iwapt 2020 workshop: acceleration of structural analysis ... · can only estimate physics under...

iWAPT 2020 Workshop:

Acceleration of Structural Analysis Simulations using CNN-based Auto-Tuning of Solver Tolerance

Amir Haderbache1, Koichi Shirahata1, Takuji Yamamoto1,Hiroshi Okuda2 and Yasumoto Tomita1

1Fujitsu Laboratories LTD.2The University of Tokyo

Copyright 2020 Fujitsu Laboratories Ltd.0

Outline

1. Introduction2. Theoretical Background3. Performance Analysis4. Proposal5. Experimental Evaluation6. Conclusion

1 Copyright 2020 Fujitsu Laboratories Ltd.

Introduction


AI for HPC There is a surge of interest in applying machine learning to

traditional HPC workloads for performance reasons.


For examples, using simulation data and deep neural networks,it was possible to:1. Estimate steady flow2. Accelerate particle physics 3. Infer rainfallat very low latency compared to traditional simulations.

(1) Guo, Li, “Convolutional Neural Networks for Steady Flow Approximation”(2) Paganini, Oliveira “Accelerating science with generative adversarial networks: an application to 3d

particle showers in multilayer calorimeters,”(3) S. Kim & .al, “Deep-hurricane-tracker: Tracking and forecasting extreme climate events,”

Simulation data

Physics simulations Neural Network

Main trends of “AI for HPC” AI models (neural networks) are used in several ways to

accelerate HPC simulations.

Copyright 2016 FUJITSU LIMITED

Existing methods Qualitative description

Pre- and post-processing of

simulation data

Example: Infer fine structure

parameter from coarse one.

Surrogate models:

Predict simulation results without

first principles at very low latency.

4

Corner Fillet

Mesh DisplacementNeural Network

Neural Network

Limitations of existing methods


Pre- and post-processing of simulation data:1. Do not accelerate directly HPC simulation.2. Are limited to specific data representation. Surrogate models:

1. Suffer from significant accuracy loss (typical error rates: 1.98% ~ 2.69%) 2. Can only estimate physics under very limited conditions.

Ground true (Simulation)

Surrogate model (prediction)

Difference

Results from AI Solver (3D model, experimental results) made by Fujitsu Laboratories of Europe, Ahmed Al-Jarro

Our approach: AI inside HPC simulation


We incorporate a CNN model inside the simulation runtime to accelerate the solver computation. The CNN is trained using internal simulation data. The CNN auto-tunes the solver convergence criteria for speed-up. The CNN does not interfere with first principle computation thus accuracy

is guaranteed.

Input model Simulation results

Simulation runtime enhanced with AI inference

Iterative solver

Internal data

Tuning control CNN

Theoretical Background: nonlinear analysis & iterative solver


Structural Analysis (SA) Simulations


Input: simulation model

SA simulations compute the effects of loads applied on physical structures. Designers test several loads (forces, heat, pressure) in order to check how the system would respond.

Output: simulation results

mesh

force

Simulation Displacement field

Geometric Nonlinear Analysis


When applied loads create large deformations on the structure: The relation between applied loads and displacement is nonlinear.Nonlinear response is computed by an incremental step-by-step analysis

where the load is applied gradually, in increments. This process is expressed by [K] Δr = ΔR (1) and rn+1 = rn +Δr (2) Equation (1) is solved iteratively because stiffness matrix K changes

during a given load step.

Displacement (m)

App

lied

load

(New

ton)

(ΔRE )0

(ΔRE )1

(ΔRE )2

r0 r1 r2 r3

load step 0

load step 1

load step 2

A nonlinear analysis with 3 load steps

Newton-Raphson in nonlinear analysis


In a given load step n, the Newton-Raphson algorithm solves iteratively the equation (1) by a recurrence relation:

r

R

rn

(ΔRE )n

(ΔRI )k-1

(ΔRE )n - (ΔRI )k-1

rn+1k-1 rn+1

k

Δrn k

iteration k

rn+1

Newton-Raphson method in a given load step n

then

Newton-Raphson iterations k proceed until the norm of becomes smaller than a specified tolerance value.

Linear solver in Newton-Raphson method


For each NR iteration k, a sparse system of linear equations of type must be solved:

NR method relies on iterative solvers (such as CG) to compute an approximate solution of such system. This is the main computational part of nonlinear analysis.

During the solving process, the solution vector x is iteratively corrected until convergence is obtained as:

P is the preconditioner matrix and , defined asis the residual error at iteration s. When the residual gets smaller than the specified solver tolerance, convergence occurs.

Summary of nonlinear analysis


Simulation computes the displacement values through several load steps.

Each load step relies on Newton-Raphson method to solve iteratively the equation .

