NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory
Authors: Navami Kairanda
Marc Habermann
Christian Theobalt
Vladislav Golyanik
Venue: NeurIPS 2024 poster (2024)
Keywords: Neural field
implicit representation
cloth simulation
consistent simulation
multi-resolution
Kirchhoff-Love theory
Primary area: machine_learning_for_physical_sciences
TL;DR: We propose a new quasistatic cloth simulator using thin shells, in which surface deformation is encoded in neural network weights as a neural field.

Abstract: Despite existing 3D cloth simulators producing realistic results, they predominantly operate on discrete surface representations (e.g. points and meshes) with a fixed spatial resolution, which often leads to large memory consumption and resolution-dependent simulations. Moreover, back-propagating gradients through the existing solvers is difficult and they hence cannot be easily integrated into modern neural architectures. In response, this paper re-thinks physically plausible cloth simulation: We propose NeuralClothSim, i.e., a new quasistatic cloth simulator using thin shells, in which surface deformation is encoded in neural network weights in form of a neural field. Our memory-efficient solver operates on a new continuous coordinate-based surface representation called neural deformation fields (NDFs); it supervises NDF equilibria with the laws of the non-linear Kirchhoff-Love shell theory with a non-linear anisotropic material model. NDFs are adaptive: They 1) allocate their capacity to the deformation details and 2) allow surface state queries at arbitrary spatial resolutions without re-training. We show how to train NeuralClothSim while imposing hard boundary conditions and demonstrate multiple applications, such as material interpolation and simulation editing. The experimental results highlight the effectiveness of our continuous neural formulation.

Neural SlothSim: Neural Deformation Fields Meet the Thin Shell Theory
Navami Kairanda
Mare Habermann
Christian Theobalt
Vladislav Golyanik
Max Planck Institute for Informatics, Saarland Informatics Campus
Abstract
Despite existing 3D cloth simulators producing realistic results, they predominantly operate on discrete surface representations (e.g.. points and meshes) with fixed spatial resolution, which often leads to large memory consumption and resolution- dependent simulations. Moreover, back-propage proo--rppggating gradients through the existing solvers is difficult, and they hence cannot be easily integrated into modern neural architectures. In response, this paper re-thinks physically accurate cloth simulation: We propose NeuralClothSim, i.e., new quasistatic cloth simulator using thin shells, in which surface deformation is encoded in neural network weights in the form of a neural field. Our memory-efficiet solver operates on a new continuous coordinate- based surface representation called neural deformation fields (NDFs); it supervises NDF equilibria with the laws of the non-linear Kirchhoff-Love shell theory with a non-linear anisotropic material model. NDFs adaptive: They I) allocate their capacity to the deformation details and 2) allow surface state queries at arbitrary spatial resolutions without re-training. We show how to train SeuralClothSim while imposing hard boundary conditions and demonstrate multiple applications, such as material interpolation and simulation editing. The experimental results highlight the effectiveness of our continuous neural formulation. See our project page: https:: //dqqv mpi inf . wwg.ppg........ DeeNeurallCothhiine
Introduction
Realistic cloth simulation is a central, long- standing and challenging problem in computer graphics. It arises in game engines, computer animation, movie production, digital art, and garment digitisation, only name a few areas. To date, has been mostly addressed with physics-based simulators operating explicit geometric representations, i.e,, meshes and particle systems. While recent simulators 24 36
can produce realistic 3D simulations that obey various types of boundary conditions and consider secondary effects, but their operational principle remains limited in several ways. First, they work on discrete surface representations such as meshes and points inherently assuming pre-defined spatial resolution that cannot be easily changed once the simulation is accomplished. Second, re-running with different meshing of the same initial template leads to different folds and wrinkles, which is often problematic for downstream applications. Third, explicit geometries require notoriously large amounts of storage for the detailed simulation: the memory size grows linearly with the number of points. Moreover, it is difficult to integrate simulators into learning frameworks and to edit the output 3D state without re-running the simulation.
33
32
38
The recent advances in physics-informen neural networks 449 25
as well as the success of neural fields 43 62 65), makes us question if continuous coordinate-bacee based representations can alleviate these limitations. All these considcrations motivate to rethink the fundamentals of physically accurate cloth simulation and we introduce a new approach for cloth quasistatics, in which the surface deformation is encoded in neural network weights. The proposed neural architecture is coordinate-
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Training
Inference
Material Space w
Neural Deformation Field
Parametric Spare
Thin Shell Supervision
Multi Resolution
Multi-Material
Figure 1: NeuralClothSSm the first neural cloth simulator representing surface deformation as a neural field. It is supervised for each target scenario with the laws of the Kirchhoff-Love thin shell theory with non-linear strain (left). Once trained, the simulation can be queried continuously and consistently enabling different spatial resolutions (center). NeuralClothSim can also incorporate learnt priors such as material properties that can be edited at test time (right).
based and has multiple advantages compared to previous simulators; see Fig. [or an overview. Our neural fields are adaptive, i.e., the parameters are used to encode the deformations as they occur As matter of efficiency, we neither need know the resolution in advance before the simulation nor do we require complex re- re-meshing schemes 445 Realistic cloth simulation requires modelling geometric non- oon-iineasssees and non- mon-linear anisotropic elasticity. It involves large bending deformations and rigid transformations leading to non-linear point displacements. To efficiently model this, we rely on neural networks as they are good universal (non-linear) function approximators.
We model cloth simulation as thin shell boundary-value problem with the deformation governed by the Kirchhoff- Love shell theory. In contrast to previous simulators using Kirchhoff-Lovw shell relying on isogeometric analysis [40] or subdivision surface algorithms [24] 20 we model thin shell deformations as implicit neural representations. i.e., 3D deformation fields encoding cloth quasistatics. During training, our formulation supervises a neural deformation field (NDF), minimising the cloth's 's
sensitive to the finite element discretisations of the initial surface, which could lead to inconsistent folds, we generate simulations with consistent drapes, folds, and wrinkles. This important for downstream applications that might query (e.g., in the case of renderer) even modify (e.g., like inverse methods) the simulation with adaptive sampling. Next, our representation memory-efficient and the simulation states are generated directly in a compressed form. In summary, our core technical contributions are follows:
potential energy functional. In contrast to classical simulators [3 36
A new continuous coordinate-based neural representation (Sec. 4.1 and a new neural solver for cloth quasistatics based on thin shell theory that accepts boundary conditions such external forces guiding motions (Sec. 4.2
Modelling of thin shell's 's deformation with non-linear Kirchhoff- theory supervising the neural deformation fields (Sec. (4.)). Upon convergence, the equilibrium state can be queried continuously and consistently.
Applications of the proposed neural simulator including material interpolation and fast editing of simulations according to updated simulation parameters (Sec. 5.4)
We want point out that do claim qualitative superiority classical cloth simulation meth- ods and completeness of our formulation (e.g., our method does not consider collisions). However, we surface representation and solver
believe that our new way of deeply integrating neural networks into cloth simulation has the potential to stimulate future research in this direction, and we show that our formulation overcomes multiple fundamental limitations of existing discrete approaches.
2 Related Work
Cloth Simulation a well-studied problem 91
26 59 38
51], with the first methods dating
back to the 1980s 2 [55] The computational flow of the modern simulation approaches includes:
Discretisation using the finite element method (FEM) [8 45), implicit time-integration []] (32]
frictional contact 33 42). and collision handling 46 54, 26 Cloth simulators model real fabric
behaviors [15] (1. which typically done by fitting constitutive material models. Liang er 38
and Li et al. [36] introduced differentiable cloth simulators, which were subsequently shown to be

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also useful in 3D reconstruction as a physics- physics-based prior [28, 37]. Zhang et al.'s 's approach [67 68 enables interactive exploration of cloth parameters with progressively consistent quasistatics. Another category of methods constitutes neural cloth simulators. Pfaff et al. [48] proposed to learn simulations
using graph neural networks. Bertiche et al. [7] is neural simulator for static draping of garments on virtual character. It is further extended with self-supervised approaches [51, [] to learning garment dynamics. They leverage physics-based loss terms and do not require simulated ground data. However, these methods are application-oriented rather than approaches for general cloth simulation, as the garments are skinned to the human body and garment deformations are driven by body shape and poses. Several methods for cloth simulation rely the Kirchhoff- Love shell theory [9 24]. The energy functionals in the theory require higher-order derivatives, which are not available for general unstructured triangle meshes. In their pioncering work, Cirak et al. [14] present Loop subdivision with control meshes that meet this additional C' interpolation requirement, which is extended dynamic cloth simulation with corotational strains [56)- NURBS isogeometry (40] also enables continuity, whereas recent methods [16 31 rely on Catmull- subdivision surfaces and model the geometric non-linearity of shells. All the aforementioned cloth simulators (traditional FEM, neural, and Kirchhoff-Love) use discrete surface representation (i.e., meshes) with several inherent limitations. The representation is not adaptive, and simulations suffer from coarse-to-fine inconsistency and are sensitive to initial discretisation.
58 offer promising alternative to meshes. As a common theme, these methods use coordinate-bl based MLP for neural field paramcterisation, which takes coordinates in the spatio-temporal domain and returns the task-specific property, e.g. occupancy or SDF values. For a detailed discussion, we refer to the survey of Xie et al. (64]. However, none of the works focus integrating such neural fields into the cloth simulation, which is the main goal of the proposed work.
Neural Fields. Recent approaches parameterising surfaces as neural fields 62 53
43
66 35 also dubbed Physics-Ifformed Informed Neural Networks (PINNs), leverage neural networks for solving tasks that are supervised by the laws of physics; we refer to recent survey from Hao et al. 25 for detailed review. Chen et al. [T] [0] use implicit neural representation to accelerate [ or replace [0] PDE solvers. However, they do not demonstrate thin-shell simulation. While previous works such as Rao et al. [50] and Zehnder et al. [66] applied neural implicit representations for volumetric clastodynamic problems, our approach focuses aspects such as geometric non- non-linearities and the integration of non-linear anisotropic models that are crucial for simulating large deformations and rotations. Another method [65] allows the processing of neural fields encoding geometric structures, Conceptually, the most closely related to ours is the work of Bastek and Kochmann (5), however, there are important differences to our work. First, they model linear small-strain regime for Naghdi shells, whereas we model the full non-linear stretching and bending behaviour of clothes. Second, we propose several architectural improvements -periodic activation functions, periodic boundary conditions, data-driven orthotropic material model- that are necessary for producing realistic wrinkles and folds, and demonstrate generalisation to point loads. different material and boundary values. Next, we present a short background on Kirchhoff-Lowe theory that enables us to model a cloth deformation as thin shell.
Neural Networks for Solving PDEs/ODES. Several recent approaches (50 49 12
realistic thin-shell and cloth simulation. It addresses important simulation
3 Kirchhoff-Love Thin Shell Theory for Cloth Modeling
Before we explain our method, we define our cloth representation. We characterise cloth as thin shell and model its behaviour with the Kirchhoff-Love theory 3g 63 A thin shell is 3D geometry with a high ratio of width to thickness. The shell continuum can be kinematically described by the midsurface located in the middle of the thickness dimension and the director, a unit vector directed along fibres in the shell that are initially perpendicular to the midsurface. The Kirchhoff hypothesis states the director remains straight and normal, and the shell thickness h E IR does not change with deformation (see inset). We provide a detailed review of Kirchhoff-Love thin shell theory in App. B
Notation. Throughout the document, we use Greek letters for indexing quantities on the midsurface, e.g., a., a, 3,... 1, 2, and Latin letters for indexing quantities on the shell, e.g., gi, j, 1,2, 3. Italic letters a, A indicate scalars, lower case bold letters a indicate first-order tensors (vectors), and upper case bold letters A indicate second-order tensors. An index can appear as superscript or subscript. Superscripts (.) refer to contravariant components of a tensor, which scale inversely with the change of basis, whereas subscripts (+), refer to covariant components that change in the same
3

