We propose a neural garment dynamics inference network powered by manifold-
aware transformers. Our approach can be directly applied to unseen garments,
bodies, as well as motions that were not included in the training data.

# Neural Garment Dynamics via Manifold-Aware Transformers

Peizhuo Li10000-0001-9309-9967, Tuanfeng Y. Wang20000-0002-8180-4988, Timur
Levent Kesdogan10009-0006-5839-9677, Duygu Ceylan20000-0001-6530-4556, Olga
Sorkine-Hornung10000-0002-8089-3974
1ETH Zurich, Switzerland 2Adobe Research, United Kingdom

###### Abstract

Data driven and learning based solutions for modeling dynamic garments have
significantly advanced, especially in the context of digital humans. However,
existing approaches often focus on modeling garments with respect to a fixed
parametric human body model and are limited to garment geometries that were
seen during training. In this work, we take a different approach and model the
dynamics of a garment by exploiting its local interactions with the underlying
human body. Specifically, as the body moves, we detect local garment-body
collisions, which drive the deformation of the garment. At the core of our
approach is a mesh-agnostic garment representation and a manifold-aware
transformer network design, which together enable our method to generalize to
unseen garment and body geometries. We evaluate our approach on a wide variety
of garment types and motion sequences and provide competitive qualitative and
quantitative results with respect to the state of the art.

††volume: 43††issue: 2

##  1 Introduction

Modeling the dynamics of garments as they interact with an underlying
collider, such as a moving human body, is a core component for many graphics
applications, e.g., animation [WCPM18], virtual try-on [STOC21], video editing
[YAP*16], etc. There are two main directions to tackle this problem, i.e.,
physically-based simulation [NMK*06, NSO12], and learning-based neural
approaches [PLP20, BME21]. Physically-based simulation provides a generic
framework to produce plausible and accurate geometric details with realistic
motion dynamics. However, acquiring and setting the physical parameters used
in the simulation is not easy and often the simulation process is sensitive to
initial conditions [ZWCM21]. To address such issues, learning-based approaches
are getting popular in recent years. A typical learning-based workflow
[WCPM18, PMJ*22] models the garment dynamics as a mapping from the encoded
garment and collider (i.e., body) motion features to a latent code
representing the garment shape in the next frame. The compact latent
representation enables efficient inference and acts as a regularizer. However,
such global approaches are difficult to generalize to unseen garment and
collider geometries during training.

In this work, we introduce a novel and generalizable learning-based framework
for predicting the garment dynamics by modeling the local interaction between
the garment and the underlying collider. Specifically, we represent the
garment deformation with a continuous deformation field [SCL*04, LSC*04] where
we treat each face of the garment geometry as a sample of this field. We
define a set of _garment and interaction features_ for each face to encode the
state of the garment relative to the underlying body as the body moves from
the current to the next frame. Such features inherently encode how each local
patch on the garment geometry interacts with the underlying body. We
accumulate such features over the past few frames to encode the dynamic
behavior and augment them with global context (i.e., the global velocity of
the garment). Finally, we train a neural network to predict the deformation
gradient of each garment face given this local and global context. Given per-
face predictions, we perform a Poisson solve to obtain the final garment
geometry in the next frame. Our network runs in an auto-regressive manner by
utilizing its past predictions when computing the garment and interaction
features that are provided as input to the network in future frames.

Central to our approach is a transformer-based network architecture [VSP*17]
that predicts the deformation of a target point (e.g., the centroid of a face)
when given tokens that represent the aforementioned features sampled on a
random set of points on the garment surface. The transformer architecture is
capable of modeling long-range correlations between how different parts of the
garment deform. While spatial proximity is a strong cue for similar
deformation behavior, in certain cases, spatially close garment parts can have
very different dynamic behavior. For example, imagine two sides of a skirt
with a cut (see Figure 8) where nearby points on two sides of the cut behave
differently. In order to handle such challenging cases, we empower our
transformer to be _manifold aware_. Specifically, we utilize the geodesic
distance matrix obtained from the rest state of the garment as part of the
attention weights. This encourages the predicted deformation to preserve the
geodesic distance in the resulting garment geometry.

Our approach generalizes to a variety of garment types and geometries. We
evaluate our method on garment and body types and motion sequences unseen
during training. Our approach produces plausible garment geometry with vivid
dynamics and performs competitively with respect to the state-of-the-art
learning-based approaches. In summary, our main contributions are:

  * •

We present a generalizable learning-based approach that predicts plausible
garment dynamics for unseen garment and body types.

  * •

We present a novel _manifold-aware_ transformer architecture that incorporates
both spatial and topological information and is agnostic to the underlying
meshing density, and can generalize to garments with unseen local connectivity
changes such as cuts.

