Emulated Digital CNN-UM Implementation of a

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Emulated Digital CNN-UM Implementation of a
3-dimensional Ocean Model on FPGAs

• Zoltán Nagy, Péter Szolgay

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Introduction

• Cellular Neural/Nonlinear Networks Universal
  Machine (CNN-UM)
• Ocean modeling
• Results
• Conclusions

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Cellular Neural/Nonlinear Networks (CNN)

• 2 or N dimensional grid
• Locally connected
• Analog processing elements
• State value is continuous in time

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Structure of a CNN cell

• uij input
• xij state
• yij output

• zij constant bias
• Aij,kl feedback template
• Bij,kl feed-forward template

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CNN-UM implementations

• Software simulation
• Easy to implement
• Slow, even if using processor specific
  instructions
• Emulated digital VLSI
• Specialized digital architecture
• Selectable computing precision (Castle
  architecture 1, 6, 12 bit)
• Orders faster than the software simulation
• Long design time
• Analog VLSI
• Huge computing power (TeraOP/s)
• Low accuracy (7-8 bit)
• Noise and temperature sensitivity

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Structure of the Falcon emulated digital CNN-UM

• Mixer
• Contains cell values for the next updates
• Memory unit
• Contains a belt of the cell array
• Template memory
• Arithmetic unit
• Processors can be connected on a grid
• Linear speedup

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Structure of the arithmetic unit

• Cell update in row wise order
• Cycle time depends on template size
• Fully pipelined

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Configurable parameters

• State, template and constant width between 2 to
  64 bits
• Number of templates
• Size of the templates
• Width of the cell array slice
• Number of layers
• Number and arrangement of the processor cores

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Example Solution of a simple PDE on CNN

• The Wave equation

• Spatial discretization
• 2 layer CNN

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Ocean models

• Barotropic model
• Baroclinic models
• z-coordinate model
• s-coordinate model
• isopycnal

• Fine resolution models
• Real-time forecast
• Fishing industry
• Search and rescue
• Coarse resolution models
• Long term predictions
• Climate modeling

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The Princeton Ocean Model (POM)

• Sigma coordinate model
• Vertical coordinate is scaled on the water column
  depth
• Second moment turbulence closure sub-model
• Provides vertical mixing coefficients

• Solution technique Mode splitting
• Internal mode (3D)
• Vertical structure equations
• Implicit solution
• External mode (2D)
• Vertically integrated equations
• Explicit solution (Leapfrog method)

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Governing equations of the external (2D) mode

• H depth of the ocean
• g gravitational acceleration
• tw, tb wind and bottom stress
• A lateral viscosity

• ux, uy mass transport
• ? free surface elevation
• O angular rotation of the Earth
• T latitude

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Solution on CNN

• Spatial discretization on a uniform grid
• 3-layer CNN structure
• Non-linear template required for advection term
• Cannot be solved on analog VLSI CNN chips
• Solvable on the modified Falcon architecture
• Support of non-linearity
• Specialized cell model

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The modified arithmetic unit of the Falcon
architecture
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Implementation on FPGA

• Complicated arithmetic unit
• Fixed-point number representation
• Configurable precision
• High level hardware description language
  required(e.g. Handel-C)

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Performance
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The Seamount problem
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Results after 72 hours
Circulation pattern
Elevation
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Error of the solution
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Error of the solution
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Memory requirements of the internal (3D) equations

• Extended memory hierarchy
• New level stores 3 cross sectional slices from
  the 3D array
• Large memory required (e.g. 512x512x64 sized
  grid, 3x512x64 elements per state variable)
• Cannot be stored on-chip
• Off-chip storage requires huge I/O bandwidth
• Processor array should be used
• The 3D array is divided between the processors
• Optimal data set for on chip storage 2048
  elements per cross sectional slice (512x32x64
  sized grid per processor)
• Each processor located on a separate FPGA

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Solution of the internal (3D) equations

• Implicit solution
• Fixed-point solution
• Requires large precision to avoid rounding errors
• Seems to be impractical
• Floating-point solution
• Requires large area (especially add/sub)
• Explicit solution
• Smaller timestep
• Simpler arithmetic unit

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Conclusions

• Ocean modeling using emulated digital CNN is very
  promising
• Moderate precision is required in 2D mode
• 1 accuracy using 24 bits
• Expected speedup (compared to an Athlon64 2GHz
  microprocessor)
• 80 times on our RC200 prototyping board
• 3700 times on the largest available FPGA