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Input
Boundasy
Model & Supervision
Neural Deformation Field
BQU
min Jp
W(u]
f- u
Outpot
Simulated x(() = x00 w(()
Reference State x(E)
Strain Computation
Parametric Domain
Cloth Material
bending Steam
Membrane Strain
External
Forces
M(3) (3)
Figure NeuralClothSim takes as input a thin shell in the reference state and its material properties, boundary motion and external forces. It then learns an NDE, i.e., coordinate-based implicit 3D deformation field. inference, NDF can continuously queried for the deformed state of the surface equilibrium using curvilinear coordinates from the parametric domain. We use the Kirchhoff-Love thin shell modelling to supervise the cloth quasistatics with the potential energy functional.
way as the basis transforms. Moreover, we use upper dot notation for time derivatives, lower comma notation for partial derivatives with respect to the curvilinear coordinates, E", and vertical bar for covariant derivatives, e.g., ú 0u/Ot x., 0x/d5", and tta la respectively. Geometric quantities with overbar notation (.) refer to the reference configuration. Additionally, Einstein summation
convention of repeated indices is used for tensorial operations, e.g., Parpp = -Pal + Pa2ip A detailed list of notations be found in Tab. in Appendix B
Midsurface
4 Method
MI
We propose NeuralClothSim, i.e., new approach for continuous and consistent quasistatic cloth simulation relying the thin shell theory. We seek to generate a complex simulation state at equilibria given a cloth geometry in reference configuration, its material properties and external forces. The physical basis for our cloth Figure Kirchhoff- shell quasistatics is the nonlinear Kirchhoff-Loov thin shell equations that model the stretching and bending of cloths in unified manner. We parameterise the cloth states as neural deformation field (NDF) defined over a continuous parametric domain (Sec. 4. We explicitly account for positional and periodic boundary conditions, incorporated as hard constraints (Sec. 4.2) NDF is optimised using loss function based on the potential energy functional (Sec. 4.3) Fig. provides method overview.
Directoc
at
x(0
4.1 Neural Deformation Field (NDF)
At the core of our approach is neural deformation field (NDF), continuous representation of cloth quasistatics, entirely parameterised by neural network. Following Sec. we model cloth geometry as Kirchhoff-Love thin shell. Given the rest state x{)) of cloth, we describe the equilibrium state x(E) of its midsurface under the action of external forces f({) and boundary constraints Ba(E) using
x({) x({) + u({), with == (s' 5 S
(1)
The curvilinear coordinate space (2) can (but does not need to) naturally correspond to the orthotropic warp-weft weft structure of woven clothes. As examples, the reference state associated with flat square cloth of side L in the zyyplane and that of a garment sleeve (radius R, length L) admitting natural parameterisation with cylindrical coordinates are:
{{E [g' -2.55. v(E", 1),
(2)
{ Ro Rsin vg' (0,2m)) L).
Analytically defining surface parameterrisatonss might not be feasible for reference geometries given as meshes. In such cases, we learn the reference parametrisation by fitting an MLP x(E; Y) with parameters Y to the reference mesh. Specifically, we learn x by supervising with the fosloss C(T) = | Y)- I|, where R IR=2 are the vertices and texture coordinates of the given reference mesh. The advantage of this preprocessing over directly using the reference mesh is that we can continuously sample in the parametric domain by querying the MI. and compute all the geometric quantities at these points, similar to analytical access to the reference surface. Our key

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idea is to regress the displacement field u({) using an MLP Fe: : RR3 and optimise its weights θ to minimise the total potential energy of the thin-shell cloth. Specifically, the NDF u is formulated as follows:
u(E: θ) (B)
(3)
where (E) and (a(E) are functions that respectively account for periodic and Dirichlet boundary conditions. In Sec. 4.2 we elaborate on encoding such conditions as hard constraints.
Apart from being parameter-differnniaabie i.e., the gradient VaJe is defined everywhere, Fe needs to be input-differentiabl,, i.e., Fe must exist likewise, in order to compute the strains required for the Kirchhoff-Loww energy functional. This restricts the activation function used in the network; only use periodic sine as the preferred activation function (3 as it can represent high-frequency signals (needed for folds and wrinkles) while allowing for computing higher-order derivatives. Note that unlike NDF u(E: θ), we use GELU (27] activations for smoothly fitting the reference shape, ( T). Sec. 4.3 describes the optimisation procedure to train the deformation field 0).
C² Co-continuous non-linearities can be used. Therefore,
Once trained, Fe provides continuous access to the cloth quasistatics. where the network can be queried at any point in the spatial domain f2. Based the requirement for downstream applications, parametric input samples during inference can be different and their number can be higher than those during training, since it does not require the expensive computations of physical quantities; see Fig. [ Thanks to our continuous formulation, at inference, different discretised meshing and texturing operations in the parametric domain & can be lifted from 2D to 3D using u({: +), which will lead to consistent result irrespective of the specific discretisation (see also Fig. 6).
4.2 Boundary Conditions
500
A practical cloth simulator allows for imposing conditions such as a user-specified corner motion; for most garments, the simulation needs to be continuous and consistent along the seams. We seek to strictly enforce these conditions in our method. We achieve this by formulating boundary conditions as spatial distance functions, and seams as pe- riodicity constraints along a curvilinear coordinate (such as the azimuthal angle of a cy cylindrically parameterised sleeve), and directly apply them to the NDF in Eq. (3).
Figure 4: Boundary conditions. In con- trast to Dirichlet conditions that alter the network output (c), we impose periodic boundaries by remapping the network in- put to its sine and cosine values (d).
Dirichlet Boundary Conditions. To constrain boundary positions, we require u(San) = O for some specified list of parameter space points Son along the boundary segment M. While we elaborate on the simpler case here, it is also possible to specify complex conditions i.e., u(E.n) b(Eag), detailed in Appendix D One solution is to sample points in the boundary segment and enforce the boundary conditions through separate loss terms. As shown in previous physics- informed neural networks (25). having competing objectives during training can lead to unbalanced gradients, which causes the network to often struggle with accurately learning the underlying solution. Further, there is guarantee that the boundary conditions will always be enforeed. Therefore, we propose to modify the NDF to embed essential boundary conditions as hard constraints [41]
0. if g E an
Specifically, a distance function Bx(E) satisfying Ba(E) = >0, otherwise if E Ω ensures that
any instance of deformation field u(E, t; Q) automatically satisfies the boundary conditions. We set == -((5- -(5' 3 - /)/ s.t. (San: E 02, v(g'. E
(4)
as distance function with small support o 0.01. Fig. provides an illustrative example.
The above formulation supports point and shape constraints in the cloth interior, i.e., an can likewise be a boundary segment inside the domain (Fig. [V appendixx. Moreover, if the initial geometry is provided as a mesh (instead of an analytical definition), point constraints can be directly provided as mesh vertices, with (Lan: (2n) corresponding texture coordinates of the vertex; see Fig. T (right).
Periodic Boundary Conditions. In contrast to the positional or dotion-de boundary conditions specified per the user's desires, additional boundary conditions can arise from the geometric
5

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cloth parametrisation. Points along the panel seams of the garment share the world-space position and velocity, though they are mapped to different values in the parametric domain. We express continuity in geometry and simulation using periodic conditions. Consider any simulation involving a sleeve: Our method needs to guarantee the additional condition due to the parametrisation, i.e. u(s", 53) =((² + 2²). Whereas the Dirichlet condition is imposed by altering the network output, we strictly impose periodic boundaries by modifying its input. Recall that any continu- ous periodic function can be written using its Fourier series. If u(E) is periodic deformation field with period P w.r.. the input coordinate ç^, u(E) can be decomposed into a weighted sum (1. sin(2n=5>/P)) cos(2nmT /P)). E N. Due to the universal approximation power of MIP, only the first cosine and sine terms need to be considered, as the others can be expressed as the nonlinear continuous functions of cos(2 /P) and sin(2 /P) [4]]. Hence, we map &^ using e² + {cos sin }^} when feeding it to the MLP, enforcing periodicity of the predicted NDF along <A. This completes the definition of boundary conditions applied during both training and inference.
4.3 NDF Optimisation
We next explain optimisation in NDF learning. Note & for u(E) and derived quantities are dropped.
Strain Computation. To compute the geometric strains due to the thin shell deformation, evaluate the NDF on samples from the curvilinear coordinate space Q. We generate N2 points using a stratified sampling approach. This ensures that the samples are random, yet well-distributed. At each training iteration, we re-sample coordinates to learn an NDF that fully explores the continuous domain over the course of the optimisation. We evaluate NDF u({) at all samples using Eq. and this prediction (i.e., it) is assumed to be in the Cartesian coordinate system, Le., 11 = u, Our further strain computations (Eq. 6 require covariant deformation components in the reference contravariant basis, i.e., u = at + uz therefore we use the basis transformation matrix T = |a a² a²] for converting from Cartesian deformation coordinates to covariant coordinates (see Appendix for detailed Kirchhoff-Lov preliminaries). While it is possible to predict in the local contravariant basis directly, the global basis is better suited for NDF training since the local basis vectors are not normalised, and the basis varies with the input position E. especially noticeable for reference geometries such as sleeve (Fig. (bb))-pppdddd
B
Next, we describe the ingredients required to evaluate the internal strain energy V. Membrane strain [ea.] and bending strain k. [saa] measure the in-plane stretching and the curvature change, respectively, and are defined as E.g 1 ( a. and Kag by, bas where (ā.,a Gog)) and (ba.g. ba,a) are the metric and curvature tensors of reference and deformed midsurface. With the assumptions of Kürchhoff-Love theory and following [3], we simplify these equations to directly operate on u and evaluate strains as
Eail
(P + Ppa + parid + po3pp3),
2
(5)
Kad Poalp - Bspas + pá (paxla + bappxx boxpua),
where the deformation gradients Par (Pa3 are the components of u,a such that
u,a = p.iāa + "puaa 3 :Pa := uxla- - b.x and paa := us.a. + 62 ux.
(6)
We do not linearise the strain. Orange and teal correspond to the linear and the non-linear components, respectively. To evaluate the derivatives of geometric quantities based on NDF u w.r.t. inputs & (as part of strain computation), we use automatic differentiation of machine learning frameworks [47].
Cloth Material Model. A thin shell develops an internal potential energy due to deformation and the material's hyperelasticity. As in the cloth simulation literature (34 67). we write the internal hyperelastic energy density as a function of the stretching and bending strains, (e(E,, K. <3; 2(x), $ h). h h is the thickness coordinate, and 2xx) Here, e are the cloth's 's material parameters and 5² E [- 2: 2 geometric quantities derived from the reference midsurace X. Our neural base- cloth simulation is orthogonal to the research on material modelling and can, thus, be formulated with many different elastic models, long as the elasticity can be represented function. For example, a linear isotropic [52] stress-strain relationship leads to strain energy of the form V HDH3A Eage 25 BHa3A Kag) where D is the in-plane stiffness and B is the E Eb. and B 12 12[1- with Young's modulus F, Poisson's
energy density
bending stiffness computed as D 1-p2

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Table 1: Quantitative evaluation. We validate the displacements obtained with our method on
the Belytschko obstacle course with analytical solutions from 6 577. Guo et al. 23 use different material and match the corresponding reference result. Below, we show the ablation. We highlight that our method outperforms prior works and baselines by a large margin.
Method
Square plate
Scordelis-Lo roof
Rigid-end cylinder
Free-end cylinder
Analytical
0.487
0.3024
1.825e-5
4.52e-4
Guo
2.566*
n/a n/a 1.81e-5
n/a
Bastek al.
n/a
0.297
n/a 4.58e-4
Ours, full
0.487
0.3018
Ours, no periodicity
3. 3.6e-9
3.13e-1
Ours, GELU
0.496
0.288
1.74c
5.7c
ratio v, and Ha8S6 == va + +1 (a + (2) with 2a being the contravariant
support the data-driven non-linear anisotropic material model of Clyde et al. (15) that has been carefully constructed to fit measured woven fabrics. We refer to Appendix B33 for the mathematical details of the non-lineat model.
metric tensors. Alternatively,
thin shell's stable equilibrium is characterised by the principle of minimum potential energy, i.e. the sum of external potential energy owing to forces f and internal potential energy V due to material elasticity. The total potential energy reads E [] = So v di dd2 Sn f-u d9,, and the stable equilibrium deformation u" can be found by minimising the energy functional subject to boundary constraints
Energy Optimisation.
= b(s < on St. We take advantage of the variational structure of E[u] and minimise it directly with gradient descent. All operations of our energy computation are naturally differentiable, and we estimate the integral as a sum over continuous parametric domain. For linear isotropic materials, arrive the following loss function to optimise the MLP weights for a physically plynccclly-priniipeedd cloth simulation encoded as u (E;
(
IN ( (L (E:: (E)E)))(((;;) BR (E:: ()H (E.))(E;:))) V((E), (7) No
|91
where E(; Rd, (): E R4 are vectorised strains computed using 5 |2
fo dg' de²
the area of the parametric domain; H(E) R4x4 depends only on the reference surface. For data-driven materials (15), the strain energy is additionally a function of thickness coordinate g². Hence, we integrate & along the thickness with the Simpson's ' 3-point rule (similar to [[6] i.e.,
Elu] [ V d d SaJ f.ū dd dΩ, where û = u + g the deformation for point on the shell continuum and w quantifies the change in the midsurface orientation (see Appendix B.2).
5 Experimental Evaluation
Refecemee
We next present the qualitative and empirical re- sults highlighting the new characteristics of our continuous neural fields, including validation (Sec. [Sec.B.I). simulation results (Sec. 5.2). compar- ison to prior works (Sec. 5.3 and applications
Deformed
(608guure plree (6) Seor delleeellelaeeee
(6) Plached cyliadez Pinched cylinder frigidl cruli) (free ende)
(Sec. 5.4
Figure 5: Belytschko obstacle course for which we generate accurate displacements (rescaled for better visualisation).
5.1 Obstacle Course
scrupulously modelled thin shell, and conse-
quently cloth, must be able to handle inextensional bending modes, complex membrane states of stress, and rigid body motion without straining. Therefore, for validation, we use the engineering obstacle course of benchmark problems from Belytschko et al. [6]. for which the analytical solutions are known for linearised functionals. Such problems were previously used in computer graphics [22] for testing the performance of finite mesh elements. Specifically, we test our method on the square plate [57], the Scordelis-Lo roof, and the pinched cylinder with rigid diaphragms and free ends examples, for which the original and our deformed shells are shown in Fig- 5 See Tab. [ converged numerical results. The results, which show that our method outperforms prior works by significant margin, demonstrate our method's 's excellent modelling ability. We further present details
7

