_id": "xV3VYN6LaLy", "reviewer": "Reviewer_RLXg", "summary": "The authors propose a neural network-based approach, which enables the simulation of a wide range of arbitrary elasticity-plasticity combinations using time step-independent, unconditionally stable optimization-based time integrators.\n", "questions": "1. In Fig.9, why the errors of sand and snow are reduced along with simulation, but the error of metal increases.\n2. How about the generalization ability to a different initial shape with the same material and a different initial shape with different material? For example, give a model trained on the sand with the initial shape of a cube (term it as 'sand cube'),  and then test it on the 'sand sphere'. Further, test the model on the 'snow sphere'.\n", "limitations": "Yes, limitations and societal impact were appropriately addressed.\n\n", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper proposes a general framework to enable the simulation of objects with a wide range of materials. The framework is novel and reliable. The paper is well-written, and the experimental setting is clear. I think this is a good paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "rUJ7l63GXyh", "reviewer": "Reviewer_LXcT", "summary": "This paper presents a neural network model to learn a plastic material’s energy function in elastoplastic simulation. The paper argues that constructing such an energy model enables optimization-based solvers for the time integration problem, which is more advantageous than solving the force equilibrium directly. Additionally, the paper introduces a few useful extensions on top of the network model: hardening effects, stability regularizer, and volume-preserving return mapping. The paper presents various 2D and 3D examples (sand, snow, and metal) to demonstrate the value of the proposed network.", "questions": "1. For solving the time integration, this paper seems to favor energy minimization (Eqn. 3) instead of solving the governing equation (Eqn. 1). However, I am under the impression that both approaches have been equally popular in deformable-body simulation, and the argument in lines 116-118 doesn’t fully convince me because we could also solve Eqn. 1 using Newton’s method plus line search/other safeguards. Maybe I am unaware of some critical reference papers here. Could you please comment more on this?\n\n2. On a related note, I am not sure I get what lines 138-140 intend to say. To me, directly feeding Z(F) to the elastic potential (followed by solving the nonlinear root-finding or energy minimization problem) seems to be a legit solution. Could you clarify this?\n\n3. I really appreciate the magic equation 8. Could you tell me more about how you derived it?\n\n4. Eqn 11: why is shear modulus used as the weight of the regularizer? Is there physical intuition behind this choice?\n\n5. One general concern that I always have with such network methods is that they don’t seem to guarantee physical plausibility. For example, a reasonable energy potential is expected to be bounded below, which is easy to guarantee for an analytical energy potential function but could be difficult to enforce for a network model (The discussion on Hessians in the limitation section might be somewhat relevant to this point). Could you give me a strong motivation for why we should appreciate these network models?\n\n6. I am confused by the experiments that compare ground-truth simulation generated using explicit time integration with your approach that solves implicit time integration. Why not use implicit time integration in both settings?\n\n7. I also didn’t find statistics about the wall clock time cost. I get that having the network to model the energy potential unlocks a larger time step than explicit methods, but doesn’t each implicit time step cost more wall-clock time? Therefore, what is the final time cost for generating each row in Fig. 5?\n\n8. For Fig 7 row 2 and row 3: not sure I understand why adjusting both time step size and Young’s modulus between these two rows.\n\n9. I also didn’t find a discussion about the generalization of the trained network model. Actually, I am not sure how generalization/training data/test data should be defined for this model. I am raising this concern because network models typically over-parametrize physics models (physics laws often only have a few parameters while networks can easily have thousands or more).\n", "limitations": "The limitation section looks good to me.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength: this paper discussed a serious physics simulation problem. The mathematical and physical background is fairly solid. Implementing MPM, FEM, and IPC are nontrivial, for which the authors deserve some credits if they coded everything from scratch. The 2D and 3D demos are also visually pleasing.\n\nWeaknesses: I feel the paper still lacks a deeper insight behind the introduction of the network model. I also think the evaluation section misses some critical points.\n\nOverall, I think this paper is very interesting and promising, but I am afraid it is not there yet. My initial score is rejection, but I am happy to re-evaluate the paper if the authors can address my questions.