##  2 Related Work

###  2.1 Physics-based Simulation

Modeling the dynamics of the garment w.r.t. the underlying collider motion has
been studied in computer graphics for more than 30 years [TPBF87, MW88].
Physics-based methods [Mül08, MHHR07, NSO12] tend to model the garment
dynamics with real-world physics based on material properties and deform them
according to laws of physics using time integration and collision response.
The focus of this community includes material modeling [BTH*03, MBT*12],
mechanical modeling [CK05, VMF09], as well as collision modeling [HVS*09,
LKJ20], and more recently, converting the whole pipeline to a differentiable
setup [LDW*22, LLK19] for inverse problems. Physics-based workflows often
produce high-quality dynamics but suffer from high computational costs and the
tediousness of tuning the material property for a desired effect.

Figure 1:  Our framework overview. We extract the garment features and
interaction features on the garment geometry from the past frames. Our
manifold-aware transformer is then applied spatially to the input features and
predicts the relative deformation gradients to the next frame. An initial
prediction is obtained with a Poisson solver. After the collision refinement,
we get the prediction for the next frame. We auto-regressively repeat this
process until the desired number of frames is reached.

###  2.2 Data-driven and Learning-based Methods

Prior to the bloom of deep learning, there have been several data-driven
approaches to model garment deformations. Aguiar et al. [DSTH10] propose to
learn a linear dynamic system on the PCA subspace of garment deformation
driven by a pre-defined body to achieve real-time performance. Guan et al.
[GRH*12] also explore statistical models to tackle how garments drape on
different body shapes and poses. Luo et al. [LSW*18] use a lightweight neural
network to transform a linear elasticity-based deformation into a non-linear
deformation. Holden et al. [HDDN19] introduce neural networks that adapt
subspace representations, making it possible to handle interactions between
multiple objects. While being efficient, subspace-based methods are often
limited to the training data and difficult to extrapolate to unseen settings.

In order to leverage the recent success of neural network architectures in the
2D domain, recent approaches have utilized 2D canonical representations (i.e.,
UV mapping) for encoding the garment deformation. DeepWrinkle [LCT18] predicts
pose-dependent wrinkles represented as normal maps in the UV space. Zhang et
al. [ZWCM21] propose to refine the details of coarse simulation using a
similar representation. In addition to pose-dependent effects, Jin et
al.[JZGF20] aim to model motion-dependent deformations. While such UV-based
representations effectively utilize 2D convolutions, they are limited in
encoding spatial neighborhoods across UV seam boundaries.

Predicting the 3D geometry of the garment directly is considered an
alternative solution. Gundogdu et al. [GCS*19], Habermann et al. [HLX*21] fuse
body and garment geometry features to predict pose-dependent effects in the
canonical pose, and deform the result with linear blend skinning (LBS). Patel
et al. [PLP20] also incorporate the dynamics and the change of garment styles,
but model the garment as a height field over the body surface which is limited
to tight-fitting garments. Zhang et al. [ZCM22] handle loose garments by first
learning a plausible deformation latent space but still training garment-
specific networks. Pan et al. [PMJ*22] utilize a virtual skeleton with
additional virtual joints to better capture the low-frequency of loose
garments with respect to the body motion. Similar to D. Li et al. [DTY*22],
high-frequency displacements are then added with a graph neural network. While
showing impressive progress, many of these methods, however, are specific to a
training garment.

The time-consuming generation of training data with physically-based
simulation often acts as a bottleneck for data-driven approaches. Hence,
several unsupervised methods have been proposed recently. Bertiche et al.
[BME21] use physically-inspired loss terms in combination with LBS deformation
to predict pose-dependent deformations. A similar approach is also used by De
Luigi et al. [DLG*22]. Santesteban et al [SOC22] extend the idea to and
introduce strain and inertia-based losses. [BME22] train a network to predict
the garment status such that it minimizes a combination of energy terms. We
provide comparisons to some of these methods in the experiments section.

While learning-based methods have shown remarkable advances in recent years,
generalization, i.e., generalizing to unseen garment types, still remains a
challenge. In most recent concurrent approaches, GarSim [TB23] and HOOD
[GTBH23] tackle this challenge by utilizing a graph-convolution-based
framework [PFSB20]. Due to the limited receptive field of graph convolutions,
their methods are sensitive to the resolution of the input meshes and require
specific designs to incorporate more global information. We provide
qualitative comparisons in the supplementary material regarding these methods.

###  2.3 Transformers

Proposed by Vaswani et al. [VSP*17] for natural language processing,
transformers have quickly been adapted to many areas, including 3D tasks
[GCL*21, DB20, CZG*22, YSW*23]. We also adapt a transformer-based architecture
and introduce a _manifold-aware_ structure to capture both local mesh
neighborhoods and long-distance dependencies effectively.

