La Latent-sp space Dynam Dynamics ics for r Re Reduced - - PowerPoint PPT Presentation

The 40° Annual Conference of the European Association for Computer Graphics

La Latent-sp space Dynam Dynamics ics for r Re Reduced Deformable Simulation

Lawson Fulton1,2, Vismay Modi1, David Duvenaud1, David I.W. Levin1, Alec Jacobson1

1 University of Toronto, Canada 2 MESH Consultants, Canada

[Ziva Dynamics]

Wh Why y de deformabl ble si simul ulation? n?

Re Research Question

Can we use machine learning to accelerate hyperelastic simulation?

Re Related Work

Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow Wiewel et al. 2019

Learn how to update the latent state of a system

Deep Fluids – A Generative Network for Parameterized Fluid Simulations Kim et al. 2019

Re Related Work

DeepWarp: DNN-based Nonlinear Deformation Luo et al. 2018

Learn correction to cheap simulation

Neural Material: Learning Elastic Constitutive Material and Damping Models from Sparse Data Wang et al. 2018

Ou Our Approach

Build on the vast literature of Model Reduction Simulate in nonlinear latent space using the true equations of motion

Fir First, t, wh why is is it it slo low? w?

u =      p1 p2 . . . pn     

Vertex Displacements

n ≈ 40, 000

F(u) = M¨ u

Solving large differential equation

Sol Solver

Elastic Potential Inertia Term New configuration

un+1 = argmin

u

V (u) + I(u, un, ˙ un)

Previous State

Fast and stable solution: Implicit Euler as a minimization problem

in E(u)

Objective Function

Solve using pre-conditioned quasi-newton solver like L-BFGS

un+1 = argmin

u

E(u)

Exi Existing ng Work: k: Mode del Reduc duction

High Dimensional System Low Dimensional System

u ∈ R40,000

q ∈ R∼60

u = Uq

Reduced Coordinates

q = UT u

Mod Model R Reduction

• n

Replace high-dimensional problem with low-dimensional

Big

Small

un+1 = argmin

u

E(u)

qn+1 = argmin

q

E(Uq)

Becomes

St Static Sol c Solve E Examp mple

Where does come from? (Uq

Mod Model R Reduction

• n - Ex

Exampl ple

Mod Model R Reduction

• n - Ex

Exampl ple

P = [u1u2u3u4]

Collect Snapshots

P = UΣVT

Perform PCA (via SVD) Keep k largest eigen values

U := U1:k

Mod Model R Reduction

• n - Ex

Exampl ple

P = [u1u2u3u4]

Collect Snapshots

P = UΣVT

Perform PCA (via SVD) Keep k largest eigen values

u = Uq

U := U1:k

u1

u2

. . .

uN

U = PCA([u1 . . . uN], k)

k = 62

Li Limi mits t to Li

• Linear R

r Reduct ction

• n

Full Space

Li Limi mits t to Li

• Linear R

r Reduct ction

• n

6 Degrees of Freedom

Ca Can we do

• better?

r?

Linear

u = nonlinear(z)

|q| > |z|

Linear: 6 DOF Nonlinear: 6 DOF

Ou Our Contribution

Many possibilities for

= nonlinear(z)

We use a neural network trained as an Autoencoder to create a unique for a given scenario

= nonlinear(z)

Au Autoencoders

encode(u) = z

Encoded “Latent” vector z

decode(z) = ˜ u

de(u

) = ˜ u

) = z

decodek(z) = activation(zT Wθ + b)

Decode is a sequence of function applications

activation(x)

θ∗ = argmin

θ N

X

i=1

k decode(encode(ui)) uik2

2

Wθ

Optimize the weights by automatic differentiation and gradient descent Minimize Mean Squared Error with ADAM

u1

u2

. . .

uN

de(u) =

) = ˜ u

Training directly on full mesh results in long training times and poor approximation

Previous work: last layer of network is linear, so just initialize it with PCA We observe you can train directly in the PCA space and get equivalent results.