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "pzaiVup7d32", "reviewer": "Reviewer_mwtb", "summary": "This paper proposes to learn a PlasticityNet to facilitate implicit Euler method for simulating plastic deformation. \n\n- Given a deformation gradient, the PlasticityNet behaves like an energy function, which gradients represents the force/stress of the plastic materials. Such formulation allows the user to write the forward step as an optimization problem and enables the implicit Euler method.\n- The plasticityNet only needs to be locally accurate around input deformation gradients for plastic deformation. Based on this property, the authors propose a fix-point iteration method with a regularization term that better exploits the locality to stablize the training.\n- The authors conducted experiments on a wide range of materials and achieve good performance.", "questions": "Regarding the stability regularization, it seems that due to the locality of the PlasticityNet, we need to solve a constrained optimization problem within the pre-defined trust regions. Isn’t it better to add the hard constrain directly? ", "limitations": "yes", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n- Simulating plastic material has a wide range of applications, from animation to robotics. This paper attacks the fundamental problem of plastic material simulation. It introduces a learning-based method to enable implicit Euler, which will have an impact on the field of learning-based simulation and facilitate research on other domains.\n- The proposed techniques are very novel to me. Learning a Pseudo-energy function for implicit Euler is a good idea and may have potential to generalize on other fields. Besides, the idea of parameterizing the function with F_0 for local approximation and the special form that embeds physical prior both are also very interesting.\n- The paper is well-written and the presentation is very clear and easy to follow.\n\nWeakness:\n\n- What I find missing is the discussion about the time complexity. Will the network and the fixed-point iteration introduce an additional computation burden, canceling out the benefits of the implicit Euler?\n- The paper could be strengthened for readers from the learning community by adding an ablation study or baseline comparisons. For example, what would happen if we directly learn a global function without inputting F_0, or if we do not introduce the “linear correction” but learn the potential energy directly? It would be interesting to have those experiments to help understand the philosophy behind the particular design. Such ablation studies are necessary to support the method. I would also be interested if the learned network can generalize to different scenes.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "iOvbiClJup", "reviewer": "Reviewer_8rPt", "summary": "The authors proposed a learning-based model to simulate plasticity effects. The method is based on optimization based time integration and trains a network to predict the energy given the deformation gradients and the hardening states. Results show that the predicted energy can successfully guide the simulation to behave similarly as ground truth ones.\n", "questions": "See above.", "limitations": "No. See above.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n - The network only tries to learn the energy mapping for a certain point, which provides the ability to scale the method to arbitrary scenes.\n - Using neural network to predict the energy can enable larger step time, which in turn speed up the simulation.\n\nWeaknesses:\n - Missing references:\n   - Hu, Y., Anderson, L., Li, T. M., Sun, Q., Carr, N., Ragan-Kelley, J., & Durand, F. (2019, September). DiffTaichi: Differentiable Programming for Physical Simulation. In International Conference on Learning Representations.\n   - Qiao, Y., Liang, J., Koltun, V., & Lin, M. (2021). Differentiable simulation of soft multi-body systems. Advances in Neural Information Processing Systems, 34, 17123-17135.\n   These two papers combines machine learning with simulation via differentiating the simulation as a whole which can enable efficient inverse problem solving as well. I suggest the authors discuss this direction and if possible, compare with DiffTaichi since it also uses MPM.\n - It is not clear how much performance gain there is after using the learned model. The step time is indeed 100x larger (from 1e-5 to 1e-3, as mentioned in the text), but to achieve convergence, the authors adopted a fixed point iteration which is not required in traditional simulations. *A few number* of iterations as mentioned by the authors are not accurate enough to evaluate the strengths of this approach. I would like to see performance tables or figures that shows significant improvement with the proposed approach so that it can be strongly supported.\n - It seems the network has to be trained all over again if the material parameters are different, which limits the generalization ability of this approach. While it is a common open problem for all learning-based simulations, I suggest the authors discuss and provide insights towards solving it in the limitation and future work section.\n - It is not clear how the 100x step time is achieved. Why 100x but not 1000x? Is there any theorectical hypothesis on how much at most a learning based model can improve? Does the error goes up with the step time, and how is the trend like? ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept