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Differentiable Cloth Simulation for Inverse Problems
Junbang Liang
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Content
- Motivation
- Related Work
- Our Method
○ Simulation pipeline ○ Gradient Computation
- Results
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Differentiable Cloth Simulation for Inverse Problems Junbang Liang - - PowerPoint PPT Presentation
Differentiable Cloth Simulation for Inverse Problems Junbang Liang - - PowerPoint PPT Presentation
Differentiable Cloth Simulation for Inverse Problems Junbang Liang 1 Content Motivation Related Work Our Method Simulation pipeline Gradient Computation Results 2 Motivation Differentiable Physics
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Motivation
- Differentiable Physics Simulation as a Network Layer
○ Physical property estimation ○ Control of physical systems
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Motivation
- Differentiable Physics Simulation as a Network Layer
○ Physical property estimation ○ Control of physical systems Yang et al. (2017) Demo of our differentiable simulation
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Content
- Motivation
- Related Work
- Our Method
○ Simulation pipeline ○ Gradient Computation
- Results
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Related Work
- Differentiable rigid body simulation
○ Formulation not scalable to high dimensionality Degrave et al. 2019 Belbute-Peres et al. 2019
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Related Work
- Learning-based physics [Li et al. 2018]
○ Unable to guarantee physical correctness
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Our Contributions
- Dynamic collision handling to reduce dimensionality
- Gradient computation of collision response using implicit differentiation
- Optimized backpropagation using QR decomposition
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Content
- Motivation
- Related Work
- Our Method
○ Simulation pipeline ○ Gradient Computation
- Results
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Introduction to Simulation
- Partial differential equation (PDE) of Newton’s law:
- Solve satisfying , where
- Discretization to ordinary differential equations (ODE):
- Solve satisfying , where
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Introduction to Simulation
- Partial differential equation (PDE) of Newton’s law:
- Solve satisfying , where
- Discretization to ordinary differential equations (ODE):
- Solve satisfying , where
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Point Cloud Simulation Flow
1. 2.
○ S ○ Newton’s method
3. 4.
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Cloth Simulation Flow
1. 2.
○ S ○ Newton’s method
3. 4. 5.
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Collision Response
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Collision Response
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Mesh Simulation Flow: Backpropagation
1. 2.
○ S ○ Newton’s method
3. 4. 5.
Gradient computation available? Handled by auto-differentiation Handled by auto-differentiation Handled by auto-differentiation
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Mesh Simulation Flow: Backpropagation
1. 2.
○ S ○ Newton’s method
3. 4. 5.
Using implicit differentiation! Gradient computation available?
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Implicit Differentiation: Linear Solve
- Formulation:
- Input: and . Output:
- Back propagation: use to compute and
○ : the loss function.
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Implicit Differentiation: Linear Solve
- Back propagation: use to compute and , where is the loss
function.
- Implicit differentiation form:
- Solution:
where is computed from , and is the solution of .
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Mesh Simulation Flow: Backpropagation
1. 2.
○ S ○ Newton’s method
3. 4. 5.
Using implicit differentiation! Gradient computation available?
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Gradients of Collision?
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Collision Handling
- Objective formulation: Quadratic Programming
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Gradients of Collision Response
- Karush-Kuhn-Tucker (KKT) condition:
- Implicit differentiation:
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Gradients of Collision Response
- Solution:
- where dz and dλ is provided by the linear equation:
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Acceleration of Gradient Computation
- Linear system of n+m
○ n: DOFs in the impacts ○ m: number of constraints/impacts
- Insight: Optimized point moves along the
tangential direction w.r.t. constraint gradient
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Acceleration of Gradient Computation
- Linear system of n+m
○ n: DOFs in the impacts ○ m: number of constraints/impacts
- Insight: Optimized point moves along the
tangential direction w.r.t. constraint gradient
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Acceleration of Gradient Computation
- Explicit solution of the linear equation:
where Q and R is obtained from:
- Theoretical speedup: O((n+m)³) → O(nm²)
○ n: number of vertices ○ m: number of constraints
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Content
- Motivation
- Related Work
- Our Method
○ Simulation pipeline ○ Gradient Computation
- Results
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Experimental Results
- Ablation study
○ Backpropagation speedup
- Applications
○ Material estimation ○ Motion control
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Ablation Study
- Speed improvement in backpropagation
- Scene setting: a large piece of cloth crumpled inside a pyramid
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Results
- Speed improvement in backpropagation
- Scene setting: a large piece of cloth crumpled inside a pyramid
The runtime performance of gradient computation is significantly improved by up to two orders of magnitude.
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Material Estimation
- Scene setting: A piece of cloth hanging under gravity and a constant wind
force.
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Results
- Application: Material estimation
- Scene setting: A piece of cloth hanging under gravity and a constant wind
force. Our method achieves the fastest speed and the smallest overall error.
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Application: Material Estimation
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Motion Control
- Scene setting: A piece of cloth being lifted and dropped to a basket.
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Results
- Application: Motion control
- Scene setting: A piece of cloth being lifted and dropped to a basket.
Our method achieves the best performance with a much smaller number of simulations.
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Application: Motion Control
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Conclusion
- Differentiable simulation
○ Applicable to optimization tasks ○ Embedded in neural networks for learning and control
- Fast backpropagation for collision response
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Future Work
- Optimization of the computation graph
○ Vectorization ○ PyTorch3D/DiffTaichi
- Integrate with other materials
○ Rigid body, deformable body, articulated body, etc
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Q&A
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