##  3 Overview

Given the motion of an underlying collider, such as a human body, our goal is
to predict the deformation of the garment as it interacts with the body. Our
key insight is that garment deformations can be predicted in a generalizable
manner by modeling the local interaction between the garment and the body
surface. We introduce a set of garment features (e.g., the deformation
gradient, velocity, and relative distance between the garment and the body)
that capture such local interactions. We form tokens from the features
obtained from a set of triangle faces sampled on the garment surface mesh. We
introduce a manifold-aware transformer network that utilizes such tokens
obtained from a set of past frames to predict the deformation of the garment
in the current frame. We empower the transformer network to be _manifold-
aware_ by encoding the geodesic information of the garment in addition to the
local interaction features. Specifically, we replace part of the learned
attention weights with the geodesic matrix obtained from the garment surface.

We represent the garment deformation using a Jacobian field [AGK*22]. We
discretize the continuous Jacobian field with random samples obtained on the
mesh representation of the garment. Given the predicted deformation gradient,
we solve a Poisson equation to reconstruct the explicit garment surface. Our
approach is agnostic to the connectivity and the density of the triangulation
of the garment surface and can handle garments with various topologies. We
provide the overall architecture of our framework in Figure 1 and next discuss
the details of our approach.

##  4 Method

Figure 2:  Our manifold-aware transformer architecture. The input features are
extracted from the faces of the input mesh. After being projected into
embedding space by a linear transformation, they are fed into the transformer
encoder consisting of $n_{l}$ identical layers. Our manifold-aware self-
attention layers explicitly involve local connectivities of the input
geometry, making it possible to predict accurate dynamics caused by seams. The
output of the transformer encoder is projected to the output features by
another linear transformation. The output features are then used to predict
the next frame of garment deformation.

###  4.1 Pipeline

We represent the garment and the underlying body geometry with their
corresponding vertex positions at a particular time $t$ and the mesh
triangulation. Specifically, let $\\{{\mathbf{V}}^{t},{\mathbf{T}}^{g}\\}$
denote the garment with triangulation ${\mathbf{T}}^{g}$ and vertex positions
${\mathbf{V}}^{t}$ and let $\\{{\mathbf{U}}^{t},{\mathbf{T}}^{b}\\}$ denote
the underlying body with triangulation ${\mathbf{T}}^{b}$ and vertex positions
${\mathbf{U}}^{t}$. Given the previous states of the garment in the past
$n_{\mathrm{hist}}$ frames, i.e.,
$\\{{\mathbf{V}}^{t-n_{\mathrm{hist}}+1},\cdots,{\mathbf{V}}^{t}\\}$, and the
state of the body in the next frame, i.e., ${\mathbf{U}}^{t+1}$, our goal is
to predict ${\mathbf{V}}^{t+1}$, i.e., the state of the garment in the next
frame. We introduce a transformer-based network that takes as input a set of
features ${\mathbf{F}}^{t}_{i}$ computed for each face on the garment surface
utilizing the past and current states of the garment and the underlying body.
The output is the relative deformation gradients $\Psi^{t+1}$ of each face in
the next frame, and global velocity ${{q}}^{t+1}$ of the entire geometry. We
then compute the absolute deformation $\Phi^{t+1}$ to reconstruct the garment
mesh at time $t+1$ via the Possion equation [SP04, SCL*04]. To reduce the
accumulation of error, we also predict the singular values $\Sigma^{t+1}$ for
$\Phi^{t+1}$ at the next frame and use it to regularize the stretching. Since
the predicted cloth geometry is not guaranteed to be collision-free with the
body geometry, we use a post-processing strategy adopted from DRAPE [GRH*12].
A pseudo-code of this process is shown in Algorithm 1. We refer the readers to
the supplementary material for a detailed discussion on the deformation
gradients and the post-process collision refinement method.

Algorithm 1 Prediction of frame $t+1$ from the past $n_{\mathrm{hist}}$ frames

Input: Garment vertex positions ${\mathbf{V}}^{t-n_{\mathrm{hist}}:t}$, body
vertex position ${\mathbf{U}}^{t-n_{\mathrm{hist}}:t+1}$

Output: Vertex positions ${\mathbf{V}}^{t+1}$

procedure
PredictFrame(${\mathbf{V}}^{t-n_{\mathrm{hist}}:t},{\mathbf{U}}^{t-n_{\mathrm{hist}}:t+1}$)

${\mathbf{F}}^{t}\leftarrow$ features extracted from garment ${\mathbf{V}}$
and body ${\mathbf{U}}$

$\Psi^{t+1}$, $\Sigma^{t+1}$, ${{q}}^{t+1}\leftarrow$ prediction from the
network with ${\mathbf{F}}^{t}$

$\bar{\Phi}^{t+1}\leftarrow\Psi^{t+1}\Phi^{t}$

$\Phi^{t+1}\leftarrow$ replace singular values of $\bar{\Phi}^{t+1}$ with
$\Sigma^{t+1}$

$\bar{\mathbf{V}}^{t+1}\leftarrow$ solve a Poisson problem with $\Phi^{t+1}$
and velocity ${{q}}^{t+1}$

${\mathbf{V}}^{t+1}\leftarrow$ collision refinement for
$\bar{\mathbf{V}}^{t+1}$

end procedure

###  4.2 Garment and Interaction Features

Our network takes as input a set of features defined on the garment geometry.
In order to capture the dynamics of the garment, we stack the features
obtained from past $n_{\mathrm{hist}}$ frames together. For simplicity, we
describe the features computed from a single frame and omit the frame index
$t$ in the following text unless otherwise specified.