Ou Our Training g Pipeline

U = PCA([u1 . . . uN], k)

Do PCA on snapshots

[q1 . . . qN] = UT [u1 . . . uN]

Project training samples Train autoencoder to reduce the PCA coefficients further

θ∗ = argmin

θ N

X

i=1

k decode(encode(qi)) qik2

2

• 5D PCA Only

• 5D PCA Only

• 5D PCA Only

• 5D PCA Only

High Dimensional System

Low Dimensional System

Tiny Dimensional System

q = decode(z)

q = UT u

z = encode(q)

Con Convergence Ra Rate

La Latent Sp Space ce D Dynami mics cs

Big

Small(er)

un+1 = argmin

u

E(u)

zn+1 = argmin

z

E(U decode(z))

Becomes

Ho How w do do we e mak ake e it it fas ast? t?

Recall our objective function:

Elastic Potential Inertia Term

E(z) = V (U decode(z)) + I(U decode(z), un, ˙ un)

Inertia Term

I = 1 2h2 ku un ˙ unhk2

M

I = 1 2h2 kdecode(z) qn ˙ qnhk2

UT MU

I = 1 2h2 kU decode(z) un ˙ unhk2

M

Precompute and only partially decode

hk2

UT MU

Save as for next timestep

qn

Ho How w do do we e mak ake e it it fas ast? t?

Recall our objective function:

Elastic Potential Inertia Term

E(z) = V (U decode(z)) + I(U decode(z), un, ˙ un)

Elastic Potential

Cu Cubature

V (u) =

# Tets

X

i=1

Vi(u)

Use [An et al. 08]’s “Optimized Cubature”

V (u) ≈ X

i∈S

wiVi(u)

Approximate with weighted sum Only fully-decode elements we need

Cu Cubature

V (u) =

# Tets

X

i=1

Vi(u)

≈ X

i∈S

Re Results: Stability

Re Results: Stability

Re Results: Stability

An And finally

Only the gradient of our objective is required since using a quasi-Newton scheme

rE(z) = JT

decode

∂E ∂q

Non-constant Jacobian matrix of our autoencoder

) = JT

decode

Automatic differentiation allows us to evaluate with equivalent complexity as a single forward evaluation

JT

decodev

rE(z) =

• Avg. Time per E Evalutation

Time (ms)

PCA Autoencoder

Time (ms)

• Avg. Time per Timestep

×28 Avg. Evaluations ×9.3 Avg. Evaluations

PCA Autoencoder

Cubature Point Update Energy/Force Evaluation

= decode(z)

JT

decodev

Precondition L-BFGS (1x)

Re Results: Performance

Re Results: Accuracy

Li Limi mitation

• ns

Su Summa mmary

§ Autoencoders can reduce system dimensionality further than linear alone. § This reduction allows faster simulation § Results are robust, even for small spaces and few cubature points.

Futur Future e Work

§ Can we incorporate cubature into our method?

Futur Future e Work

§ Can we incorporate cubature into our method? § One network, many shapes? § Automatic training data generation?

Ac Acknowledgements

§ NSERC Discovery Grants (RGPIN–2017–05235, RGPIN-2017-05524, RGPAS–2017–507938, RGPAS-2017-507909) § Connaught Funds (NR2016–17) § Canada Research Chairs Program § Gifts from the Fields Institute, Adobe Systems Inc, Autodesk Inc, and MESH Inc. § Sarah Kushner for help with figure creation

Thank you for listening!

Contact: lawson@cs.toronto.edu Project Page: bit.ly/2V3U9Kv

La Laten ent-space ce Dynamics cs for Reduce ced De Defor

• rmab

able le Sim imula lation ion

Lawson Fulton, Vismay Modi, David Duvenaud, David I.W. Levin, Alec Jacobson University of Toronto

Training Error×10-2 Step Time (ms) # Hidden Layers

• Avg. Time per E Evalutation

Time (ms) Time (ms)

• Avg. Time per Timestep

cubature evaluation cubature decode Uq vjp(z,v) ϕ(z)

×28 Avg. Evaluations ×9.3 Avg. Evaluations

preconditioner cost PCA Autoencoder PCA Autoencoder

Tr Training Data

Ch Choi

• ice of
• f Ac

Activation

• n

Pr Preconditioner

˜ H = JT

zn ˜

K0Jzn

˜ K0 = U T K0U

K0 = ∂2V(0) ∂u2