Solving Differential Equations Using Neural Networks

7/29/2019 Solving Differential Equations Using Neural Networks

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(n+1,1) (n+1,j) (n+1,n+1)

(1,1)

(i-1,j)

(i,j)

(i+1,j)

(i,j+1)(i,j-1)

(n+1,0)

(i,0)

(1,0)

(0,0)(0,1)

(i,n+1)

(1,n+1)

(0,n+1)(0,j)


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(i,j+1)(i,j)(i,j-1)

(i+1,j)

(i-1,j)

1 1

1

1

-4


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ewi,j

n

s

i,j

i,j

i,j

u

u u

u

i,j+n

i,j-s

i,j

i,j

i-w i+ei,j i,j,j ,ji,ju


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d2

u

dx2

d2

u

2dy0=+

in the interior

Heated side

u(x,0)=100

u(0,y)=100(1-y)

Cooled side

Heated side

u(1,y)=100

u(x,1)=0

Partially heated side

x

0

0

y

1

1


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Heated side

u(i,0)=100

u(5,j)=100

Heated side

u(i,5)=0

Cooled side

u(0,j)=20(5-j)

Partially heated side

0

0

ih

jh

(i,j)

1h

2h

4h

5h

3h

1h 2h 3h 4h 5h


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5V

5V

5V

5V

5V 5V 5V 5V

4V

0V

0V

0V

0V

3V 2V 1V

(1,1)

(2,1)

(3,1)

(4,1) (4,2) (4,3) (4,4)

(3,4)(3,3)(3,2)

(2,2) (2,3) (2,4)

(1,2) (1,3) (1,4)


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R

RR

vi,j

vi,j

-

+

R

- -

++

u v

v

v

v

v

vi+1,j

i,j i,j

i,j+1i,j

i,j-1

i-1,jI

i,j


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Local Memory

WeightCell

LocalMemory

Weight

LocalM

emory

Weight

LocalMem

ory

Weight

Loca

lMemory

Weig

ht

1

2

3

4

n

rnr1 r2

r1 r2 rn

0

Select lines for each resistance value.

Out

In


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r1 rn0

Two Phase

Clock

Two Phase

Clock

Two Phase

Clock

Shift In Shift Out


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Local Memory

(Shift Register)-

+

-

+

Shift OutsShift Ins


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Clk1

Clk1

Clk1

Clk1

4,1

Bottom Shift In

Top Shift In Shift Out to next cells Shift In

3,1

2,1

1,1

4,4

1,4


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Element

Processing

Local Memory

Element

Processing

Local Memory

Element

Processing

Local Memory

Element

Processing

Local Memory

Element

Processing

Local Memory

Global

Memory


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4,1

3,1

2,1

1,1

4,4

1,4

Row Output

On/Off

Selects


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Vss

VddResistor

Select

Shift In Shift Out

Clock


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Vin

-+ Vout

Vdd

Vss

Vbias

Vss

M2 M1

M5M7

M6

M4 M3


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out

in

Vgg

Vdd!

Vss


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in

ab

GND

Vdd

c

1

out

432 5


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Vss

Vdd


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Output of

Node (3,2)

in Volts

Row 1

Row 2

Row 3

Row 4

Average value between lines


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3 2.2 1.6 1

3 3 3 3

3

3

3

3

1

1

1

1

3 2.2 1.6 1

3

3

3

3

3

1

1

1

1

2.22.22.2

3 2.2 1.6 1

3

3 1

1

1

1

2.22.22.2

1.6

1.6

1.6

3 2.2 1.6 1

3 1

1

1

1

2.22.22.2

1.6

2.2

1

1.83

Voltage Set 1 Voltage Set 2

Voltage Set 3 Voltage Set 4


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in

GND

Vdd

out

clk clkb


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GND

Vdd

clk

clkb8 7 6 5 4 3 2 1

in


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S0

S0B

S1

S1B

Vdd

GND

OUT0 OUT1 OUT3OUT2

S


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5

4

3

2

37

36

1

40

39

38

3534333231302928272625

24

23

22

21

20

19

18

17

16

15 14 13 12 11 10 9 78 6


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Solving Differential Equations Using Neural Networks

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