In addition to the deformation gradient $\Phi_{i}$, derived from the
predictions of preceding frames, in an auto-regressive fashion into the
present frame, we also define a set of features for a given triangle $i$ to
encode the current state of the garment as well as its interaction with the
body. The following features encode the state of the garment geometry.

Orientation. The deformation gradient $\Phi_{i}$ records the deformation
_relative_ to the rest state. Hence, the network is not aware of the
orientation of the surface. To mitigate this issue, we include the normal
direction ${\mathbf{n}}_{i}$ in the world-coordinate of triangle $i$ as part
of the input feature.

Centroid. The input to the transformer network is permutation invariant.
Similar to the positional encoding used in the original transformer, we
include the centralized centroid coordinate ${\mathbf{c}}_{i}=c_{i}-z$ of each
triangle to preserve the spatial order information. $c_{i}$ is the centroid of
triangle $i$ and $z=1/|T|\sum_{i\in T}c_{i}$ is the average centroid of the
garment at a given frame.

Global velocity. The solution of the Poisson equation is not unique up to a
translation. We thus incorporate the per-frame global velocity
${{q}}^{t}=z^{t}-z^{t-1}$ as part of the input.

In order to capture the interaction of the garment with the underlying body,
we also define a set of _interaction_ features as follows:

Signed distance. We encode the relative position of the garment with respect
to the body. For every triangle $i$ in the garment, we record its signed
distance $d_{i}$ to the body. In addition, we also record a direction
$\vec{v}_{i}$ of the nearest point on the body to the centroid $c_{i}$ of
triangle $i$. This constitutes the singed distance feature
${\mathbf{s}}_{i}=\\{d_{i},\vec{v}_{i}\\}\in\mathds{R}^{4}$. It serves as the
local coordinate of the collider, enabling us to encode collider deformation
in the garment space.

Collider deformation. For the network to predict the deformation at frame
$t+1$, we incorporate the deformation of the collider, i.e., the body, at
frame $t+1$ as part of the input feature. To this end, for each face $i$ in
the garment in the current frame, we compute its nearest face on the body in
the current frame. We define a _collider deformation feature_ ,
${\mathbf{d}}_{i}=\\{d_{i},\vec{q}_{i}\\}\in\mathds{R}^{12}$, where
$d_{i}\in\mathds{R}^{3\times 3}$ represents the relative deformation gradient
to the next frame of its nearest body face and $\vec{q}_{i}$ is the velocity
of the centroid of the nearest body face.

Note that, unlike commonly used SMPL [LMR*15] body and pose parameters, the
proposed interaction features are not limited to a specific body model and can
be directly applied to any other type of collider geometry. We conduct
experiments showing the versatility of the proposed interaction features in
Figure 11.

###  4.3 Manifold-aware transformer networks

Network input and output. We denote the concatenation of the aforementioned
per-triangle features as
${\mathbf{f}}_{i}=\\{\Phi_{i},{\mathbf{n}}_{i},{\mathbf{c}}_{i},{\mathbf{s}}_{i},{\mathbf{d}}_{i},{\Sigma}_{i}\\}$.
Furthermore, we collect the features of past $n_{\mathrm{hist}}$ frames
together as
${\mathbf{F}}^{t}_{i}=\\{{\mathbf{f}}_{i}^{k}\\}_{k=t-n_{\mathrm{hist}}+1}^{t}$.
The input to our network is ${\mathbf{F}}^{t}$ along with the global velocity
of the garment in the past $n_{\mathrm{hist}}-1$ frames denoted as
${\mathbf{Q}}^{t}=\\{{{q}}^{k}\\}_{k=t-n_{\mathrm{hist}}+2}^{t}$. The network
then predicts the relative deformation gradient $\Psi_{i}^{t+1}$ and the
singular value ${\Sigma}_{i}^{t+1}$ for every face $i$ and the global velocity
${{q}}^{t+1}$ of the garment in the next frame.

