1. A Dynamic System Analysis of Simultaneous Recurrent Neural Network
Dr. Gursel Serpen and Mr. Yifeng Xu
Electrical Engineering and Computer Science Department
University of Toledo
Toledo, Ohio, USA

2. Introduction Motivation for research
Simultaneous Recurrent Neural Network (SRN)
Local Stability of SRN
Global stability of SRN
FOR MORE INFO...
Werbos, Paul  many recent articles – some unpublished! Serpen et al., The Simultaneous Recurrent Neural Network for Addressing the Scaling Problem in Static Optimization, Neural Systems, Vol. 11, No. 5, 2001, pp

3. Motivation
Simultaneous Recurrent Neural (SRN) network has been shown to have the potential to address large-scale static optimization problems: located relatively high quality solutions.
SRN is trainable: implies that it can learn from prior search attempts (A Hopfield net cannot do this!)
Computational complexity for SRN simulations is much less again compared to the Hopfield and its derivatives.

4. Research Goals
Understand the stability and convergence properties of the SRN dynamics.
Establish stable dynamics following initialization.
Establish stability while training the SRN with a fixed-point algorithm (recurrent backprop).
Apply the SRN to (large-scale) static optimization problems.

5. Research Goals - detailed
Initialization of weights to guarantee existence of at least one or more fixed points in the state space of the SRN dynamics.
Stability as weight matrices are being modified while learning with a fixed-point algorithm, i.e., recurrent backpropagation.
Assessing the computational power of the SRN as a static optimizer for large-scale problem instances.

6. Simultaneous Recurrent Neural Network
Topology
Feedforward Network
Input
Output
Typical propagation delay exists on the feedback path as dictated by physical constraints!
Delayed Feedback

7. SRN Detailed Structure – 3 Layersx y W U z V
Propagation Delay
outputs
z = f (Uy) and y = f (Wx+Vz)

8. Analysis of SRN Dynamics
Local stability analysis
SRN dynamics linearized at hypercube corners (that are also equilibrium points)
(Local) stability conditions for hypercube corner equilibrium points
A theorem...its proof (elsewhere)

9. Equilibrium Points in State Space of SRN Dynamics
Output layer
Hidden layer
SRN dynamics in matrix form in terms of z:

10. Equilibrium Points of SRN Dynamics
Hypercube Corners
zk = 1 or zk = 0
for k = 1, 2, …, K
Points interior, on the surface, and on the edges of the hypercube

11. Stability of SRN Dynamics - Linearized
Eigenvalues of SRN dynamics linearized at hypercube corners (equilibrium points):
for k = 1, 2, … , K.

12. Stability of SRN Dynamics - linearized
Set of inequalities derived from eigenvalues through stability condition for equilibrium points.
= 0
for
for
= 1

13. A Stability Theorem for SRN Dynamics
For any given hypercube corner in the state space of SRN dynamics,
which is configured for static combinatorial optimization, i.e. with high-gain neurons, one hidden layer, one output layer, no external input, and operating in associative memory mode,
it is possible to define the weight matrices U and V to establish that hypercube corner as a stable equilibrium point in a local sense.

14. Simulation Study Fixed points (hypercube corners) 48 Case 1 - 2X4 SRN
Methodology: Create 75 instances of U & V then, observe set of stable
equilibrium points thru multiple random initializations.
Fixed points (hypercube corners) 48
Limit cycles - cycle length of 2 12
Stable non-hypercube points 19
* all 16 hypercube corners appeared as
stable equilibrium points for 75 instances
of weight matrices.
Case 2 - 2X100 SRN
Case 3 - 5X10000 SRN
Case 4 - 5X25000 SRN

15. Global Stability of SRN?
The SRN paradigm is closely related to a number of globally asymptotically stable recurrent neural network algorithms!
BAM is a special case of SRN.
ART 1 & 2 cores are similar topologies.
Significant simulation-based empirical studies conducted by authors suggest global stability.
However, no Liapunov function yet!

16. Conclusions
A theorem demonstrates that there exist real forward and backward weight matrices for SRN dynamics, which will induce stability of any given hypercube corner as an equilibrium point.
This theoretical finding was also validated by extensive simulation-based empirical studies, some of which was reported in a recent journal article:
Serpen et al., The Simultaneous Recurrent Neural Network for Addressing the Scaling Problem in Static Optimization, Neural Systems, Vol. 11, No. 5, 2001, pp

17. Thank You !
Questions ?
This research has been funded in part by the US National Science Foundation grant ECS