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of the experiments with the square plate and pinched cylinder, including the experimental setup and visualisations of the full displacement fields, in App.
Scordelis-Lo Roof is a non-flat reference shape subject to complex membrane strains, i.e. an open cylindrical shell with radius R 25 m, length L. = 50 m and subtends angle of 80' It is supported with two rigid diaphragms at the ends and loaded by gravity f = [0, -90, 0J The shell's 0.25 m. We obtain the maximum
material is given as E' 4.32e8 Pa, V 0 and thickness vertical displacement U2 at the centre of the edge (averaged over the two sides) as 0,3018, closely
approximating the analytical U2 0.3024 [6]
5.2 Qualitative Results
We next present our simulation results. The experiments are performed with the values E 5000 Pa, V = 0.25, / = 0.0012 m for the linear isotropic material, and with parameters from Clyde al. [5] for the nonlinear orthotropic material. For the supplemental video, we extend the method to visualise the deformation trajectory (i.e., transition from the reference to the equilibrium state). Details on boundary conditions, external forces, and time-stepping can be found in Appendix [D]
Napkin. We first consider a square napkin of length = 1 m,, falling freely under the effect of gravitational force. The napkin has a flat reference state in the xy-plane given by (2) and the gravitational force field is applied along the negative y-axis, i.e., external force density f = [0, 99,8p. 0] We specify fixed boundary condition at the top left corner to constrain the napkin movement. The meshes extracted from the trained NDF are visualised in Fig. [ceettr)) Note that apart from its realism, one can also query the simulation arbitrary resolution in the case of NeuralClothSim. Next, we perform another experiment with napkin subject to gravity and dynamic boundary condition, i.e. in which the corners move inwards. This leads fold formation at the top. as visualised in Fig. (a) and in Fig. [foor varying fabrics such as cotton and silk.
Sleeve. We also consider cylindrical shell and perform sleeve compression and twisting. In both cases, we consider the reference state (2) with L - m and R = 0.25 m. See Figs. (b) and (b) for visualisations. In the first case, we apply torsional motion the sleeve, Le., total rotation of 3n around the y-axis to both the top and bottom rims. The optimised NDF forms wrinkles at the centre as expected [24). In the second case, we compress the sleeve to produce the characteristic buckling effect. There are no external forces here and the compression is entirely specified by boundary conditions. We achieve total displacement of 0.2 m due to compression with the inward motion of the top and bottom rims along the cylinder axis; see Fig. (right). The demonstrated simulation is a representative example of strain localisation, with noticeable diamond patterns of shell buckling.
5.3 Comparisons to Previous Methods
In this section, we compare NeuralClothSim to state-of-the-an FEM cloth simulators and physics- informed neural networks for shell structures. We do not compare to other neural simulators 18511, as they do not support simulating non-garment cloths, whereas ours is general neural cloth simulator.
Cloth Simulators. Next, we validate the consistency of cloth simulations at different discretisations of the reference state. We consider two scenarios: 1) A napkin with fixed corner under gravity simulated with our approach and DiffARCSim [8] (Fig. (eett), and 2) a flag with two fixed corners deforming under wind and gravity simulated with our approach and DiffCloth [36] (Fig. (rright). In both scenarios, we simulated ours and compared methods thrice, starting with a marginally perturbed meshing of the same initial geometry resulting in different mesh discretisations. In the case of NeuralClothSim, we learn the reference parameterisation by fitting an MI. x({: ') for each initial discretisation followed by NDF optimisation. We find that the simulated meshes extracted from NDF are consistent for all discretisations. In contrast, for competing FEM-base methods, simulation is sensitive the discretisation; while multiple simulations with the same initial mesh produce identical results, slightly different meshing generates inconsistent simulations. Theoretically, well-defined FEM-based cloth solver should lead to consistent simulation results under different discretisation at high mesh resolutions. To investigate this, we perform an additional experiment where we increase the resolution of the ARCSim simulation (10k vertices) so that the computation time roughly matches ours. However, the results still contain noticeable inconsistencies (Fig. XIV appendix), possibly due to several operations that are highly deeeeeee 61 (such as the bending model relying on the dihedral angles). In contrast, our method leads to consistent results already at much
8

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Discretisation 1
Discretisation II
Discretisation II
Discretization
Discrelisation
Discretisation
Initial
accoratent
DVE ARCSim
(Laang a1)
Incossisenn
Sunalated
ioosiident
mncomisleut
Neunie
Neural
(Chrz))
Figure 6: Simulation consistency. At different initial state discretisations, FEM-based simulators lead inconsisies with often differences in the folds or wrinkles. In contrast, ours overfits an MLP to the reference mesh and encodes the surface evolution using another MLP (continuous neural fields). lead to inconsistent results with often occurring differences the folds or wrinkles.
We next evaluate the memory efficiency for simulations generated by NeuralClothSim, DiffAR Cii and DiffCloth. The simulations are chosen to be of similar complexity, and qualitative results visualised in Fig- appendxx. In Fig. XV appendixx, we then plot the memory requirement as function of spatial resolution. Memory is recorded for the simulated mesh states for the compared methods and weights of the NDF network for ours. While the memory requirement of mmite-elementtbaseee methods grows linearly as the function of the number of vertices in the simulated cloth, our approach requires constant and comparably small memory volume to store the quasistatic simulations. For better memory efficiency, existing simulators offer adaptive refinement (such ARCSim 451 by re-meshing at each time step (coarse triangulation used at smooth regions, and fine meshes are used for wrinkles). However, this requires additional computation and loss of important characteristics, such as differentiability. contrast, our approach is adaptive without the overhead and without losing correspondence and differentiability due to re-meshing.
coarser discretisations (400 vertices, Fig. 6
PINNs for Shells. Bastek et al. [5] focus on engineering scenarios, and we compare the solutions the Scordelis- Lo roof and a square plate in Tab. [ both ours and theirs closely match the reference solutions. Bastek er al. note instabilities during training a neural network trained on point load, therefore, define the Gaussian force kernel in their pinched hemisphere example. In contrast to theirs, we propose a new loss function for point loads (see App.[C C) addressing the pinched cylinder obstacle course. While Bastek al. show converged results on engineering ex- amples, their method cannot capture the high- figh-frreuency signal (folds and wrinkles) required for cloth simulation; Fig. 7 for illustrative example. The main reasons for their failure are !) the linear strain and 2) that their acti- vation function (GELU) capture only smooth signals.
Bustek
(bi NeuralClthhSm (Oums
Figure Comparison to Bastek al.
sleeve twisting. While the cylinder in (a) twists without wrinkles, our result (b) is correctly wrinkled, similar to 24
5.4 Ablation and Applications
Ablation. evaluate the following ablated versions our approach: 1) Contravariant coordinate system for NDF components, 2) Using linear approximation of the strains instead of our model, 3)

---

Variants of boundary constraint imposition and 4) Choice of activation functions for NDF. For the latter two, we present the numerical results in Tab. T See Appendix E for further details.
NeuralClothSim can incorporate learnt priors: Our NDF can be directly extended by making it dependent the material properties, i.e., it can accept the material parameters as an extra input. This is possible since the material parameter space is typically low-dimensional. Once such NeuralClothSim modification is trained, we can edit the simulated geometry at test time. as shown in Fig. [-bottoo right). We provide implementation details of conditioning in Appendix F
cnnteria-conditionnng
Simulation editing. For nigh-dimmensioal scene parameters such reference pose and external forces, we can edit simulations: The user can interrupt the training of NDF at any point, change the parameters and continue the training. Moreover, editing also be done after full convergence (aka pre-training) and then fine-tuned with gradually modified design parameters. Editing an NDF has multiple advantages over NDF training from scratch: It is computationally and memory efficient, and provides access to interpolated simulations. We provide further details and results in Appendix [F)
Discussion and Conclusion
NeuralClothSim closely matches reference values in challenging cloth deformation scenarios (e.g. the Belytschko course), thanks to compact NDF representation governed by the non-linear Kirchhoff- Love shell theory with (non-)linear orthotropic material. An extended NDF allows test-time interpola- tion of material properties and simulation editing. In contrast to the previous mesh-based simulators, NeuralClothSim enables querying continuous and consistent equilibrium cloth states. The shown results are physically plausible in different scenarios under time-varying external forces and boundary motions. We also see multiple avenues for future research, such as adding dynamic effects, i.e. inertia and damping. Our simulator currently does not support contacts and friction necessary for many applications beyond what is demonstrated here (cf. Appendix on this standalone research problem).
In conclusion, we see NeuralClothSim as an exciting step towards neural-field-one based continuous and differentiable cloth simulation. Inverse problems in vision could benefit from its multi-resolution consistency. While there is a long way until other functionalities such as collision handling are unlocked, we believe it can pave the way towards new generation of physics simulation engines.
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NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory
Appendices
Navami Kairanda
Marc Habermann
Christian Theobalt
Vladislav Golyanik
Max Planck Institute for Informatics, Saarland Informatics Campus
Table of Contents
Implementation Details
15
Kirchhoff-Love Thin Shell Theory
15
Geometric Preliminaries
16
Kirchhotf--ove Shell Kinematics
16
B.3
Material Elasticity Model
18
B.4
Equilibrium Deformation
19
B.5
Tensor Algebra
20
B.6
Proof of Strain Computation
20
Belytschko Obstacle Course
22
Square Plate
22
Scordelis-Lo Roof
22
Pinched Cylinder
23
D Simulation Details
23
D.I Cloth Trajectory Visualisation
24
D.2 Napkin
24
D.3 Sleeve
25
D.4 Skirt
25
Ablations
25
Activation Function
25
NDF Coordinate System
26
E.3
lin-aiiy of Strains
26
E,4
Boundary Constraints
26
Applications
27
Material-conditioeed NDFs
27
NDF Editing
27
G
Performance
28
Runtime
28
G.2
Sampling Strategy
29
G.3
Simulation Reproducibility
29
14

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Table II: Notations. We omit separation of quantities in undeformed (overbar, e.g. x) and deformed configurations. Moreover, we list the tensors with their covariant components but omit the contravari- ant and mixed variant versions. Instead of subscripts (covariant components), they are represented with superscripts or mix of superseripts and subscripts.
Symbol
Description
Symbol
Description
8" E ( (E', - e 21 h),
Curvilincar coordinates
1 [0
Time
Thickness coordinate
i: 0t x 0,7] R3
Velncity
x: RR
Midsurface representation
I: 0, R
Initial distance function
a, R²
Midsurface tangent vectors
B: B:
Boundary condition
a Ra
Unit normal to midsurface Metric tensor midsurface
Fe: >B
Neural deformation field (NDI)
Oug
PER
Mass density
boz Q +
Curvature tensor midsurface Shell representation Tangent base Metric tensor on shell External force Midsurface deformation
BER
Young" modulus
r:f [-7
R
VER
Poisson's ratio
8: E. R
shell
111, G12
Infinitesimal strain parameters
Bg 15:2x 4
Pn.
Nonlinear material response
RS
d1.d2
Warp/weft material directions
u:
Pag :a::20 (-2.
Orthotropic strain
u x 1- - R
Shell deformation
Vala: ala:0
Covariant derivative
w R
Midsurface urientation change
D i R
Christoffel symbol
Pai Q >
Deformation gradients
E: >R
Potential energy
Fag f x [-4 + R
Green-Lagrange strain
V:
Hyperelastic strain energy
Ead 12
Membrane strain
Mad + R
Bending strain
H Additional Comparisons
29
Runtime
H.2 Multi-Resolution Consistency
31
Collision
32
J
Extended Discussion and Limitations
33
Sections referenced with numbers refer to the main matter. All referenced figures and equations are per default from this document, unless they are followed by the "(main matter)" mark.
A Implementation Details
We implement NeuralClothSim in PyTorch 47 and compute the geometric quantities on the reference shape and on the NDF using its tensor operations; the first and second-order derivatives are calculated using automatic differentiation. Our network architecture for NDF is an MLP with sine activations (SIREN) [53] with five hidden layers and 512 units in each layer. We empirically set SIREN's frequency parameter to wp 30 for all experiments (we observed that choosing Wo does not permit folds). Although we sample from (5' (r)) E Ω, t E [0, 27,7 T 1, we normalise samples to (E1, f) (0, 1]3 when feeding the input MLP per the initialisation principle of SIREN. Note that all physical quantities are computed in the original domains Ω, [o, T and the gradients are tracked in their scaled versions. For training, use N2 20 and N, 20. At test time, sample much higher for visualisation, usually with Nn 100 100 and N, 30. For material conditioning. single random material sample per training iteration. NeuralClothSim's training time amounts to 10 - 30 minutes for most experiments, and the number of training iterations equals 2000 5000. We use ADAM [30] optimiser with a learning rate of 10-1 and run our simulator on single NVIDIA Quadro RTX 8000 GPU with 48 GB of global memory.
B Kirchhoff-Love Thin Shell Theory
In this section, we briefly review the Kirchhoff-Love thin shell theory following 14 29]: detailed treatment of the subject can be found in (3). We already introduced physical and mathematical
15