Each NR iteration relies on an iterative solver to compute an approximation of the solution.

The outer loop (NR) and inner loop (solver) convergences are

controlled by theirs corresponding tolerance

parameter.

Nonlinear analysis is a nested loop.

Performance analysis


Performance analysis


We conducted a performance analysis of SA simulations based on nonlinear algorithm.

The goal is to clarify the impact of solver, preconditioner and tolerance value on simulation time and accuracy.

Our experimental environment: Intel Xeon machine (72 CPU cores)Main memory (RAM): 128 GBOperating system: CentOS 7.2 Simulation software: FrontISTR v5We use OpenMP multithreading for running simulation with either single

or multiple threads.

Performance analysis: Application models


We used 3 different FrontISTR application models: Plastic Can (a), Hyper elastic Spring (b) and Ball Grid Array (c)

(a)

(b)

(c)

Default simulation parameters for each application model:

Performance analysis: best solver/preconditioner


First we evaluate the performance of simulation using different combinations for solver method and preconditioner.

0200400600800

100012001400160018002000

CG BiCGSTAB GMRES GPBiCG

Exec

utio

n tim

e [s

ec]

SSOR_1 SSOR_2 Diag_Scal AMG ILU_0 ILU_1 ILU_2

Results for Plastic Can model,using 72 threads

fastest



0

200

400

600

800

1000

1200

1400

1600

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2000


Exec

utio

n tim

e [s

ec]


Simulation which fail to converge:

fastest

Results for Elastic model,using 72 threads

0200400600800

100012001400160018002000


Exec

utio

n tim

e [s

ec]




fastest

Results for BGA model,using 72 threads

CG-AMG provides the fastest convergence for our 3 models. Thus, we keep fixed this combination for our remaining study.

Performance analysis: different solver tolerance


Then, we evaluate the performance of simulation using different solver tolerance values using fixed model and CG-AMG.

0

10

20

30

40

50

60

70

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Exec

utio

n tim

e [s

ec]

Solver tolerance

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02

1.60E-02

1.80E-02

2.00E-02

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Mea

n A

bsol

ute

Erro

r

Solver tolerance

Results for Plastic Can model,using CG-AMG and 72 threads

Best time-error trade-off

0

50

100

150

200

250

300

350

400

450

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Exec

utio

n tim

e [s

ec]

Solver tolerance



The Mean Absolute Error (MAE) is computed from the exact solution provided by a direct solver.

Results for Elastic model,using CG-AMG and 72 threads


0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Mea

n A

bsol

ute

Erro

r

Solver tolerance



0

200

400

600

800

1000

1200

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Exec

utio

n tim

e [s

ec]

Solver tolerance

OthersNewton Raphson post-processingIterative solver (CG)Newton Raphson pre-processing

0

0.01

0.02

0.03

0.04

0.05

0.06

1.00

E-08

1.00

E-07

1.00

E-06

1.00

E-05

1.00

E-04

1.00

E-03

1.00

E-02

0.02

50.

050.

075

0.1

0.25 0.

50.

75

Mea

n A

bsol

ute

Erro

r

Solver tolerance

Performance analysis shows there is an optimal tolerance value considering simulation time and error trade-off.

Results for BGA model,using CG-AMG and 1 thread


Performance analysis: residual evolution


The shape of residual error curves are changing with respect to the current solver tolerance. Each slope is a NR iteration

while each point is a single solver iteration.

1E-08

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

Res

idua

l Err

or

Solver iteration1.00E-08 1.00E-04 0.025

0 100 200

Solver tolerance:

Performance analysis: the take away


1. Iterative solver in NR method is the main computational part of nonlinear analysis.

2. For our three models, CG-AMG is the fastest combination.

3. Low tolerance value long execution time, small error.4. Big tolerance value small execution time, big error.

5. There is an optimal tolerance value which optimize solver convergence time and accuracy of results.

6. Residual error is directly correlated to the solver tolerance and is continuously generated by the solver.

Important to accelerate

Need autotuning

Data is available

Proposal


Proposal overview


Based on performance analysis, we developed an auto-tuning of solver tolerance inside NR runtime to optimize simulation performance.

Our proposal is based on 3 core components:1. A CNN model aware of time-error tradeoff.2. A quantification of tolerance update based on Softmax probability output.3. An in-memory data transfer for minimizing the overhead of AI inference.