Architecture. The overview of our network architecture is demonstrated in
Figure 2. For memory and computational efficiency, we evenly split the faces
into $n_{s}$ disjoint subsets $\\{{\mathbf{T}}_{i}\\}_{i=1}^{n_{s}}$. For each
of the split ${\mathbf{T}}_{i}\subset{\mathbf{T}}$, we gather the features
$\\{{\mathbf{F}}_{j}^{t}\\}_{j\in{\mathbf{T}}_{i}}$ of every triangle in this
split, concatenated with the global velocity feature ${\mathbf{Q}}^{t}$, as
the input to the network. Note that the features $\\{{\mathbf{F}}_{j}^{t}\\}$
are calculated before the splitting, and no downsampling is involved for
feature calculation. A linear transformation first maps the input features
into an embedding space of dimension $n_{e}$. The embeddings of the features
are then passed through $n_{l}$ transformer [VSP*17] encoder layers, where the
self-attention mechanism learns to capture the global context. Unlike graph
convolution-based networks with only limited receptive fields, our
transformer-based architecture is capable of learning long distance
correlation. The output of the last encoder layer is passed through a linear
layer to predict the relative deformation gradient $\Psi^{t+1}$ and singular
values ${\Sigma}^{t+1}$, as well as the global velocity ${{q}}^{t+1}$ for the
next frame. Note that this framework is triangulation-agnostic and is by
nature not limited to a single garment. The network can be trained with
multiple types of garments and colliders, and can be used to predict unseen
clothes. We demonstrate the versatility of our network in Figure 5.

Manifold-aware self-attention. For the cases with complex garments or
challenging body poses, close-by spatial locations over the garment surface
may have very different dynamic behavior, e.g., the sleeve opening for
left/right arm of a shirt can be spatially close under a crossed arm pose
while their dynamic behavior might be different as shown in Figure 7. To
prevent such spurious correlations, we consider the connectivity of the
garment surface. Specifically, we propose a manifold-aware self-attention
mechanism. For $n_{\mathrm{conn}}$ heads, we use the pairwise geodesic
distance between the centroids of the triangles as the attention score as
below:

${\mathbf{A}}_{i\cdot}=\mathrm{softmax}(-{\mathbf{D}}_{i\cdot}^{p_{\mathrm{geo}}}),$
(1)

where ${\mathbf{D}}_{ij}$ denote the matrix of pair-wise geodesic,
$p_{\mathrm{geo}}$ is the exponential index to control the attention range. We
visualize the geodesic attention weights for a triangle in Figure 3. For an
in-depth study of the effect of manifold-aware self-attention, we refer the
readers to Figure 7.

\begin{overpic}[width=433.62pt]{figures/geodesic-fig.pdf}
\put(11.0,-5.0){{\small$p_{\mathrm{geo}}=1$}}
\put(45.0,-5.0){{\small$p_{\mathrm{geo}}=5$}}
\put(78.0,-5.0){{\small$p_{\mathrm{geo}}=20$}} \end{overpic} Figure 3:
Geodesic attention weights. We use $p_{\mathrm{geo}}$ to control the attention
range.

####  4.3.1 Singular value prediction

Our network predicts the relative deformation between two consecutive frames
and auto-regressively predicts further frames based on existing predictions.
This can lead to error accumulation of the absolute deformation gradient, and
cause severe area distortion of the geometry. As singular values indicate the
scaling of the deformation gradient along the three principal directions, to
mitigate this issue, we also predict the singular values of the deformation
gradient. During inference time, we accumulate the predicted relative
deformation gradients to absolute deformation gradients. We perform SVD and
replace the singular values with the predicted ones before using them to
reconstruct the deformed mesh with Poisson equation. We refer the readers to
Section 5.2 for the study on the effect of singular value prediction.

####  4.3.2 Loss functions and training

We adopt a fully-supervised training scheme to learn the deformation field
from a set of physically-simulated training data. The training is supervised
via the following loss terms:

Deformation gradient loss. The L1 norm is employed to measure the difference
between the predicted relative deformation gradients $\Psi_{i}^{t+1}$ and the
ground truth $\tilde{\Psi}_{i}^{t+1}$:

$\mathcal{L}_{\text{def}}=\frac{1}{|{\mathbf{T}}_{i}|}\sum_{j\in{\mathbf{T}}_{i}}\|\Psi_{j}^{t+1}-\tilde{\Psi}_{j}^{t+1}\|_{1}.$
(2)

Singular value loss. Similarly, the L1 norm is utilized to measure the
difference between the predicted singular values ${\Sigma}_{i}^{t+1}$ and the
ground truth $\tilde{\Sigma}_{i}^{t+1}$:

$\mathcal{L}_{\text{sv}}=\frac{1}{|{\mathbf{T}}_{i}|}\sum_{j\in{\mathbf{T}}_{i}}\|{\Sigma}_{j}^{t+1}-\tilde{\Sigma}_{j}^{t+1}\|_{1}.$
(3)

Global velocity loss. The L1 norm is applied to measure the difference between
the predicted global velocity ${{q}}^{t}$ and the ground truth
$\tilde{{q}}^{t}$:

$\mathcal{L}_{\text{vel}}=\|{{q}}^{t}-\tilde{{q}}^{t}\|_{1}.$ (4)

Our full loss used for training summarizes as:

$\mathcal{L}=\mathcal{L}_{\text{def}}+\lambda_{\text{sv}}\mathcal{L}_{\text{sv}}+\lambda_{\text{vel}}\mathcal{L}_{\text{vel}}.$
(5)