---

notations in Sec. 3 (main matter). A detailed list of notations can be found in Tab. II We next present concepts from the differential geometry of surfaces to explain the midsurface and director (Appendix B.T). We then follow with the shell parameterisation and computation of strain measures on and off the midsurface (Appendix B.2). Further, we present the hyperelastic material models that relate the strains to the internal stress (Appendix (B.3) and finally review the energy principles for equilibrium deformation (Appendix B.4). Moreover, we provide a proof the simplified strain formulation in Appendix B. well as additional results from tensor algebra that are relevant for the computations (Appendix B.5
B.1 Geometric Preliminaries
In Kirchhoff-Lowe shell theory, the shell midsurface completely determines the strain components throughout the thickness. Therefore, we review those aspects of the differential geometry of surfaces that are essential for understanding the shell theory.
Let represent the midsurface described by a smooth map. x f2 C R + Rs on the parametric domain Ω. Any position x(51, 5²) 5²) on the surface is uniquely identified using the convective curvilinear coordinates (2) Ω. As
2D manifold in the 3D space, as shown in Fig. ! It can be
positions can be specified using Cartesian coordinates x = x,,, it follows that the invertible maps I; (E < and 5 (1111 22. 2z) exist. We define local covariant basis conveniently express local quantities on the surface. Such a basis is constructed using ag. the set of two vectors tangential to the curvilinear coordinate lines ça:
a X
(8)
To measure the distortion of length and angles, compute the covariant components of the symmetric metric tensor (also known as the first fundamental form):
o.B 0,3 a, a,3.
(9)
The corresponding contravariant components of the surface metric tensors denoted by ad can be obtained using the following identity:
axs 8ay
(10)
where 6ag stands for the Kronecker delta. a ad can be used to compute the contravariant basis defined
a ag- Jag follows:
ax.
(11)
While the covariant base vector a., is tangent to the go line, the contravariant base vector a is normal ag when a * B. Generally, a, and a" need not be unit vectors.
The shell director coincides with ag, the unit normal to the midsurface, and, therefore, computed as the cross product of the tangent base vectors:
aj
X
ag
a² ag.
(12)
a
a2
The second fundamental form which measures the curvature of the midsurface can be defined with ag
bag == .a. a3,3 = -a a3,a a,,8 ag.
(13)
Finally, the surface area differential dl2 relates the reference coordinates via the determinant of the metric tensor:
dn vadg dg², where vā v=:= |a a2
(14)
B.2 Kirchhoff-Lovee Shell Kinematics
The Kirchhoff-Love model proposes reduced kinematic parameterisation of thin shell characterised by a 2D midsurface and shell director. It relies on the Kirchhoff hypothesis, i.e., the director initially perpendicular to the midsurface remains straight and normal, and the shell thickness h does not change with deformation; see Fig. M
16

---

Reference Shell
Deformed Shell
Contravariant basis on reference midsurfaces
sin
a(({)
000
(E'
ax())
('(
g2
co
a (E)
a4)
x(5* x
x(5
12 ga
2
4 10.
[02m] [0.1]
Parametric Domain
(x)
(b)
Figure I: (a) Kirchhoff-Love thin shell. thin shell can be kinematically described by the midsurface (here: reference and deformed midsurfaces) and the director (here, āz). Any material point P the midsurface is then parameterised with curvilinear coordinates (5' <²), whereas point on the shell continuum requires an additional thickness coordinate ş Geometric quantities on the midsurface (off the midsurface or on the shell continuum) are coloured red (blue). (b) Contravariant basis for midsurfaces in the reference configuration. While local contravariant basis coincides with the global Cartesian coordinate system for planar reference shell, such basis varies in magnitude and direction across any circular section of the cylinder. Local basis relies on the surface parameterisation, therefore the derived basis vectors need not be normalised (notice how a (E) scales inversely with the radius).
The position vector of material point in the reference configuration of the shell continuum can be parametrised with curvilinear coordinates (2) and thickness coordinate ç² as:
(, = x3)
,32),,), s.t. S
(15)
2
where x(E', { represents the midsurface.
The shell adopts deformed configuration under the action of applied forces f. Analogously, the deformed posítion vector r is represented as
h
r(g' 2.) + <?), < 3
(16)
2
where the deformed director ag coincides with the unit normal.
As a consequence, the overall deformation of the Kirchhoff-Love shell is fully described by the displacement field u(g", (²) of the midsurface, i.e.,
x(5' {²) x(E (('',3)) + !
(17)
Analogous to the deformation field == x - X of the midsurface, we define w as the difference vector of unit normals to the midsurface. i.e..
wya + w. w>ax + w
(18)
Using this formulation, deformations on the shell continuum can be described by the field ū({² 32 2 u(E'.53 w The difference vector w describes the change in the orientation of the midsurface, enabling us to quantify bending. A simplified way to compute the components w; of is provided in (5T).
The tangent base vectors at a point on the shell continuum are denoted by gi = r,; and expressed by those of the midsurface as:
ga an 3 ag...
(19)
g3 ag.
17

---

Linear
StVK
Canvas
Silk
Cotton
Figure II: Material model. Simulation of stable equilibria of m i napkin with corners held 60 em apart. From left to right, we visualise linear isotropic, linear anisotropic St. Semant-Kirchhol (canvas), and non-linear anisotropic canvas, silk and cotton materials from Clyde al. [15]
The corresponding covariant components of the metric tensor are then obtained using
Dij
g
(20)
To measure strain, the symmetric Green- Lagrange strain tensor E Eng &g' since it discards rotational degrees of freedom from tangent base vector gi while retaining the stretch and shear information. It is defined as the difference between the metric tensors on the deformed and undeformed configurations the shell, i.e.,
Eij == 2 9 - 9s)).
(21)
Using (99 and (20), note that transverse shear strain measuring the shearing of the director vanishes (Fa3 = 0) and the stretching of the director is identity, i.e, E33 1; hence, the strain simplifies to
Eag Eag + 5 Kads
(22)
with membrane strain measuring the in-plane stretching defined as
Eag (0aB Tas),
(23)
and bending strain measuring the change in curvature defined as
KaB = bag bag-
(24)
B.3 Material Elasticity Model
Our NeuralClothSim orthogonal to the research on material modelling and can, thus, be formulated with many different elastic behaviours. We demonstrate results with simple linear isotropic
model [4], and the data-driven anisotropic non-linear model from Clyde et al. 15] as well as the St Venant-Kirchhoff variant of the Clyde's model.
Linear Isotropic Material. Given the material Young's 's modulus E, Poisson's ratio v, a linear isotropic stress-strain relationship leads to hyperelastic strain energy density [52] of the form
V 2
(DH Xs BH ka
(25)
where D is the in-plane stiffness and B is the bending stiffness computed as
Eh D 1 1
Eh and B = 12(1 - v2)
(26)
and
1 + 2 + a ga)).
88M = 8
(27)
2
Here, V is the sum of the membrane strain energy density (the first term) and the bending strain energy density (the second term).
18

---

Non-lineas Orthotropic Material. While the linear isotropic model is simple and sufficient to demonstrate our NeuralClothSim formulation, a data-driven model with estimated fabric material parameters can generate highly realistic cloth simulations. Therefore, we additionally demonstrate our method with the non-linear anisotropic material model from Clyde et al. [15, and its simplification to St. Verant-Kirchhoff model A We show the simulation results with the material model with varying materials, such as cotton and silk in Fig. !i and describe the model next.
Clyde et al. present an orthotropic constitutive model that accurately represents the anisotropy introduced by the warp and weft structure of woven cloth. More concretely, they write the hyperelastic strain energy density as y (E, D, ② where E is the Green-Lagrame the reference configuration warp/weft orthotropy (d1, dg) and normal (dg) directions, and &
strain (21). D [d1. d2, dg)
being the fabric parameters. We follow the technique of [4] to determine the material directions D. Orthotropy directions are computed as tangents to the midsurface with the warp direction d1 coinciding with the normalised covariant base vector, i.e,,
āg
dj
d3 = ā3, and d2 = -d d1.
(28)
ag
Next, the orthotropic components Fay of the strain are obtained by expressing E in the material basis, ED. Due to the Kirchhoff-Love kinematic assumptions, any stretches and shears in
with E = D the out-of- out-of-plane direction d3 vanish, i.e. Eg3- = Ezi O. Finally, the Clyde model's strain energy density intuitively separates the distinct deformation modes and defined as,
11 m (ET)) + k1272( E Ez)) 2
R:22 nu (Eg)) + G12ta
V
(29)
2
where (k11 ( 122, G12) describe the cloth's infinitesimal (linear) strain behaviour, whereas the function my describes the nonlinear response to larger strains with,
di
(5(
Pj ((x + 1)0se - 1).
(30)
Orj
We obtain the values for material parameters + = {k11, k12, k22, G12, Pyi agjs dz d
i=.je.. from
T7 and model the silk, canvas and cotton fabrics. Additionally, we can arrive at the orthotropic (linear) St. Konant-Kirchhoff model [4] by simply choosing ny ()(x)) for all (see Fig. I..
When optimising the NDF, the non-lineat response (30) gives unpredictable results for strains outside the fitting dataset. For reasonable strain extrapolation. the closest valid strain, as proposed in [17] (see [54]-supplement for the derivatives). Towards this, we leverage the strain cutoffs Emin and Æmax provided as part of the material dataset.
quadratic Taylor expansion around
B.4 Equilibrium Deformation
Under the action of external forces and boundary conditions, thin shell deforms and achieves an equilibrium configuration. Its stable equilibrium state characterised by the principle of minimum potential energy, which is the sum of external potential energy owing to applied forces and internal potential energy due to material elasticity.
While all the geometric quantities in g 27 are defined at each material point (5' .²2) E Ω, the energy is integrated over the parametric domain Q2. Considering the total potential energy of the shell Elu] is given by the sum of clastic potential energy &i and the potential energy due to the external force density f. we obtain:
Elu] = S v du
/ fu.
(31)
Next, the stable equilibrium deformation of the shell can be found by minimising the potential energy functional subject to boundary constraints:
u" = arg min E(u], subject to u
(32)
u((' - = b . on A2.
The above definition of hyperelastic energy the shell requires the displacement field E H2 R23) that must necessarily have square-integrable first and second derivatives.
19

---

B.5
Tensor Algebra
We provide additional results from tensor algebra shells (see Sec. [.6].
63 that are relevant for strain computations on
Based on the coordinate system for the tensor components, tensor can be covariant (e.g., A,,)), contravariant (e.g., Aa3) and may even have mixed character, i.e., partly contravariant and partly covariant in different indices (e.g., A2). For computing 3.. and yo in (5) matter), we use the following rule from shell theory that transforms covariant tensor to a mixed one:
A AsxaAa
(33)
Ag = AAzā Aa
tensor of n-th order has n indices. For example, Va first-order, and _ is fourth. For computing the covariant derivatives of the first-order tensor upla and the second-order tensor Paxla in ( we use the following rules:
Vals g Va, exT ais and
(34)
Aasly = Ang.7 Axst an Ana By.
where I aB is the Christoffel symbol given by
T = a.a aa.g.
(35)
Some tensors arising in the kinematic description of Kirchhoff-Love thin shells are symmetric with respect to indices a and 8, i.e., Aas = Asa- We exploit the symmetry for efficient computations of the following tensors: agg, bag, Eag. Kag, and r2g--
In the of linear elastic material, also exploit the symmetry of fourth-order symmetric tensor H:
HaBA8 H HaoA 1a Aó
This property means that only six independent components (after applying symmetry) need to be computed (i.e. 111111, 11112 1122 11 1212 H 1222 and H| 22222))
B.6 Proof of Strain Computation
According to the Kirchhoff-Love theory, the Green-Lagrange strain associated with the deformation of thin shell is decomposed into the stretching and bending strains of the midsurface. One could compute them using Eqs. (23) and (24), written in terms of the reference state X and the deformed state of the midsurface. As easier alternative, directly evaluate strains with the NDF of the midsurface using
matter). Next, we prove that the two formulations are identical
following 3
Lemma B.1 (Deformation gradient). Deformation gradient u can be written as u,a = Paaa
Paaā where the components of the gradients Pax: Paai are defined as
Pat uxla baxUs, and
(36)
Pa3: == U3.a b2ux.
Proof. Given deformation field u of the midsurface described in contravariant basis as u = u.aa Wz we compute the deformation gradient as follows:
u = u/a = =xxlaā= + uyā a la + uslaaa + uya
(37)
= =xxlaa> + ux + us -uyda
where we use the following identities from the shell theory arrive the bottom part of the previous equation:
a l/ a²
-baxa b. aA.
(38)
20

