The voids of computing
3
Arithmetic heavy
Data heavyNonlinear dynamics
Floating point arithmetic
Boolean logic
Weather prediction
Gene sequencing(+ other NP-class)
Real-time regex matching
Image processing
Present digital computers
Major limitations of digital computers:
• End of Moore’s law
• Von Neumann architecture• Boltzmann tyranny• Boolean logic• Turing limit
The post-Moore’s law physics-driven computer
4
Materials discovery
(new behaviors)
Functional devices
(new functions)
Interacting devices
(new algorithms)
Outline of the talk
1. Neuronic devices – information processing or communication
2. Synaptic devices – information storage
3. The fundamental connection between synaptic and neuronic devices
4. Synaptic + neuronic NP-hard accelerators
5
Mott Memristors
7
Volatile resistive switching can emulate neuron-like spiking!
NbO2 Mott insulator
Nature Materials 12, 114 (2013)
T
R
Devices and static behavior
8
NbO2 devices
Diameter ~70 nm
Local activity
?
Kumar et al., Nature Communications, 8, 658 (2017)
The cause of NDR in NbO2
9Kumar, Nature Comms. 8, 658 (2017)
0.0 0.5 1.00.0
0.5
1.0
1.5
Cu
rre
nt (m
A)
Voltage (V)
NDR-1
NDR-2
Crystalline
Thermal anomaly in NbO2
10
𝜎𝑡ℎ𝑀𝑒𝑡𝑎𝑙 < 𝜎𝑡ℎ
𝐼𝑛𝑠𝑢𝑙𝑎𝑡𝑜𝑟
Counter Wiedemann Franz postulate
Lee et al., Science 355, 371 (2017)
Kumar et al., Nature Communications 8, 658 (2017)
Model for the electrical behavior
11
𝑖𝑚 = 𝛼e−Ea2kBT
kBT
β 𝑣𝑚
2
1 +β 𝑣𝑚/t
kBT− 1 e
β 𝑣𝑚/t
kBT +𝛼e−EakBT
2t𝑣𝑚 … Equation (1)
dT
dt=𝑖𝑚𝑣𝑚
Cth−T−Tamb
CthRth T… Equation (2)
Rth T = 1.4 × 106 for T ≤ TC2 × 106 for T > TC
… Equation (3)
Modified 3D Poole-
Frenkel Equation
Temperature
dynamics
Mott transition in Rth
Kumar et al., Nature, 548, 318 (2017)
Kumar et al., Nature Communications, 8, 658 (2017)
Temperature-controlled instability (Chua Corsage)
12
Kumar et al, Nature Communications, (2018)
Kumar et al., Nature Communications, 8, 658 (2017)
Mannan, Int. J. Bifurcation Chaos, 27, 1730011 (2017)
NDR + Local activity
Mott transition + Local activity(Chua Corsage)
New type of current-voltage behavior!
Theorem: Chua corsage leads to chaos in dynamics
Modeling the dynamical behavior
13
𝑑𝑇
𝑑𝑡=𝑣𝑚2
𝑅 𝑇 𝐶𝑡ℎ−𝑇−𝑇𝑎𝑚𝑏
𝑅𝑡ℎ𝐶𝑡ℎ
𝑑𝑣𝑚
𝑑𝑡=𝑣𝑖𝑛
𝑅𝑠𝐶− 𝑣𝑚
1
𝑅 𝑇 𝐶+1
𝑅𝑆𝐶
For chaos, where is the third state variable or oscillating force?
η T = 𝑇𝑘𝐵𝐶𝑡ℎ
12 4π
𝑅𝑡ℎ𝐶𝑡ℎcos2π𝑡
𝑅𝑡ℎ𝐶𝑡ℎ
What does it take to produce chaos?
1. Local activity
2. A minimum of three state variables; OR two state variables + a coupled dynamic signal
Kumar et al., Nature, 548, 318 (2017)
0 5 10
0.0
0.5
1.0
Cu
rre
nt
(mA
)
Time (s)
+η(𝑇)
Fluctuation-dissipation theorem of local activity
14
Local Activity:
The biology-based theory that now allows us to predict dynamics of electronic devices using their static behavior.
Local activity leads to chaos
Kumar et al., Nature Communications, (2018)
15
What is chaos?
– Extreme sensitivity to variations in initial conditions
– In between perfectly ordered and completely random behavior
– Very difficult to predict (but not impossible)
– Why is edge of chaos behavior favored in computing systems?
– Units communicate/cooperate with to compute
– But no global synchronization
– Applies to neurons in a brain
Thermal fluctuations
16
η T = 𝑇𝑘𝐵𝐶𝑡ℎ
12 4π
𝑅𝑡ℎ𝐶𝑡ℎcos2π𝑡
𝑅𝑡ℎ𝐶𝑡ℎ
0 5 10
0.0
0.5
1.0
Curr
ent (m
A)
Time (s)0 5 10
Time (s)
a
b cNo fluctuations With fluctuations
400 800 1200-2
0
2
dT
/dt (K
/ps)
Tave
(K)
0
2
dTstatic
/dt
T
|T
| (K
)
Simulations
0 5 10
0.0
0.5
1.0
Cu
rre
nt
(mA
)
Time (s)
dWith fluctuations, large device
Double pendulum with a large driving force!