During training, we supervise only one step of prediction. We random sample
$n_{\mathrm{hist}}+1$ consecutive frames from a physically-simulated dataset
and use the features of the first $n_{\mathrm{hist}}$ frames as input of the
network, and supervise the prediction on frame $n_{\mathrm{hist}}+1$ using the
loss functions in Equation 5. Besides, all input features are normalized to
have zero mean and unit variance. To prevent error accumulation in our auto-
regressive workflow, we add noise to the input features of the network during
the training. Specifically, we add a random noise
$\varepsilon\sim\mathcal{N}(0,\sigma_{n})$ to the normalized input features
${\mathbf{F}}^{t}$. The noise is sampled independently for each feature
dimension and each triangle. We find that this is a simple yet effective
solution to stabilize the long-term generation. For a detailed description of
the layers in our network and the specific values of the hyper-parameters, we
refer to the supplementary material.

##  5 Experiments

Figure 4: Breakdown of inference performance.

We evaluate our approach on various garment and body types to demonstrate the
generalization ability of our method. We also compare with other neural
techniques and provide an ablation to evaluate the effectiveness of various
components in our design. Please refer to the supplementary video for
qualitative results.

###  5.1 Implementation details

Our framework is implemented in PyTorch [PGM*19], and the experiments are
performed on an NVIDIA GeForce RTX 3090 GPU. We optimize the parameters of our
network with the loss term in Equation 5 using the Adam optimizer [KB14]. It
takes about 48 hours to train our network. We refer the readers for a detailed
description of our network architecture and hyper-parameters to the
supplementary material.

Running time. We show a breakdown of the running time of each component in
Figure 4. The most expensive operation is collision refinement, which involves
solving a sparse linear system of size $3N\times 3N$, where $N$ is the number
of vertices. The feature collection contains the computation of the network
input ${\mathbf{f}}_{i}$ excluding signed distance function (SDF), which is
listed separately. Our efficient SDF calculation is implemented on GPU by
combining minimum pairwise distance and winding numbers [JKS13]. The Poisson
solver is implemented with a pre-computed Cholesky decomposition using CHOLMOD
[CDHR08] and solving on the GPU [Nau11] using the implementation by Nicolet et
al. [NJJ21].

\begin{overpic}[width=433.62pt]{figures/different-garments-fig.pdf}
\end{overpic} Figure 5: A general model for different garments. Our model is
capable of predicting the dynamics of different garments driven by various
motions.

Dataset. We train our model with the CLOTH3D dataset [BME20] which contains 7
categories of garments (shirt, shirt, top, trousers, skirt, jumpsuit, and
dress). Within each category, garment shape is augmented using cutting and
resizing. The garments are then simulated in 3D on a body that is animated
with a motion sequence from the CMU human motion dataset [CMU19]. We randomly
select 200 simulated sequences (around 50,000 frames) and train our model with
6 garment categories, excluding the “skirt” category. Since the dataset also
contains several material configurations, we use the “cotton” configuration as
our training data.

At test time, we evaluate our model with unseen augmentations (i.e., cutting
and resizing) of the seen garment types and with garments in the unseen
“skirt” category, driven by 30 unseen motion sequences.

Evaluation Metrics. We evaluate our method with respect to ground truth
simulation results using the mean vertex error (in cm) and Chamfer Distance
[WPZ*21]. We also measure the geodesic distortion by calculating the L1 error
between the pair-wise geodesic distance of the generated results and ground
truth.

###  5.2 Results

As shown in Figure 5 and our supplementary video, our method can synthesize
realistic results on different types of garments and correctly capture the
subtle dynamics. In the following, we provide a more thorough evaluation.

Robustness to remeshing.

\begin{overpic}[width=433.62pt]{figures/resolution-fig.jpg}
\put(10.0,-5.0){{\small 1032 faces}} \put(44.0,-5.0){{\small 2067 faces}}
\put(78.0,-5.0){{\small 5168 faces}} \end{overpic} Figure 6: Robustness to
remeshing. Our network predicts consistent results for different meshing,
thanks to our feature representation and network architecture.

The input and output features of our network are triangulation-agnostic.
Furthermore, the manifold-aware self-attention module is also robust to
changes in triangulation. We evaluate our model on the same garment with
different meshing, specifically containing 1032, 2067, and 5168 faces,
respectively. It can be seen in Figure 6 that our method generates consistent
results across different mesh resolutions, which is not possible to handle
with graph-convolution-based methods [GTBH23, TB23].

\begin{overpic}[width=433.62pt]{figures/manifold-aware-fig.pdf}
\put(9.0,-5.0){{\small w/o manifold awareness}} \put(68.0,-5.0){{\small Full
approach}} \end{overpic} Figure 7: Manifold awareness. The manifold aware
component is able to resolve instances where spatial proximity happens to
unconnected cloth segments. While without the manifold-aware component, it
fails to distinguish the two segments.