---

Finally, we rewrite them as
u,a = P + Pasa
(39)
Theorem B.2 (Membrane strain). Membrane strain (guantfying/measuring in-plane stretching) can be written as a function of the deformation gradient in the following form:
1
Eas = (Pas - Pea + PaxiA + Paalfas)
(40)
2
Proof. We start with membrane strain given the difference of metric tensors (first fundamental form)
Eag == 2 a0, āas),
(41)
EaB 2 aa
a ān -g)).
Substituting the tangent basis vectors acx
an X 8, U,, an + U,a
(42)
gives us updated strain in terms of deformation W.
Eaf 2 aa u, + ag u.o + Usa U,
(43)
Assuming the following identities from Kirchhoff-Love shell hypothesis [3
āa a = -80- a² = ā3, ā.
a =5 th. a3 0, a = =1 and
(44)
ã = a. A3 Asxara
and considering the above lemma for the deformation gradient u,o. we finally obtain the target formulation for strain:
1
Eas
2 (Pas = Psa + Parp3 + Paslsps)
(45)
Theorem B.3 (Bending strain). Bending strain (measuring the change in curvature) can be written as a function of the deformation gradient in the following form:
Nag = -Paals B3Fan + pa + 1 J. b3xPaa).
(46)
Proof. The bending strain of the midsurface is defined as the difference of curvature tensors (second fundamental form) in the reference and deformed configurations:
Kaß := bug bag
(47)
Nag a. a3,8 a.. a3,6
Using 42 and (18). we rewrite strain using deformation gradients as: Kaf (ä. + U,a) (ā3, + w.s) a. a
(48)
Further simplification and applying identity
38
leads to:
Ka a. W, + U.a W,8 bzl a*
(49)
Using Lemma B. for deformation gradients u,a and w.g, f.e. w (wxls brows)a + (Wa.8 + fywx)aa and with the shell identities of (44), we arrive at:
Kxg Wa
bagua DYyar + pa(lloo bxows)
(50)
+903(W3,6 + +bB
21

---

433
431
6200
+13
('rrddera)
2'Vecianoi
Visualisation of the deformation fields predicted by our NeuralClothSim. From the left to the right: Square plate, Scordelis-Lo roof, pinched cylinder with fixed boundary conditions and pinched cylinder with free boundary conditions. For the pinched cylinder, the results are re-scaled versions with E' = 30.
Figure III: Belytschko obstacle course:
With the Kirchhoff-Love normal hypothesis and neglecting cubic terms, we can approximate the components of w as the following:
Wi 2 1x "xW0 -
2 Pa333
(51)
Wa -Pa3 + pappa.
First eliminating the component wy and subsequently wa. we arrive at the target strain formulation:
Kag R wals - Pgp + 2 w + w laPPa + BBWAPas
(52)
1
Kag R -Pa3ls B3Fas + ps(paala + 2 bapp1 bas))
C Belytschko Obstacle Course
In the following, we provide detailed information for reproducing the Belytschko obstacle course experiments from Sec. Sec.5.TT (main matter). We visualise the NDF along the direction of applied load in Fig. that closely matches the reference solutions [6].
C.1 Square Plate
In the first test case, consider simple bending problem of flat square shell [14] It is simply supported at all edges and is subject to a uniform load. The plate has a side length of L = 100 m and thickness h 1 and, therefore, falls under the scope of Kirchhoff-Love thin shell theory. The material parameters are given as E = 1e7 Pa and v = 0. We represent the reference geometry with (main matter) and impose Dirichlet boundary constraints by constructing a distance function the plate edges. This is followed by training an NDF to solve for quasi-static displacement minimising the total potential energy (3T) subject to uniformly distributed external load f [0. 0, -1]T. With the simply supported constraints along the boundary defined by än = {(5) 0), (0. (L.
we define NDF as follows:
u(E: (E))) B(E) : -
(53)
We train with the loss 7 (mmmin matter) for 2500 iterations and illustrate the solution Fig. 5 (main matter) where the displacement is scaled up by factor of 50. The maximum displacement ug at the centre of the plate is found to be 0.487 after convergence and exactly matches the reference
solution 57))
Fig. (left) shows the obtained NDF along the z-axis for the square plate.
C.2 Scordelis-Lo Roof
The reference geometry of the Scordelis-Lo roof is given by the following parametric expression: x{E) = [Rt cos(E" + 50"), Rsin + 50%) T
(54)
vel E [0, S [0, LJ, with R = 25 m, L = 50 m, and h 0.25 m.
22

---

Concerning the boundary conditions, the structure is supported with rigid diaphragm along the
edges, i.e., an {(E', 0). (5' L)}. The material properties are set as E 4.32e8 Pa, v 0 and a uniformly distributed load f [0, -90, 0J is applied to it.
We optimise NDF under boundary conditions as follows:
"((: 0) = F
(55)
s.t. ( := -²)
Fig. III (second on the left) visualises the computed NDF along the y-axis.
C.3 Pinched Cylinder
Pinched Cylinder. Finally, we consider the pinched cylinder problem, i.e., one of the most severe tests for both inextensional bending modes and complex membrane states. As shown in Fig. 5 (main matter), cylindrical shell is pinched with two diametrically opposite unit loads applied
at the middle of the shell. We consider two cases: First, a shell with ends supported by rigid diaphragms [6] (similar to Scordelis-Lo roof), and second, cylinder with free ends [57]. We define the cylinder geometry with (2)-n matter), where R = 300 m. L = 600 m and the thickness is set h 3 m; the material properties are given E = 3e6 Pa, v 0.3. In contrast to the previous test geometries, which required specifying only the Dirichlet boundary conditions, we additionally account for the periodicity constraint along the circular cross-sections. To model this, we define NDF as Fe (cos E sin ,' E²2). instead of the default case Fe ( A crucial challenge of pinched cylinder test case is modelling load at singular points in the sample space. To achieve this, we adapt the potential energy functional (31)-(main matter) deserred previously for uniformly distributed forces -o the point load setting, rewriting it as [P] = -n ds Ene f-u, u, where 2o is the set of points of the load application. We apply point loads f E {[0, 0. T] -0]] at diametrically opposite points 2o = {(90), 300), (270), 300))- In the case of distributed load, previously proposed computing external and hyperelastic strain energy at an identical set of stratified samples in the parametric domain. We depart from this setting for point loads: At each training iteration, we sample all points from Go for external energy, whereas random stratified samples are used for computing strain energy. To speed up the convergence, we E' = 30 Pa. instead of the original value E = 3e6 Pa; this simply scales the displacement field in the linear setting as shown in (5). As mentioned. the constrained cylinder is supported with a rigid diaphragm along the edges, i.e., On = {(5) ,), (E' L)). therefore, we optimise NDF using
u(E: θ) (Fe) Fez, Fes
(56)
s.t. ((g) =- <L
Next, we consider a pinched cylinder with free ends, i.e. 0n 0. Without any boundary constraints, the cylinder can move rigidly due to the applied force, and such rigid body motion should be factored out. Therefore, to suppress it, we restrict the displacement of the point under the load directions other than the direction of the force vector. We achieve this by enforcing új 0, ue = O at load points. The NDF parametrisation factoring out the rigid motion reads as:
u(E: 9) = B ( (E (E) s.t. (E) 1 e ((6' (62")=(((==000") and B2(E) = 1 - e ((6' 220 -
(57)
In both examples with the pinched cylinder, we monitor the displacements under the loading point. As shown in Fig. (main matter) and Tab. (main matter), it qualitatively and quantitatively converges to the reference solution. Fig. [II (second from the right, and the rightmost) shows the obtained NDFs along the z-axis for the two cases of the pinched cylinder.
D Simulation Details
We first describe the extension of NeuralClothSim for visualising trajectory to equilibria. Then, similar to the previous section, we provide boundary and loading conditions for all experiments here.
23

---

D.1 Cloth Trajectory Visualisation
We next visualise the transition from the ref- erence to the equilibrium state: We extend the NDF 3 to u(Ş. 1; )) modelling time- dependent deformations, Vt E [0, 7) with
Initial state
Initial state
1. For smooth and physically-plausible interpo- lation from the initial state (() to the deformed xE,,T) T),, we impose initial conditions and tem- poral regulariser.
(a) Point constraint
(b) Edge constraiol
Initial Cloth Configuration. If we optimise the time-dependent NDF only with the potential energy loss ( the model finds the converged equilibrium states of the underlying cloth model
Figure IV: Napkin simulation upon convergence
under gravity with non-buundrry constraints.
Vt [0, .). To start from the initial undeformed cloth state, initial conditions leading to zero dis- placement and velocity need to be explicitly incorporated. Hence, we use the function (t) == as an additional multiplying factor in (3 leading to u(E, 0) = 0, and ú(ç. O.
Temporal Smoothing. Without any temporal prior, the transition from the reference to the equilibrium
state will be too swift and not smooth. Therefore, we use an additional regularisation loss ()) :=
constraining the cloth velocity ú. Specifically, the smooth optimised trajectory u (E, (; Q) is obtained with the final loss C L, where, similar (+)), the physics loss [(O) is now evaluated over the entire parametric-
No., Lim1
|2] No ش pp
0)) ty;
domain. Additionally, for some examples, such as sleeve compression/torsion we drive changes in the deformation trajectory by imposing time-varying Dirichlet boundary conditions. We note that the time-stepping is performed purely for visualisation, and we do not model the simulation dynamies that would require taking into account inertial and damping effects. For linear isotropic material, we set 0.144 kg m -2 and for the non-linear orthotropic material from Clyde et al. (15)
D.2 Napkin
The force in this experiment is defined as f = [0, 99.8p, T and the boundary conditions read
a2 = {00 0), (0, L)}. The result of a napkin droop with fixed corner is shown Fig. | (main matter). NDF in this experiment is parametrised as follows:
u( = (iop:)))((BBoo.eeeecooo (E),
(58)
s.t. _efo__ett_cormer(( .s',,
.52) == 1 - e ((6))"((("""
The experimental result of a napkin droop with moving corners is shown in Fig. (main matter). The boundary condition read ön {(0, L), (L, L)). The NDF parametrisation in this scenario is as follows:
u =FFIBrop Joft Boo Brop_ right + (1 - Joft) Joft Bmotion (1 - Brop_right B Bmotions S.t. Beop loft (5) <2) = - (6"
(59)
Biggtt (5+ (2) = (16))=))))
and
Buotion (t) = [0.2t, 0, 0)².
The experimental result for napkin droop with fixed edges is shown in Fig. VII The boundary conditions are defined as a {(5) 0). (0, {²)), v(El, [0, LJ2 and the NDF parameterisation reads
u = Brggebtt edge
s.t. Biettedge (E = c- (E') and
(60)
Bright_odgg edge (E2) = 1-
24

---

D.3 Sleeve
In the experiment with sleeve, no external force is exerted: = (0, 0, 0J". The boundary region is defined by: 0n = {(E' 0), L)). VgL E 2T). The NDF is parametrised as follows:
u = FelBbotom rim Brop + (1 Bbottom rim Bmotion (1 Beop @top_rim/ Buotion:
Bbottom_rim (5²) = 6-
(61)
Btop (5²) 1 -5---LL)) and
Bmotion (t) (0, 0
Sleeve twist is achieved by introducing rotation displacement / 3x parametrised as follows:
The NDF in this scenario
4
u == FoI([ Bhottom_
rim) Bottom Biimmmrrmmmiieee Bottom motion + Beop rim Beop motion:
s.t. Bbottom rim (52) e
Biop (52) = e -EE--1
Bbattom motion ()),1)= = R(coo(E) 8t) cos
and
R(sin(E' - ft) - sin
Btop_ (t), := [R(cos(5 0t) cos ) 0 R(sin(s' + 0t) sin )
(62)
demonstrate sleeve torsion Fig. (left) and buckling in V (right).
D.4
Skirt
See Fig. XIII for the experimental results with skirt. The reference skirt geometry is defined as:
x(E) cos _2 -²,,, riin Ag* 2m; E [0, LJ.
s.t. r({2)= (Roop Rhottom) )
(63)
Rbbotoo
L
The skirt deform in this experiment under gravity, iee,, = 10, 9.8pp the boundary region is given by U2 {(5', L)}, VE [0, 2T). NDF is parametrised as follows:
FeI(1 Seop_rim)
(64)
s.t. Biop_rin rim (g = -5 -
The conditions for skirt twisting (angular displacement) are similar to those of the sleeve twist (applied at the top rim) in Sec. D.3
E
Ablations
E.1
Activation Function
Experimental results for sleeve twist with different activation functions in the NDF network shown in Fig. V While ReLU lacks support for higher-order derivatives leading to artefacts, network with GELU activation can only represent low-frequency deformations. Our usage of sine activation 53 overcomes these limitations and successfully represents fine folds.
25

---

A ReLU
GELU
Sine
Contravariant Basis
Cartesian Basis
Figure V: Activations (left). Results of our method with different activation functions (ReLU, GELU and Siren). Contravariant vs Cartesian basis (right). Prediction of NDF output in the Cartesian coordinate system well conditioned compared the local contravariant coordinate system.
Dirichlet Boundary Condition
Periodic Boundary Condition
(a) Soft Constraint
(b) Hard Constraint
(a) No constraint
Connected
Periodic
Figure VI: Ablation study for boundary conditions, with Dirichlet (top) and periodic (bottom) boundary conditions.
E.2 NDF Coordinate System
In the Kirchhoff- formulation, strain energy computation is performed in the local contravariant (or covariant) basis. This leaves us with an obvi- choice of predicting covariant components of the NDF in a locally varying contravariant basis (Fig. [ (b)). Hence, (a) predict NDF in contravariant basis and use it directly in strain calculation (ablated version), and (b) predict NDF in global basis and transform its components to local basis before strain calculation. The second case leverages the knowledge of local basis (which is not guessed) and leads better convergence (Fig.
Linear
Non-Linwer Strain
Figure VII: Linear vs non-linear strain. We demonstrate napkin drooping under down- ward force. Kirchhoff-Love strain is inher- ently highly non-linear.
E.3 Non-linearity of Strains
In the small-strain regime, linearised kinematics is often employed. However, accurate simulation of cloth quasistatics requires modelling of both rigid motion and non non-linear deformation. Kirchhoff- Love membrane and bending strains are non-linear functions of the displacement field and non-linear strain calculation is decisive for obtaining realistic results. Thus, we evaluate the linear approximation of Kirchhoff-L.on strain by omitting the non-linear terms in (main matter). In Fig. VII show that a linear approach leads to significant inaccuracies in modelling cloth bending under gravity.
E,4 Boundary Constraints
perform ablations on the Dirichlet and periodic boundary conditions. We try a soft constraint variation, in which impose the boundary condition a loss term in addition to the Kirchhoff--ove
26