Takes hundreds of transistors to emulate!
Kumar et al., Nature, 548, 318 (2017)
Chaotic oscillations
17
0 10 20Time (s)
1.02 V 1.03 V 1.04 V
0.95 1.00 1.05 1.100.01
0.1
1
(
s-1,x1
07)
Voltage (V)
b
c
0 10 200
150
300
Curr
en
t (
A)
Time (s)0 10 20
Time (s)
a
Are temperature fluctuations the only route to chaos?
Kumar et al., Nature, 548, 318 (2017)
Controlled chaos in a single
electronic device!
Nonlinearity and bifurcations
18A different route to chaos in nonlinear devices.
Bifurcations due to instabilities The missing pieces of device models:
1. Minimization of internal energy or ΔH
2. Spontaneous symmetry breaking during nonlinearity
3. Amplification and coupling of ambient thermal fluctuations in nanoscale devices
Kumar et al., Nature Communications, (2018)
Kumar et al., Advanced Materials, 28, 2772 (2016)
Ridley, Proc. Phys. Soc., 82, 954 (1963)
)𝑗U. 𝐴 = 𝑗L. 𝐴 − 𝑥. 𝐴 + 𝑗H. (𝑥. 𝐴
Ion migration in Hf and Ta oxides
20O K-edge
Pristine/Virgin
HfOx+Hf device
Material stack:
Pt/HfOx/Hf/Pt
I570eV I570eV
Operated
Fundamental requirement for S-NDR non-volatility
23
𝑖 = 𝐺𝑣
𝐺 = 𝜓𝑇𝜉d𝑇
d𝑡=𝑖𝑣
𝐶𝑡ℎ−𝑇 − 𝑇amb𝐶𝑡ℎ𝑅th
𝑇 = 𝑇𝑎𝑚𝑏 + 𝑅𝑡ℎ𝑣2𝜓𝑇𝜉
d𝑇
d𝑖= 𝑅th𝑣
2𝜓𝜉𝑇𝜉−1d𝑇
d𝑖+ 2𝑅th𝑣𝜓𝑇
𝜉d𝑣
d𝑖
𝑅th𝑣2𝜓𝜉𝑇𝜉−1 = 1
𝑇 = 𝑇NDR =𝜉 𝑇amb𝜉 − 1
ξ >1
Steady state: d𝑇
d𝑡= 0
Estimate only this material parameter and one can predict most of the nonlinear and information storage behaviors.
The traveling salesman problem
25
Objective:
Find the shortest path
Constraints:
1. Visit every city once2. Visit every city no more than once3. Do not visit more than one city in a given stop
“Hard” problems
26
It is non-deterministic polynomial (NP) complete.
Other NP-complete/hard problems:
Gene sequencing/traveling salesman
Sudoku
Vehicle routing
Open shop scheduling
Bandwidth allocation
NFL scheduling256 games20k variables50k constraints
Takes ~3 months on a 1000-core system to solve!
1M100
10
10M
#co
nstr
ain
ts#vars
Hopfield network
27
E = −1
2 𝑖 𝑗𝑠𝑖,𝑗
𝑘 𝑙𝑠𝑘,𝑙𝑤 )𝑖,𝑗 ,(𝑘,𝑙 +
𝑖 𝑗𝑠𝑖,𝑗θ
Synapses + neurons
Energy function:
Kumar et al., Nature, 548, 318 (2017)
US Patent App. 15/141,410
Recursive feedback neural network
An incredibly compact transistorlessall-analogue implementation of a
Hopfield network
Example of one solution with chaos
28
The traveling salesman problem
Kumar et al., Nature, 548, 318 (2017)
US Patent App. 15/141,410 (2017)
1 mm
Statistics of many solutions with and without chaos
29
Literally annealing the system into its solution!
Kumar et al., Nature, 548, 318 (2017)
We only want better solutions quickly.
High precision prohibitive slow downs
Room temperature analogue of quantum
adiabatic annealing
Mem-HNN outperforms digital hardware and quantum annealing
10,000x better Solutions/sec/Watt !!
Benchmarking on a suite of 60 node max-cut problems
Analog Digital Quantum
Mem-HNN GPU D-wave 2000Q
Clock frequency 1 GHz 1.5 GHz
Time-to-solution 0.3 µs 10 µs 104 s
Power 792 mW <250 W 25,000 W
Solutions/s/Watts 4.6 x 106 >400 4 x 10−9
Major Caveat: Circuit level analysis only – no scalable architecture analyzed yet
Reason for optimism: NP-hard problems are compute-intensive, not data-intensive
arXiv:1903.11194: Kumar et. al. Harnessing Intrinsic Noise in Memristor Hopfield Neural Networks
for Combinatorial Optimization
Summary
–Analog, brain-like computing can outperform any digital alternative in solving certain classes of intractable problems
–But we need new device physics to do that
– Controlled chaos
– Nano-size coupling with ambient thermal fluctuations
– Enthalpy minimization and symmetry breaking
– Super-linearity and non-volatility (the connection between neurons and synapses)
31
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