Effect of manifold-aware self-attention. In certain motion sequences, two
different parts of the garment (e.g., the sleeves and the torso) can spatially
come close together while their dynamics are still significantly different. As
shown in Figure 7, our manifold-aware attention module can effectively handle
such cases and generate plausible results.

\begin{overpic}[width=433.62pt]{figures/dress-cut-fig.pdf}
\put(8.0,-5.0){{\small w/o Manifold wareness}} \put(67.0,-5.0){{\small Full
approach}} \end{overpic} Figure 8: Local connectivity changes, such as a cut
as seen here, are incorporated into our network’s prediction with the help of
the manifold-aware transformer.

Since our model explicitly encodes the geodesic information, it can generalize
to garments with unseen seam cuts. Due to the lack of connectivity changes in
the CLOTH3D dataset, we created a dataset from a given dress with 10 different
seam cuts as the training set and 2 different seam cuts as the test set. It
can be seen in Figure 8 that our model is able to synthesize plausible results
reflecting the unseen seam cuts without re-training. Please refer to the
accompanying video for a complete result.

Table 1: Ablation study. | Mean vertex error (cm) | Geodesic distortion
---|---|---
w/o manifold-aware | 4.54 | $2.83\times 10^{-2}$
w/o singular prediction | 12.2 | $9.33\times 10^{-2}$
Full approach | 3.19 | $2.05\times 10^{-2}$

We conduct an ablation study over the manifold-aware component by using only
learned attention weights. As shown in Table 1, the geodesic information not
only leads to better visual performance, but also better preserves the
geodesics.

\begin{overpic}[width=433.62pt]{figures/pgeo-cmp-fig.pdf}
\put(10.0,-5.0){{\small Small $p_{\mathrm{geo}}$}} \put(43.0,-5.0){{\small
Suitable $p_{\mathrm{geo}}$}} \put(78.0,-5.0){{\small Large
$p_{\mathrm{geo}}$}} \end{overpic} Figure 9: Choice of $p_{\mathrm{geo}}$. We
show that with a suitable choice of $p_{\mathrm{geo}}$, our method can handle
the problem of spatial proximity, while a too small or a too large
$p_{\mathrm{geo}}$ can lead to artifacts.

Choice of $p_{\mathrm{geo}}$. As can be seen in Figure 9, a too small or too
large $p_{\mathrm{geo}}$ yields inferior results. We show that our model is
robust to a large range of $p_{\mathrm{geo}}$ values in Table 2. When
$p_{\mathrm{geo}}$ ranges between 1 and 50, the mean vertex error (MVE)
remains small. We use $p_{\mathrm{geo}}=20$ in our other experiments.

Table 2: Ablation study on the choices of $p_{\mathrm{geo}}$. $p_{\mathrm{geo}}$ | 0.01 | 0.1 | 1 | 10 | 20 | 50 | 100
---|---|---|---|---|---|---|---
MVE (cm) | 4.34 | 3.79 | 3.23 | 3.31 | 3.19 | 3.20 | 4.72

Effect of singular value prediction. We evaluate the effectiveness of our
singular value prediction term in Equation 3 by removing it from the loss
function and directly using the predicted deformation gradient without
replacing the singular values. As shown in Table 1 and Figure 10, the singular
value prediction term helps prevent the accumulation of stretching and
produces more accurate results.

\begin{overpic}[width=433.62pt]{figures/sigma-cmp-fig.pdf}
\put(5.0,-5.0){{\small w/o singular value prediction}} \put(68.0,-5.0){{\small
Full approach}} \end{overpic} Figure 10: Singular value prediction. With the
help of predicting singular values of absolute deformation gradient, our
method does not suffer from the over-stretching problem.
\begin{overpic}[width=433.62pt]{figures/unseen-body-fig.pdf}
\put(4.0,-5.0){{\small Training body}} \put(31.0,-5.0){{\small Unseen body}}
\put(70.0,-5.0){{\small Our result}} \end{overpic} Figure 11: Unseen body
model. The mannequin wearing the dress is not seen by our model during
training. However, it is still able to predict a plausible result.

Unseen body model. While our training data is driven by the SMPL [LMR*15]
model, our model can be applicable to different human body geometries. As
shown in Figure 11, we test our model with a mannequin, a significantly
different body model from the SMPL and demonstrate plausible results. Please
refer to the accompanying video for additional results.

###  5.3 Comparisons

Table 3: Quantitative comparison to the supervised method. | Mean vertex error (cm) | Chamfer distance
---|---|---
SSCH [STOC21] | 2.93 | $3.84\times 10^{-4}$
Ours | 2.69 | $3.30\times 10^{-4}$
\begin{overpic}[width=433.62pt]{figures/comp-supervised-fig.pdf}
\put(5.0,-5.0){{\small
SSCH~{}\cite[cite]{[\@@bibref{}{santesteban2021self}{}{}]}}}
\put(46.0,-5.0){{\small Ours}} \put(75.0,-5.0){{\small Ground truth}}
\end{overpic} Figure 12: Comparison to SSCH. Our network is able to faithfully
capture the dynamics of the garment.