---

energy. This requires empirically determining the optimal loss weight, takes much longer to train and does not guarantee satisfying boundary constraints, as shown Fig. (l.fft...... Our approach with hard constraints avoids all these problems. In the second example, we simulate the compression of cylindrical sleeve as described in Appendix D.3 As seen in Fig. (right), at gl = n, cylinder (a:) is disconnected if no constraint is specified; (b:) is connected with ç <> cos 5 (c:) fully models continuity and differentiability forming folds with ç² + {cos 5 s
F
Applications
F1
Material-conditionee NDFs
For simplicity, we choose the linear elastic materials, i.e. + = (p, h, v). Conditioning on + allows us to adjust at test time mass density e. cloth thickness h. as well as the linear isotropic elastic properties of the material, i.e. Young's modulus and the Poisson's ratio The updated NDF-which is now a function of material as well-reads, u(E,t, t, Φ; Q)
((), where E I (min: max] the continuous range of material parameters. At each training iteration, we uniformly (at random) re-sample & to explore the entire material domain. At test time, novel simulation can be generated with a single forward pass for any material & in the valid material range. Unlike latent space conditioning in other fields and problems, the material space conditioning in NeuralClothSim has a direct physical (semantic) interpretation.
As an example, we train an NDF conditioned on cloth thickness and 4b E (p. E, x [Dmins hmaax with Amin 0.0005 m and hmax 0.0025 m. We visualise the simulated result for h {0.0005, 0.0015, 0.0025), in Fig. (bottom right).
F.2
NDF Editing
In movie and game production, 3D artist's workflow includes updating design parameters, which requires multiple repeated simulations from scratch. Such scene parameters include reference state geometry, external forces, and material properties. Material parameters typi- cally constitute low-dimensional space, propose to condition the NDF on material prop- erties. However, other inputs such shape and pose of reference state, as well as external force, high-dimensional. Instead of learning sim- ulations over the entire scene space, we offer
Pre trraining
Fine tuning
Scene parmmeters
(a) Editing external Force
(b) Editing vole ennce good
Pre-triinet eimulsion..
12+3 mins
12+3+3 mins
2k ieerations micutes
2.2k der
2,6 iter
12+1
12+1-2 mins
simulation editing the following way: the user can interrupt the training of NDF at any point, change the scene parameters and continue train- ing for successive improvement. On the other hand, editing can also be done after full con- vergence (aka pre-training) and then fine-tuned with gradually modified design parameters. Edit- ing an NDF provides multiple advantages over training a new NDF from scratch: It is computa- tionally and memory efficient and allows access to interpolated simulations.
Figure VIII: Simulation editing with Neural- ClothSim. We show an example of a simulation pre-trained with a fixed reference state and external force. Once converged, we fine-tune the NDF with smoothly varying external force (top) the pose of the reference geometry (bottom) in each itera- tion. Fine-tuning a pre-trained NDF with updated design parameters is faster and offers querying of paysically-plausiii intermediate simulations.
In the following, we demonstrate editing of the following scene parameters: (a) external force, and (b) reference state geometry. The key idea is use the modified scene parameters the loss function and update the NDF weights with gradient-based optimisation. Specifically, given cloth geometry x, external forces f, we train an NDF to obtain a simulation u" parameterised by network weights O", as described in the main method. As an editing objective, we would like to arrive at novel simulation corresponding to external force p and/or reference geometry x with E N training iterations. Here, / is much smaller than the iterations needed for the convergence of the original simulation. We can then fine-tune the pre-trained NDF over iterations E (0, ..., 7} by minimising the loss function, L(O; ft x²) to obtain edited and interpolated simulations u', O'. Here, assume
27