An application of proposal is structural design optimization:• Spatial representation• Physical properties• Condition of experiment

Computer simulation results

manufacturingstandard ?

yesprototyping

noComputer Simulation is the bottleneck of design iteration

Training data for time-error tradeoff Training data are generated while running simulations using

different solver tolerance (~70 samples per simulation).Residual values are accumulated and converted into binary image in a

periodical manner (e.g: every 2 NR iterations). According to tolerance value used during simulation, we assign a label to

each image: “increase” ([1,0]) or “decrease” ([0,1]). At inference, the CNN predicts if the current tolerance value should be

increased or decreased for performance improvement. The choice of label depends on specific policy: speed or accuracy concern

1e-08 1e-07 1e-06 1e-05 1e-04 1e-03 0.01 0.025 0.05 0.075 0.1 0.25 0.5 0.75Speed

concernAccuracy concern Increase Decrease

DecreaseIncrease

Example of labelling from BGA simulation results of page 21fastest: 0.025, best trade-off: 1e-04

CNN architecture and hyperparameters


We trained a CNN using Keras 2.2.4 (Tensorflow backend) and two Nvidia Tesla GPUs.

Conv1

FC1Conv2

Conv3

• AlexNet like architecture (Conv-Pool-Relu)• Input layer: single channel image (256x256)• Training algorithm: Adam optimizer• Loss function: Binary Cross-Entropy• Final activation layer: Softmax

FC2

• Samples: 6706• Epochs: 100• Accuracy: 98%• Val accuracy: 97%

Input data256x256 image

Two classes:* increase: p* decrease: 1-pSoftmax

System: Auto-Tuning of Solver Tolerance


We modified the conventional Newton-Raphson algorithm and incorporate an AI-based auto-tuning of solver tolerance.Our proposal does not interfere with nonlinear-level (NR) convergence,

thus consistent and correct simulation results are guaranteed.

while ( |ΔRn | ≥ NR_tolerance ) do //NR_loop

if ( period is true ) then[p, 1-p] AI_inference()adjust(solver_tolerance, [p, 1-p])

while ( rs ≥ solver_tolerance ) do //solver_loop

send residual data

• probability of increase (p)• probability of decrease (1-p) CNN

CPU

GPU

k-1

…

……

Modified simulation algorithm

Control of tolerance tuning


After inference, the returned probability values are used to update the current tolerance value, between NR iterations. This is implemented by our custom ‘adjust’ function (see page 28). The modification (increase or decrease) is proportional to the level of

confidence expressed by the probability value.

Very low probability strong decrease

low probability decrease a little bit

high probability increase a little bit

Very high probability strong increase

Incorporate AI inference at minimal overhead


AI inference is very fast due to efficient accelerators. However, we must minimize the overhead of data exchanges between simulation and neural network processes.

To transfer data at low latency, we developed a in-memory data path relying on shared memory using memory-mapped file (mmap). This approach is faster than I/O (read/write).

Experimental Evaluation


Test model for evaluation


We evaluate our proposal using the BGA model which is a real application model used in chip-carrier design.

Test model

The simulation conditions used at test phase are different from the one used in simulation which has generated training data.

We evaluate the capability of the model to

extrapolate tolerance tuning as value are out of the scope of the range

used at training.

Performance results on real application


0

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Sim

ulat

ion

time

[sec

]

[A] Baseline: simulation without AI [B] Proposal: accuracy concern [C] Proposal: speed concern0

0.0010.0020.0030.0040.0050.0060.0070.008

Mea

n A

bsol

ute

Erro

r

x1.20

x1.58

Baseline simulation use default and fixed tolerance at 1e-08 Auto-tuned simulations (B,C) use AI to update the tolerance

every 2 NR iterations, starting from default value at 1e-08.Difference between case B and C is the tuned policy: both AI

models used different training labels. Achieve 1.58x speed-up with an error rate of 0.279% (case C).

0.279%

0.00193%

Solver tolerance auto-tuning


Auto-tuned simulations (B,C) use AI to update the tolerance every 2 NR iterations, starting from default value at 1e-08.

The tolerance evolution during auto-tuned simulation: fast increase at the beginning then a smooth convergence towards optimal tolerance value around 1e-04.

1E-08

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

Solv

er to

lera

nce

NR iterations

Fast & Smooth: Adjustment with Softmax probability

Conclusion


We proposed an AI method to accelerate HPC simulation based on Newton-Raphson algorithm, a numerical method used in several engineering fields.

We incorporated an AI inference inside simulations and show how AI models can replace a part of parameter tuning.

On real application, we achieved: 1.58x faster simulation with 0.279% error rate (speed concern) 1.20x faster simulation with 0.00193% error rate (accuracy concern)

HPC simulation generate large amounts of internal data, so AI algorithms represents an opportunity to accelerate HPC applications such as design space exploration.

Thank you for listening !

For any questions, please send email to:[email protected]


iwapt 2020 workshop: acceleration of structural analysis ... · can only estimate physics under...

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