Supervised methods. We compare out method to SSCH [STOC21] as a baseline
supervised method. For a fair comparison, we retrain our model with the same
VTO dataset this method is trained on. The results generated by our method
faithfully reconstruct the dynamics of the garment as demonstrated in Figure
12. We also report superior quantitative results in Table 3.

\begin{overpic}[width=433.62pt]{figures/comp-snug-fig.pdf}
\put(12.5,-5.0){{\small
SNUG~{}\cite[cite]{[\@@bibref{}{santesteban2022snug}{}{}]}}}
\put(73.0,-5.0){{\small Ours}} \end{overpic} Figure 13: Comparison to SNUG.
The global self-attention of our model allows the cloth to deform naturally in
front of the torso, yielding better visual quality, while the baseline fails
to generate the same effect.

Unsupervised methods. We compare our method to the state-of-the-art of
unsupervised learning method SNUG [SOC22]. We use the version of our method
trained on the VTO dataset that this method is also trained on. Note that SNUG
requires new training for every new garment while our method provides a
general model. It can be seen in Figure 13 that our model is able to generate
plausible results and provides better dynamics, while the garments are not
seen during training. Please refer to the accompanying video for a complete
result.

\begin{overpic}[width=433.62pt]{figures/comp-hood-2-fig.jpg}
\put(12.0,-5.0){{\small
HOOD~{}\cite[cite]{[\@@bibref{}{grigorev2023hood}{}{}]}}}
\put(72.0,-5.0){{\small Ours}} \end{overpic} Figure 14: Comparison to HOOD.
The meshes shown in the first and second rows contain 5,169 and 31,008 faces,
respectively. It can be seen that our method produces consistent results
across different resolutions, whereas HOOD exhibits unnatural dynamics and
artifacts when processing high-resolution input.

Generalizable methods. Finally, we compare our method to the concurrent works
[TB23, GTBH23] that tackle the generalization problem. As the code/date of
GarSim [TB23] is not publicly available yet, we are limited to providing
qualitative comparisons in the supplementary material. To compensate for the
limited receptive field of graph convolution, HOOD [GTBH23] constructs a
hierarchy of simplified meshes. As demonstrated in Figure 14, when provided
with high-resolution input, HOOD struggles to capture correct global dynamics
due to the slow propagation of these dynamics and generates undesired
artifacts caused by simplified graph construction. Please refer to the
accompanying video for detailed results.

##  6 Conclusion

In this paper, we introduce a learning-based generalizable solution for
predicting garment dynamics with respect to an underlying body in motion.
Previous learning-based approaches have been limited to garment-collider
combinations present in the training dataset, often requiring refinement when
applied to unseen cases. To address this limitation, we propose a deformation
field-based garment representation combined with a transformer-based neural
network. This combination enables us to handle different garment types and
body models as colliders, contrary to existing methods, which are usually
limited to a predefined parametric model (e.g., SMLP [LMR*15]). Additionally,
we incorporate pairwise geodesic distances to weight the self-attention heads
in our network, resulting in a manifold-aware transformer capable of capturing
not only spatial but also topological correlations.

Our experiments demonstrate that our method can handle challenging scenarios
involving complex garments and dynamic motions. We believe that our approach
represents a novel direction toward modeling realistic and captivating
clothing behavior for general garment dynamics and digital human modeling.

Limitations and Future work. To focus on our key insights, we have made
several assumptions to simplify the problem setup. However, these assumptions
also set limitations for our work and may stimulate potential future work.

_Self-collision handling_ We do not explicitly handle the garment-to-garment
collisions. Thanks to our manifold-aware transformer, garment self-
interpenetrations do not affect our auto-regressive workflow and are visually
hard to observe according to our experiments. However, this could still be a
problem for some downstream applications and may eventually lead to physically
incorrect garment modeling.

_Material variation._ Our training dataset is produced by the same set of
parameters, so the fabric material variation is not formulated into our
current model. Allowing the users to control the material property can be a
bonus feature for many scenarios [WCPM18].

_Fine details in predicted geometry._ Although our method can handle global
dynamics well, the results lack fine details such as wrinkles when compared
with state-of-the-art methods, as can be seen in the collar region in Figure
13. We conjecture that this is due to the downsampling strategy and could be
improved by reducing the downsampling rate and training over a longer time.

_Unsupervised learning._ Last but not least, recent advances in unsupervised
garment dynamics learning based on physical properties [BME22, SOC22] show a
promising direction with no data generation burdens. We believe having the
terms from the physical constraints can potentially elaborate our approach in
an unsupervised manner as well.

## Acknowledgements

We thank the anonymous reviewers for their valuable feedback. This work was
supported in part by the ERC Consolidator Grant No. 101003104 (MYCLOTH), Adobe
Research Internship Program, and Adobe University Gift Fund.

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