---

Refercuce Sunulated
Simnlation Buotime
Start
Loes
mane7.tk Her
Mexm deplaseennt
Sleere Comprtsiion
Befoee
Truned for
Cacveeged
Trainig fizee (aninnne))
Toining (cliouted)
Figure IX: Runtime analysis of NeuralClothSim. On the left, we visualise the evolution of the last frame (T 1) over the training iterations. On the right, the plot shows NDF convergence function of training time leading to refined simulations.
a smooth transition of external force or the reference shape from the initial to the edited value, which can be obtained, for example, by linear interpolation, i.e., f lerp(f, ft,
We next demonstrate results for dynamic editing of the pre- pre-trained simulations. We conduct two experiments, i.e., editing external forces and editing the 6DoF pose of the reference geometry; the results are visualised in Fig. VII First, a short simulation of napkin pre-trained as an NDF with fixed reference state and external force, which takes s=12 minutes. In the first example (top row), we gradually vary the direction and magnitude of the external force by linearly interpolating between the original and the final forces. This leads to the motion of cloth towards the instantaneous force direction. the second example (bottom row), we smoothly vary the reference poses and the corresponding position of the handles, generating novel edited simulations. Editing reference pose leads to the motion the cloth towards fixed force direction but originates from varying initial poses. Note how the change in the input Notably, fine-tuning is much faster and takes R=2 minutes, leading to a time-saving of =083% We show two intermediate simulations in Fig. VIIII and other edited simulations corresponding to each iteration can be queried as well. As the simulation is parameterised by the weights a neural network (instead of meshes), our proposed way of simulation editing memory-efficient.
parameters propagates to the entire simulation.
G Performance
The performance of cloth simulator-sute as computation time-is crucial aspect of its ability. This work focuses on the fundamen- tal challenges of developing ral quasistatic simulator with new characteris- ties. Our method does not outperform the clas- sical simulators in all aspects as they are well- engineered and highly optimised. Next, we pro- vide a detailed analysis of NeuralClothSim's
sumples
implicit
Loss
Metm leglacrntntt
8.50 act-
performance.
My
No
G.1 Runtime
Tratning theattonss
Figure X: Analysis of the sampling strategies. We show the influence of the number of training
NeuralClothSim encodes the cloth equilibrium state an NDF, and, consequently, the bulk of computation time lies in the NDF training (i.e., optimisation of the network weights). At inference, extracting the simulated states from NDF as meshes or point clouds requires a single forward pass and is, therefore, fast. In Fig. [X] we provide runtime analysis of three representative simulations function of training time. On the left,
points on the performance of our method.
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we visualise the evolution of the last frame (i.e., equilibria state), showing the refinement of the simulated state with increased training time. Before training, the simulation state is the sum of the reference state and random noisy output from NDF. Within few minutes of training, NDF generates a reasonable simulated state, which then converges within 30 minutes to one hour; see our supplementary video for the evolution of simulation states over training iterations. On the top-right of Fig. [X] we plot loss values as a function of training time, which shows that our training is stable. As NeuralClothSim is instance of a physics-informed neural network with physies loss only (but no data term), the loss is not expected to converge to zero. We monitor the mean NDF over all sampled spatio-temporal points (Fig. IX (bottom right))
additional cue the simulation refinement. Along with the loss, saturation in mean NDF can be used as a stopping criterion. Note that all our experiments are carried out on single NVIDIA Quadro RTX 8000 GPU.
Similar to classical methods 13 simulation with our approach is not unique, as bifurcation due to buckling can lead to solutions with different folds and wrinkles. Among them, the selection of the simulation outcome depends on the NDF convergence. Specifically, the randomness in training samples and weight initialisation introduces desirable optimisation path variations. In all cases, observe NDF training to be numerically stable.
G.2 Sampling Strategy
Next, we study the influence of the number of training points on the performance of our method. Input samples NDF include curvilin- ear Ng and temporal stratified N₂ coordinates (for trajectory visualisation) over which the loss computed at each training iteration. For a nap- kin of size Ω [0, 1]2, simulate for [o, 1] by training NDF for 10k iterations with num- ber of sampling points Nn (5, 10, 15, 20, 25). Computation times for all experiments are com- parable (and slightly higher for the higher num- ber of samples) as they share the GPU memory and are processed in parallel. Fig. X shows the qualitative and quantitative performance. We observe that higher Nn leads to faster learning, as seen in the qualitative result in the top row and the mean displacement plot in the bottom row. Furthermore, it leads to stable optimisation, as seen in the loss plot. Future work could explore advanced sampling techniques for improved performance.
Reprocluibility with identical random seed
Multiple equilibria with varying random seeds
Figure XI: NDF weight initialisation allows us to control the simulation outcome. We can generate multiple valid equilibrium solutions or reproduce simulation.
G.3 Simulation Reproducibility
Next, we investigate whether NeuralClothSim simulations are deterministic. Cloth simulation does have a single ground truth; rather, can have multiple equilibria solutions under the same input parameters (template, material, and boundary conditions). While FEM-based cloth simulators are designed to deterministic, practice, there are several factors-such as numerical precision and parallel computing- that can lead to slight variations in the simulation results between runs. We note that a mesh-based simulator running the same simulation scenario on different machines generates non-identical results (but reproducible ones on the same machine). Interestingly, we can replicate such behaviour by employing the sensitivity of our method to the initialisation the neural network weights. We conducted two experiments leading to the following observations: 1) We can obtain reproducible results we set the random seed leading to the same network initialisation (Fig. [X] (left)), and 2) We observe non-identical results if we do not set the random seed (Fig. XI (rightt)))
H
Additional Comparisons
NeuralClothSim is the first step towards neural implicit cloth quasistatics. Although less mature compared to FEM-based simulators, it offers several desired characteristics; See Table ||| for comparison between existing cloth simulators and our approach.
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Computation minute Budget
2 minutes
minutes
10 minutes
(Max # of vertices)
Diff' ARCSim
(Fixed of vertices)
Diff ARCSim
n/a
NeuralC lothhm (Ours)
Figure XII: Runtime comparison DiffARCSii (3 and our approach. Like most classical simulators, DiffARCSim integrates forward in time, solving for 3D deformation field at each time step, in contrast to our approach which optimises for the 4D spatio-temporal NDF. With decreasing computational budget, Difi produces converged simulated states of the cloth at low resolutions or only early frames at high resolutions. On the other hand, NeuralClothSim offers partially converged simulations at arbitrary resolutions the computational budget decreases.
Table III: Conceptual comparison of our NeuralClothSim to previous state-of-the-art cloth simulators. Our approach enables highly desired properties such as surface continuity, and consistent simulations (folds/wrinkles) at different discretisations of the initial mesh, material conditioning and simulation editing for updated parameters.
Continuous
Consistency
Sim. Editing
Mat. Interpolation
Narain et al.
45
X
X
Liang al.
38
x
XX
Li et al. 36
X
Zhang al. 67
X
Ours
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Increasing spatial resolution
Increasing spatial resolution
Diff ARCSim
(Liang et al.)
DiffClott (L) et al.)
inconsistent
insansistent
NeuralClothSinn (Ours)
NeuralClothse
(Ours)
Figure XIII: Spatial and temporal surface consistency of state-of-the-art differentiable simula- tors and our approach. Classical simulators such as ARCSim (38 and DiffCloth [36] reproduce simulation outcomes when re-running the resolution. However, changing spatio-temporal resolution requires multiple runs and generates possibly different folds or wrinkles instead of re- fining (or previewing) the geometry. Since we learn continuous neural parameterised model, converged (or partially converged) NDF provides consistent simulation when queried at different spatio-temporal inputs. Note that NeuralClothSim does not provide consistent refinement func- tion of computation time (no speed vs fidelity trade-off), but rather consistent simulation with respect the spatio-temporal sampling (at a given computational budget).
H.1 Runtime
We compare the runtime of method to those of the FEM-based simulator DiffARC-
Disrretisation
Discretisation II
Discretisation
Sim 38 45). Since approach does not support collisions, we turn off collision han- dling in DiffARCSim due to the computational overheads for a fair comparison. We simu- late napkin sequence, and our quasistatic re- sult and the dynamic simulated state (after 8) from DiffARCSin
Initial
inconsistent
visualised in Fig. XII For the same computation budget (runtime), we show the best simulated states for both methods. Therefore, we present two sets of results for Dif- fARCSim, i.e., simulated states for the given computational budget 1) with maximum mesh resolution (Fig. (top (top row)) and 2) with fixed mesh resolution (Fig. X (middle row)). We notice that both methods refine the simulated states with increased runtime. With decreasing computational budget, DiffARCSim produces converged simulated states the cloth at resolutions or only early frames high resolutions. the other hand, NeuralClothSim offers partially converged simulations arbitrary resolutions the runtime decreases.
inconsistent
Simulated
Figure XIV: Visualisation the inconsisten- cies observed in the results by FEM-based ARC- Sim [45], even at high resolutions. Our method leads consistent results for much coarser dis- cretisations (Fig. mainn.
H.2 Multi-Resolution Consistency
Next, we show the comparison of NeuralClothSim to DiffARCSim, and DiffCloth 3d in terms of the multi-resolution simulation consistency. We simulate napkin with a fixed corner under gravity, with our approach and ARCsim (Fig. XIII top two rows) and 2) twisting and twirling motion of the skirt with our approach and DiffCloth (Fig. XIII two bottom rows). The compared simulators operate on meshes of pre-defined resolution (as provided initially). Hence, they need to run from scratch for different mesh resolutions, and the simulation outcome are not guaranteed to be the same across these runs under different discretisations. Thus, increasing (or decreasing) spatial resolution result in
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Figure XVI: Limitations. Our approach does not handle collisions, contacts and frictions at moment, since the focus of this work is the fundamental challenges of developing neural cloth simulator. These examples show inaccuracies due to the simplifications made in one possible extension (65)
different folds wrinkles instead of refining simulations at coarser resolutions. Unlike DiffARCSim and DiffCloth, our method provides consistent simulation at arbitrary resolutions. The same 3D points remain unaltered in meshes extracted from NDF different resolutions. We emphasise that do not claim consistent refinement as a function runtime but rather a consistent equilibrium state with respect spatial sampling (at given computational budget). This means that both converged or partially converged NDF provide consistent quasistatics when queried at different spatial inputs.
Our comparison deviates from the literature, the primary reason for using different spatio-temporal resolutions is to adjust runtime and memory usage. For example, the recent method [67] produces artefact-free previewing geometries (at various ap- proximation levels) by biasing their solutions with
12
arcsmm PapRin
ours naplin
dificiath, skirt
File Size (MB)
shell forces and energies evaluated on the finest-level model. This approach offers a trade-off between runtime resolution while maintaining simulation consistency. contrast, with NeuralClothSim, sim- ulation is consistent arbitrary resolutions any moment during the NDF training, which, we believe, is still beneficial for many downstream tasks. Of course, ARCSim and DiffCloth also support very high resolutions, which eventually enables browsing the simulations at different mesh resolutions (while maintaining mesh consistency across the levels); how- ever, at the cost of high memory consumption. More- over, their case, methods for inverse problems that estimate the simulation parameters from simulated states cannot use adaptive, e.g., coarse-to-fine and importance sampling. In contrast, our continuous formulation offers clear advantages in this regard.
500
1000
Vertices
Figure XV: Memory efficiency. We plot the memory requirements for simulations gener- ated by ours and DiffARCSim (38), and Dif- fCloth (36). The simulations are chosen to be of similar complexity and visualised in Fig. XIII The constant memory requirement of our approach due to the compressing property of the MLP weights that encode the simulations.
I Collision
In a preliminary experiment, we model collisions with external objects following earlier neural methods [51]8) i.e., define an additional loss term Lcollision (A) that penalises collisions, leading
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to:
== physics (0) + X (0), with
[22| No
No N2
(655)
Ccalission (0)
ΣΣ ==
max(e SDF (DF(
where SDF(x) is the signed distance to the object, is small safety margin between cloth and object ensure robustness, À the weight for the collision term, and (physica( is our main thin-shell loss in Eq. (mmann matter). We set À 1000, e 0.001, and use pre-trained SDF network encoding signed distance function. Specifically, we employ the method of Sitzmann al. [53] fit an SDF network on an oriented point cloud, where an Eikonal regularisation is used in addition to the SDF and normal loss. Fig. (left visualises simulation result for piece of cloth falling on the Stanford bunny; see our supplementary video for the full simulation. We observe that the cloth coarsely respects the object contours, although constraints Eq. (65) are soft and do not guarantee physically realistic deformations. Difficulties in training PINN with multiple loss terms were previously reported the literature (25] and future research necessary to further investigate collision handling in the context of NDFs.
J Extended Discussion and Limitations
This article addresses the fundamental challenges of cloth simulation with NDFs. All in all, we find the proposed design and the obtained experimental results very encouraging and multiple avenues for future research. Our current quasistatic approach is the first step towards implicit neural simulation. It would be a promising direction add dynamic effects such inertia and damping in this setting. Moreover, our simulator does not handle contacts, friction and collisions which will be necessary for many potential applications beyond those demonstrated in this article. This is, however, a standalone research question in the new context. Several limitations of NeuralClothSim originate from NDF modelling whereas the movement of mesh vertices affects only the local neighbourhood. While this global nature offers continuity and differentiability, we believe exploring alternative network parameterisations that bring the best of both representations could bring improvements in future. Second, our results are currently empirical: While we observe expected results in all our experiments, there are no convergence guarantees or upper bounds on accuracy. Finally, periodic boundary conditions aid mainly with simple geometries; the extension to more complex garments needs further exploration. Future work could also explore modelling different types of human clothing with the help of the proposed implicit neural framework.
single MLP: First, MLP weights have a global effect on the simulation,
Summa summarum, NeuralClothSim is the first step towards neural implicit cloth simulation, which we believe can become a powerful addition the class of cloth simulators. Inverse problems in vision and graphics could also benefit from its consistency (e.g., multi musti-resolution data generation). and adaptivity.
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Reviews:
[{"review_id": "GpvyIclKRW", "reviewer": "Reviewer_bD8m", "summary": "The paper proposes to model the cloth as a fixed parameter domain embedded via a function encoded as a neural network. The network weights are then optimized to minimize a Kirchoff-Love free energy, thus implementing a quasistatic cloth deformation model without a mesh discretization.", "questions": "- The second spatial derivatives of $\\mathcal{F}_{\\Theta}$ are required to exist (164), but what about mixed second partials in $\\Theta$ and $\\mathbf{\\xi}$? Presumably this is not a major problem.\n- The paper proposes representations for rectangular cloth patches with point constraints as well as cylindrical sleeves. Can this framework be extended to garments of arbitrary rest shape and topology?\n- How simple would it be to extend the trajectory model to accurately model dynamics?\n- How do you sample the surface", "limitations": "The authors freely point out that they \"do not claim qualitative superiority over classical cloth simulation methods\" (64).", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The method is technically sound, and the appendices document extensively the choices made. The results are compelling, and they show the benefits of mesh-independence.", "weaknesses": "### Discretization-independence\nThe neural cloth simulation is not \"sensitive to the finite element discretisations\" (49-50). But this is because discretization is used only in a post-simulation evaluation step. Perhaps a fairer analogue of discretization independence would be whether the results are sensitive to the initialization of the neural network weights.\n\n### Generality\nThe paper proposes representations for rectangular cloth patches with point constraints as well as cylindrical sleeves. Can this framework be extended to garments of arbitrary rest shape and topology? Is it possible to support non-boundary point constraints or shape constraints?\n\nThe NDF editing application allows editing the scene parameters after simulation, but it seems like it might be hard to adapt this method to real-time editing of point constraints, which would be desirable for artists.\n\n### Minor details\nThe paper needs general copy editing, but here are some specific points:\n- \"therefore, inherently assume\" => \"inherently assuming\" (26)\n- \"a detailed\" => \"detailed\" (227)\n- \"stretching, and\" => \"stretching and\" (232)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer bD8m, for their comments. The reviewer notes that our \"method is technically sound\", and the \"results are compelling\". We will update the draft to include minor comments, such as copy editing. We now address the points raised in the review. \n\n\n### **Discretisation-independence**\nOur statement at L49-50 needs more elaboration, and we will include the following in the revised version. The statement includes two observations regarding consistency that our method offers, but the finite-element methods lack: 1. consistency with respect to the discretisation/meshing of the initial geometry (Sec. 2, Figs. 6, XII), 2. consistency with respect to multi-resolution simulation (Sec. H.2, Fig. XI), also well-explored in [Zhang et al., 2022]. Regarding the latter, there are different paradigms for speed vs. quality trade-offs for FEM-based cloth simulators and NeuralClothSim. FEM-based methods can have increased speed by reducing the spatial resolution. We highlight that such a trade-off is conceptually not possible for NeuralClothSim as we model the cloth as a continuous surface throughout the entire training. Instead, we can reduce the training time for partial convergence as we did in Fig. X. Thus, a one-to-one comparison is conceptually difficult, which we also elaborate on in Sec. H.2 (supplement). \n\n\nRegarding sensitivity to initialisation, while FEM-based cloth simulators are designed to be deterministic, in practice, there are several factors (such as numerical precision and parallel computing) that can lead to slight variations in the simulation results between runs. This inconsistency is not problematic as cloth simulation doesn't have a single ground truth; rather, it can have multiple equilibria solutions under the same input parameters (template, material, and boundary conditions). We observed that a mesh-based simulator running the same simulation scenario on different machines generates non-identical results but leads to reproducible results on the same machine. This is indeed the case for ours as well. We conducted two experiments: 1) We can obtain reproducible results if we set the random seed leading to the same network initialisation (Fig. 5-(left) in rebuttal pdf), and 2) we observe non-identical results if we do not set the random seed (Fig. 5-(right)). Our results are indeed somewhat sensitive to the initialisation of the neural network weights, similar to the classical simulators are sensitive to hardware- and parallelisation-related effects. \n\n\n### **Arbitrary rest shape and topology**\nKindly see our general comments G3.\n\n\n### **Non-boundary constraints**\nYes, we support point and shape constraints in the interior of the cloth. For those, no change in the method is necessary, i.e., $\\partial \\Omega$ in Eq. (4) can well be a set of boundary points inside the domain. In Fig. 4 of the rebuttal pdf, we show a simulation of non-boundary constraints. \nMoreover, when the initial geometry is provided as a mesh (instead of analytical definition), mesh vertices can be specified as point constraints $\\partial \\Omega$ where $(\\xi^1_{\\partial \\Omega}, \\xi^2_{\\partial \\Omega})$ now correspond to curvilinear coordinates of the fixed vertex. We have demonstrated such results in Fig.6/Fig. XII. \n\n\n### **Derivatives**\nWe agree with the observation that the mixed partial derivatives of $\\mathcal{F}_\\Theta(\\boldsymbol{\\xi})$ with respect to $\\Theta$ and $\\boldsymbol{\\xi}$ are required to exist. We will update the draft.  \n\n\n### **Extension to dynamics**\nKindly see our general response G1.\n\n\n### **Sampling** \nDuring training, we sample the surface with a stratified/jittered sampling technique (points perturbed within a uniform grid). We resample at each training iteration to continuously explore the parametric domain. The number of training samples is typically determined by the available  GPU memory. At test time, we sample regular grid points so that it is easy to triangulate a deformed mesh. Moreover, samples at inference can be generated at much higher resolution as it requires a single forward pass, unlike the expensive derivative computations of physical quantities that are required during training. \n\n\n### **References**\n[Zhang et al., 2022] Zhang, Jiayi Eris, et al. \"Progressive simulation for cloth quasistatics.\" _ACMTOG_ 2022.\n\n[Müller et al., 2022] Müller, Thomas, et al. \"Instant neural graphics primitives with a multiresolution hash encoding.\" _ACMTOG_ 2022.\n\n[Xie et al., 2024] Xie, Tianyi, et al. \"Physgaussian: Physics-integrated 3d Gaussians for generative dynamics.\" _CVPR_ 2024.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dz0HOX32YH", "reviewer": "Reviewer_BLQv", "summary": "The paper proposes a cloth simulation model based on the Kirchoff-Love thin shell theory, using a neural network (SIREN activations) to parameterize a deformation field (NDF) from a base parameterization. The model can handle periodic and Dirichlet boundary conditions, and uses the network to calculate the necessary higher-order derivatives. This continuous NDF allows the model to be discretized at any resolution via sampling more coarsely or finely, and also allows for material conditioning. It is tested against the Belytschko obstacle course to show validity, and in a few simple scenarios against DiffCloth and DiffArcSim, two mesh-based differentiable simulators, and shows comparable performance with superior memory performance.", "questions": "1. Can the framework handle multiple panels joined at seams? If so, did you try any such models?\n2. For the initial fitting to an input mesh, do you have any examples of this? If so, it should be put in the main text, as this represents a nice demonstration of applicability beyond simpler test scenarios. \n3. Can the method be used for inverse design scenarios with respect to material parameters, or with determination of external forces to achieve a trajectory? These were applications considered in comparison methods, and I'd be curious to know if this was attempted at all.", "limitations": "The authors have acknowledged the limitations of the method throughout their text.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The use of an NDF allows for simulation without knowledge of the necessary resolution beforehand.\n    * It also lessens the memory footprint.\n2. The method leverages a principled and sophisticated thin shell theory and thus is able to reproduce several anisotropic and buckling effects that are challenging for simpler traditional mesh-based models.\n3. The work may spur further work in neural cloth simulation by educating readers about the potential advantages of a continuous NDF representation, and a working implementation of it. \n4. There are many details and additional validations in the supplementary material. It is a pretty thorough presentation of the work at hand.", "weaknesses": "1. No collision detection or friction, which are approximated by DiffCloth and DiffArcSim, as acknowledged by the authors.\n2. The training time is quite high, and I believe it's slower than simulation time for state-of-the-art FEM systems (see App. A).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer BLQv for their comments. The reviewer nicely summarizes our paper and notes that our method \"leverages a principled and sophisticated thin shell theory\", \"may spur further work in neural cloth simulation\", and that our presentation is \"pretty thorough\". We now address the points raised in the review.\n\n### **Initial fit and multiple panels**\nKindly see our general comment G3.\n\n### **Inverse design** \nThanks for suggesting this interesting point, which could further expand the usefulness of NeuralClothSim. Although we did not attempt the inverse design scenario of accurately estimating forces/materials, we made initial attempts for NeuralClothSim as a physics-based prior for ill-posed inverse problems. Using the thin-shell hyperelastic energy as a loss function in addition to data terms will have a spatial regularisation effect for improving the physical plausibility (force-material ambiguities would remain in this ill-posed setting). Such ideas are used in earlier works [Kairanda et al., 2022; Yang et al., 2023] with mesh-based physics simulators. These could benefit from our continuous representation and memory adaptivity. While a thorough investigation of this direction would be a standalone research project, we already have promising results. In Fig. 3 of the rebuttal pdf, we visualise fine-grained surface reconstruction from monocular video using NeuralClothSim as a thin-shell physical prior. \n\n### **References**\n[Kairanda et al., 2022] Kairanda et al. \"f-sft: Shape-from-template with a physics-based deformation model.\" _CVPR_ 2022. \n\n[Yang et al., 2023] Yang, Gengshan, et al. \"Ppr: Physically plausible reconstruction from monocular videos.\" _ICCV_ 2023.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "U6nJeTndYE", "reviewer": "Reviewer_Gekm", "summary": "The paper uses Physics-Informed Neural Networks (PINNs) to solve cloth quasistatics. The cloth is represented by a neural implicit function, which provides infinite resolution. The cloth elasticity is modeled using Kirchhoff-Love thin shell theory. The equilibrated displacement field is obtained by minimizing the potential energy of the system. Boundary conditions are strictly enforced through reparameterization tricks.", "questions": "The framework presented in the paper shares many components with the work, [Physics-Informed Deep Learning for Computational Elastodynamics without Labeled Data](https://arxiv.org/abs/2006.08472). However, this mentioned paper is not cited. The primary difference lies in the type of elasticity: the mentioned paper addresses volumetric elasticity, while this paper focuses on thin shell elasticity. Both papers use the same trick to strictly enforce boundary conditions. Additionally, the mentioned paper solves dynamics by incorporating a temporal term in the governing PDE, where the displacement field is a neural implicit function that depends on $(\\boldsymbol{x}, t)$, which is more expressive than this paper. The approach of using potential minimization to solve quasistatic elasticity is also explored in [NTopo](https://proceedings.neurips.cc/paper/2021/hash/55d99a37b2e1badba7c8df4ccd506a88-Abstract.html). These facts significantly weaken the technical contributions of the paper. Please consider differentiating between background knowledge and your contributions. I think the solver framework is not novel, while the continuous shell representation and the introduction of shell elasticity to the PINN community are new.\n\nThe inconsistency in classical simulators can be alleviated by increasing the resolution. A well-defined FEM-based cloth solver should exhibit convergence under refinement. I am interested in whether increasing the resolution of a FEM solver, so that the computation time is roughly the same as the proposed method, will make the inconsistency negligible.\n\nIn Fig.6, why does the proposed method have the concept of discretization? That is, what do discretizations I, II, and III mean for the proposed method?", "limitations": "Limitations are well discussed.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "PINNs can offer infinite resolution, ensuring that the cloth does not suffer from numerical locking issues that arise from spatial discretization in mesh-based simulation methods.", "weaknesses": "The paper is targeted at computer graphics applications.  But it only solves quasistatics and doesn’t consider collisions, making it actually a less suitable candidate for computer graphics applications where realistic dynamics and collision resolution are vital. The paper proposes a visualization of the trajectory from the rest shape to equilibrium, but it appears very damped.\n\nThe proposed method seems to run much slower than the classical simulators.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer Gekm for the detailed comments. The reviewer notes that our cloth modelling “offers infinite resolution”, and the method “does not suffer from numerical locking issues” like classical mesh-based methods. We now address the remaining concerns: \n\n### **Distinction from PINN-approaches for elastodynamics**\nThanks for suggesting the paper by Rao et al. It is relevant, and we will include it in the related works. We agree that we are not the first (or the only) ones to apply neural implicit representation for elastodynamic problems. At the same time, our setting addresses thin-shell and, in particular, realistic cloth simulation that the earlier volumetric elastic works do not [Rao et al. 2021; Zehnder et al. 2021]. Volumetric modelling will often lead to ill-conditioned optimisation when one dimension is thin, necessitating the modelling of thin-shell kinematics. Moreover, to the best of our knowledge, prior works do not tackle geometric nonlinearity: Formulating non-linear strain (Eq.(5)) is crucial for large deformations and rotations arising in cloths (Sec. E.3 ablation). In addition, separating bending and stretching deformation modes allows us to better integrate data-driven non-linear anisotropic models (Secs. 4.3, B.3), unlike the linear elasticity used in Ref. [Rao et al. 2021; Zehnder et al. 2021]. Further, our extensions, such as material conditioning, non-analytical reference geometry and simulation editing, present many opportunities for computer graphics and vision.\n\nWe agree that we did not clearly highlight the differences. In the revised version, we suggest to add the following statement: “While previous works like Rao et al. and Zehnder et al. applied neural implicit representations for volumetric elastodynamic problems, our approach focuses on realistic thin-shell and cloth simulation. It addresses important cloth simulation aspects such as geometric non-linearities and the integration of non-linear anisotropic models, which are crucial for simulating large deformations and rotations.”\n\n### **Figure 6**\nIn this experiment, we show that the classical mesh-based simulators are sensitive (produce different folds and wrinkles) to the discretisation of the reference geometry. We use the discrete meshes as inputs for both competing methods [36,38] and ours for a fair comparison. Thus, for NeuralClothSim, we use the discrete meshes (I, II, and III) to train the initial geometry MLP using the process described in L152-158. The exact discretisations of the initial shape and additional comparison to DiffCloth[36] are visualised in Fig. XII-appendix; Fig. 6 is a shortened version of Fig. XII-appendix due to space constraints of the main paper. \n\n### **Consistency evaluation of classical simulators**\nFor FEM-solvers, we agree and expect the convergence to a continuous solution upon refinement. We tried ARCSim[46] for three slightly perturbed initial mesh discretisations (a similar setup as our Fig. 6 comparison) at increasing resolutions and observed improvements. For a simulation of the napkin with 10k vertices and a runtime of 18 minutes (a few minutes higher than our napkin example), we visualise the results in Fig. 1 of the rebuttal pdf. Indeed, there is an improvement in the degree of consistency in the higher-resolution case (Fig.1-rebuttal) compared to Fig. 6/XII's result of 400 vertices. However, the results still contain noticeable inconsistency, we think it could be due to several operations that are highly discretisation-dependent (such as the bending model relying on the dihedral angles) [46,60]. In contrast to FEM-based solvers, our method produces consistent results and is much less sensitive to discretisations of the initial states as shown in Fig. 6 and Fig. XII.\n\n### **References**\n[Rao et al., 2021] Rao, Chengping, Hao Sun, and Yang Liu. \"Physics-informed deep learning for computational elastodynamics without labeled data.\" _Journal of Engineering Mechanics_ (2021).\n\n[Zehnder et al., 2021] Zehnder, Jonas, et al. \"Ntopo: Mesh-free topology optimization using implicit neural representations.\" _NeurIPS_ (2021).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all reviewers for their valuable feedback, which will help us improve our work further. The reviewers note that our "results are compelling"(bD8m), our proposed method "may spur further works" in neural cloth simulation (BLQv), and, in contrast to classical mesh-based simulators, ours "does not suffer from numerical locking issues" (Gekm).


