Neural Inverse Space Mapping EM-Optimization J.W. Bandler, M.A. Ismail, J.E. Rayas-Sánchez and Q.J. Zhang Simulation Optimization Systems Research Laboratory McMaster University
Bandler Corporation, www.bandler.com
[email protected]
presented at 2001 IEEE MTT-S International Microwave Symposium, Phoenix, AZ, May 23, 2001
Simulation Optimization Systems Research Laboratory McMaster University
Neural Inverse Space Mapping EM-Optimization (NISM) outline ANN approaches for microwave design NISM optimization statistical parameter extraction inverse neuromapping the NISM step vs. the quasi-Newton step examples conclusions
Simulation Optimization Systems Research Laboratory McMaster University
Conventional ANN-Based Optimization of Microwave Circuits (Zaabab, Zhang and Nakhla, 1995, Gupta et al., 1997, Burrascano and Mongiardo, 1998) step 1
ω xf
step 2
fine model
ANN w
Rf
≈ Rf
ω xf
ANN
≈ R*
many fine model simulations are usually needed solutions predicted outside the training region are unreliable
Simulation Optimization Systems Research Laboratory McMaster University
ANN-Based Microwave Optimization Exploiting Available Knowledge EM-ANN approach (Gupta et al., 1999)
ω
fine model
xf
coarse model
neural space mapping approach (Bandler et al., 2000)
ω
Rf ∆R
fine model
xf
ωc
Rc
neurox mapping c
Rf
coarse model
w ANN w
≈ ∆R NSM optimization requires 2n+1 upfront fine simulations
Rc ≈ R f
Simulation Optimization Systems Research Laboratory McMaster University
Objectives develop an aggressive ANN-based space mapping optimization avoid multipoint parameter extraction and frequency mappings
Simulation Optimization Systems Research Laboratory McMaster University
Neural Inverse Space Mapping Optimization Start i=1 COARSE OPTIMIZATION: find the optimal coarse model solution xc* that generates the desired response
PARAMETER EXTRACTION: Find xc(i) such that
INVERSE NEUROMAPPING: Find the simplest neural network N such that
Rcs(xc(i)) ≈ Rf s(xf (i))
xf (l) ≈ N (xc(l))
xf(i) = xc* Calculate the fine responses Rf s(xf (i))
l = 1,..., i i=i+1
no
xf(i) ≈ xf(i+1) yes
End
xf (i+1) = N(xc*)
Simulation Optimization Systems Research Laboratory McMaster University
Statistical Parameter Extraction
(1)
(2)
xc (i ) = arg min U PE ( xc ) xc U PE ( xc ) = e ( xc ) e ( xc ) = R fs ( x f
(i )
∆max =
2 2
) − Rcs ( xc )
(3)
δ PE ∇U PE ( xc* )
∆ xk = ∆max (2randk − 1) k = 1K n
∞
begin * solve (1) using xc as starting point
(i )
while e ( xc )
∞
> ε PE
calculate ∆x using (2) and (3) solve (1) using xc* + ∆x as starting point end
Simulation Optimization Systems Research Laboratory McMaster University
Inverse Neuromapping
(4) w * = arg min U N ( w ) w
U N ( w ) = [ L el
el = x f
(l )
T
L]
T
2
begin
2
solve (4) using a 2LP
(l )
− N ( xc , w )
h=n
l = 1,K, i
while UN (w*) > εL solve (4) using a 3LP
ANN (2LP or 3LP) xc
N
w
xf
h=h+1 end
Simulation Optimization Systems Research Laboratory McMaster University
Nature of the NISM Step xf (i+1) = N(xc*) evaluates the current ANN at the optimal coarse solution is equivalent to a quasi-Newton step departs from a quasi-Newton step as the nonlinearity needed in the inverse mapping increases does not use classical updating formulas to approximate the Jacobian inverse
Simulation Optimization Systems Research Laboratory McMaster University
Termination Condition for NISM Optimization x f (i +1) − x f (i ) ≤ ε end (ε end + x f (i ) ) 2
2
∨
i = 3n
Simulation Optimization Systems Research Laboratory McMaster University
HTS Quarter-Wave Parallel Coupled-Line Microstrip Filter (Westinghouse, 1993)
L0 L1 L2 L3
S1
we take L0 = 50 mil, H = 20 mil, W = 7 mil, εr = 23.425, loss tangent = 3×10−5; the metalization is considered lossless
S2 S3
L2 L0
the design parameters are
L1 W εr
H
xf = [L1 L2 L3 S1 S2 S3] T
specifications |S21| ≥ 0.95 for 4.008 GHz ≤ ω ≤ 4.058 GHz |S21| ≤ 0.05 for ω ≤ 3.967 GHz and ω ≥ 4.099 GHz
S2
S1
Simulation Optimization Systems Research Laboratory McMaster University
HTS Microstrip Filter: Fine and Coarse Models fine model:
coarse model:
Sonnet’s em with high resolution grid
OSA90/hope built-in models of open circuits, microstrip lines and coupled microstrip lines
2
1
Simulation Optimization Systems Research Laboratory McMaster University
NISM Optimization of the HTS Filter responses using em (o) and OSA90/hope (−) at the starting point 1
|S21|
0.8 0.6 0.4 0.2 0
3.95
4
4.05
frequency (GHz)
4.1
4.15
Simulation Optimization Systems Research Laboratory McMaster University
NISM Optimization of the HTS Filter (continued) responses using OSA90/hope (−) at xc* and em (o) at the NISM solution (after 3 NISM iterations) 1 0.8
|S21|
0.6 0.4 0.2 0
3.95
4
4.05
frequency (GHz)
4.1
4.15
Simulation Optimization Systems Research Laboratory McMaster University
NISM vs. NSM Optimization
after NISM (3 fine simulations)
after NSM (14 fine simulations) (Bandler et al., 2000)
1
1
0.95
0.95
|S21|
|S21|
HTS filter optimal responses in the passband
0.9
0.85
0.8
0.9
0.85
3.98
4
4.02
4.04
frequency (GHz)
4.06
4.08
0.8
3.98
4
4.02
4.04
frequency (GHz)
4.06
4.08
Simulation Optimization Systems Research Laboratory McMaster University
NISM vs. Trust Region Aggressive Space Mapping (TRASM) Exploiting Surrogates fine model minimax objective function after TRASM Exploiting Surrogates (8 fine simulations) (Bakr et al., 2000)
after NISM (3 fine simulations) 1.0
fine model minimax objective function
fine model minimax objective function
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
2
iteration
3
0.8
0.6
0.4
0.2
0
1
2
3
4
5
iteration
6
7
8
Simulation Optimization Systems Research Laboratory McMaster University
Conclusions we propose Neural Inverse Space Mapping (NISM) optimization up-front EM simulations, multipoint parameter extraction or frequency mapping are not required a statistical procedure overcomes poor local minima during parameter extraction an ANN approximates the inverse of the mapping the next iterate is obtained from evaluating the ANN at the optimal coarse solution this is a quasi-Newton step NISM optimization exhibits superior performance to NSM optimization and Trust Region Aggressive Space Mapping Exploiting Surrogates