The reviewers have suggested additional experiments to better showcase the strengths and limitations of our work. **We are pleased to report that we have successfully conducted most of the proposed experiments, yielding favourable results that we are excited to include in the paper.** We now provide clarifications on some of the points shared by the reviewers.


We are happy that the reviewers generally find our method technically sound and interesting while identifying its strengths and weaknesses, which can spur future works in this direction. Our method fundamentally changes the surface representation in cloth simulation and deeply intertwines continuum physics with learning. As cloth simulation is an established field, we do not claim superiority over the existing methods and focus on the fundamental challenges of developing a neural physics-based simulator with new characteristics. We agree regarding the shortcomings; advanced features (e.g. dynamics, full garments) are not in scope for now, but all these ideas highlighted by the reviewers indeed suggest a strong rationale for pursuing this direction.


### **G1: Dynamics (bD8m, Gekm)**
We leave modelling of dynamics as a future work as there are two key aspects that need to be addressed in this regard: 1. physical modelling of inertial & damping effects, 2. a suitable network architecture to model long-term/scalable simulations. Next, we sketch two potential solutions for how one could extend our method (particularly trajectory modelling) to model dynamics. 
One possible solution is to model the _conservation of energy over time_ assuming conservative forces. In our trajectory model for visualisation (Sec. D.1), we minimised the total energy, $\mathcal{E}=\mathcal{E_p} + \mathcal{E_k}$ (L837-846), the sum of and potential $\mathcal{E_p}$ (L257) and kinetic energy $\mathcal{E_k}$. To model true dynamics, one should instead minimise $\frac{d\mathcal{E}}{dt}$.  We attempted this, but the NDF struggles to converge. More specifically, cloth simulation as an initial value problem struggles to propagate the deformation to future states. It is more easy to see this for a toy example. We consider an 1D elastic spring with total energy  $\mathcal{E}=\frac{1}{2}ku^2 + \frac{1}{2}m(u')^2$. Energy constant with initial conditions $u(0)=0,u'(0)=1$ should yield the solution dynamics $u(t)=\sin t$. The learned solution achieved with NDF after 5k iterations is shown in Fig. 2-(a) (rebuttal pdf). This leads us to the alternative solution, the strong form of dynamic equilibrium. It can be modelled as $ku + mu'' =0$, yielding a more accurate solution with fewer 1k iterations (Fig. 2-(b)).


Like the spring example, governing equations as strong form can be modelled for NeuralClothSim. The main changes required would be replacing hyperelastic strain energy with the stresses derived from it, which is well explored in the literature for thin shells [Clyde et al., 2017; Wempner et al., 2023]. Additional things to take care of include enforcing free boundary conditions and ensuring that a higher order of gradients does not hurt the NDF optimisation [Rao et al., 2021].


In summary, we see a clear path to extending our method to dynamics, and we believe it should be treated as a standalone research question.


### **G3: Arbitrary rest shape and topology (bD8m, BLQv)**
We showed examples of initial fitting to meshes in Fig. 6. A detailed version of Fig. 6 is presented as Fig. XII-appendix, which shows the discretisations of the input meshes. In Fig. XII-(right), we additionally showed examples of fitting to non-flat initial geometries. We can move these to the main paper. Regarding arbitrary topology, our current framework does not support multiple panels. We believe this is an important future work. A potential solution is modelling seams as soft constraints. Alternatively, the cloth could be modelled as a Kirchoff-Love thin shell with signed distance functions as the representation [Schöllhammer et al, 2019].


### **References**
[Clyde et al., 2017] Clyde et al. "Simulation of nonlinear Kirchhoff-Love thin shells using subdivision finite elements." _SIGGRAPH SCA_ 2017.


[Wempner et al., 2023] Wempner, Gerald, Demosthenes Talaslidis, and J. Petrolito. "Mechanics of solids and shells: theories and approximations." _Appl. Mech. Rev._ 56.5 (2003).


[Rao et al., 2021] Rao, Chengping, Hao Sun, and Yang Liu. "Physics-informed deep learning for computational elastodynamics without labeled data." _Journal of Engineering Mechanics_ (2021).


[Schöllhammer et al., 2019] Schöllhammer, Daniel, and Thomas-Peter Fries. "Kirchhoff–Love shell theory based on tangential differential calculus." Computational mechanics (2019).

Meta-review: The paper introduces a quasistatic cloth simulation model grounded in the Kirchhoff-Love thin shell theory. It employs a neural network with SIREN activations to parameterize a deformation field (NDF) from a base parameterization. The model adeptly handles periodic and Dirichlet boundary conditions and utilizes the network to compute the necessary higher-order derivatives. This continuous NDF enables the model to be discretized at any resolution by adjusting the sampling density and supports material conditioning. Validation is performed using the Belytschko obstacle course and comparisons with two mesh-based differentiable simulators, DiffCloth and DiffArcSim, demonstrating comparable performance with enhanced memory efficiency.

The method is technically sound, leveraging a sophisticated thin shell theory to capture anisotropic and buckling effects that are challenging for traditional mesh-based models. The results are compelling, showcasing the advantages of mesh-independence. This work could pave the way for further research in neural cloth simulation. However, there are some weaknesses: the method does not incorporate collision detection or friction, which are approximated in DiffCloth and DiffArcSim, as noted by the authors. The training time is notably high, potentially exceeding the simulation time of state-of-the-art FEM systems.

Overall, the paper is thorough and well-presented, with many details and additional validations provided in the supplementary material and videos. The rebuttal addresses all the questions raised by the reviewers, and outlined the future research for dynamics. All reviewers unanimously agree to accept the work.
Decision: Accept (poster)