There are a number of ways to create surrogate models of microwave and radio- frequency (RF) devices and structures. They can be classified into two groups:
functional and physical surrogates. Functional models are constructed from sam- pled high-fidelity model data using suitable function approximation techniques.
Physical surrogates exploit fast but limited-accuracy models that are physically re- lated to the original structure under consideration.
Functional surrogate models can be created using various function approxima- tion techniques including low-order polynomials [40], radial basis functions [40], kriging [31], fuzzy systems [41], support-vector regression [42], [43], and neural networks [44]-[46], the last one probably being the most popular and successful approach in this group. Approximation models are very fast, unfortunately, to achieve good modeling accuracy, a large amount of training data obtained through massive EM simulations is necessary. Moreover, the number of data pairs neces- sary to ensure sufficient accuracy grows exponentially with the number of the de- sign variables. Practical models based on function approximation techniques may need hundreds or even thousands of EM simulations in order to ensure reasonable
accuracy. This is justified in the case of library models created for multiple usage but not so much in the case of ad hoc surrogates created for specific tasks such as parametric optimization, yield-driven design, and/or statistical analysis at a given (e.g., optimal) design.
Physical surrogates are based on underlying physically-based low-fidelity mod- els of the structure of interest (denoted here as Rc). Physically-based models de- scribe the same physical phenomena as the high-fidelity model, however, in a simplified manner. In microwave engineering, the high-fidelity model describes behavior of the system in terms of the distributions of the electric and magnetic fields within (and, sometimes in its surrounding) that are calculated by solving the corresponding set of Maxwell equations [47]. Furthermore, the system perform- ance is expressed through certain characteristics related to its input/output ports (such as so-called S-parameters [47]). All of these are obtained as a result of high- resolution electromagnetic simulation where the structure under consideration is finely discretized. In this context, the physically-based low-fidelity model of the microwave device can be obtained through:
• Analytical description of the structure using theory-based or semi-empirical formulas,
• Different level of physical description of the system. The typical example in microwave engineering is equivalent circuit [7], where the device of interest is represented using lumped components (inductors, capacitors, microstrip line models, etc.) with the operation of the circuit described directly by im- pedances, voltages and currents; electromagnetic fields are not directly considered,
• Low-fidelity electromagnetic simulation. This approach allows us to use the same EM solver to evaluate both the high- and low-fidelity models, however, the latter is using much coarser simulation mesh which results in degraded accuracy but much shorter simulation time.
The three groups of models have different characteristics. While analytical and equivalent-circuit models are computationally cheap, they may lack accuracy and they are typically not available for structures such as antennas and substrate- integrated circuits. On the other hand, coarsely-discretized EM models are available for any device. They are typically accurate, however, relatively expensive. The cost is a major bottleneck in adopting coarsely-discretized EM models to surrogate-based optimization in microwave engineering. One workaround is to build a function- approximation model using coarse-discretization EM-simulation data (using, e.g., kriging [31]). This, however, requires dense sampling of the design space, and should only be done locally to avoid excessive CPU cost. Table 8.1 summarizes the characteristics of the low-fidelity models available in microwave engineering. A common feature of physically-based low-fidelity models is that the amount of high-fidelity model data necessary to build a reliable surrogate model is much smaller than in case of functional surrogates [48].
Table 8.1 Physically-based low-fidelity models in microwave engineering
Model Type CPU Cost Accuracy Availability
Analytical Very cheap Low Rather limited
Equivalent circuit Cheap Decent Limited (mostly filters) Coarsely-discretized
EM simulation Expensive Good to very good
Generic: available for all structures
Consider an example microstrip bandpass filter [48] shown in Fig. 8.3(a). The high-fidelity filter model is simulated using EM solver FEKO [18]. The low- fidelity model is an equivalent circuit implemented in Agilent ADS [49]
(Fig. 8.3(b)). Figure 8.4(a) shows the responses (here, the modulus of transmission coefficient, |S21|, versus frequency) of both models at certain reference design x(0). While having similar shape, the responses are severely misaligned. Figure 8.4(b) shows the responses of the high-fidelity model and the surrogate constructed using the low-fidelity model and space mapping [48]. The surrogate is build using a sin- gle training point – high-fidelity model data at x(0) – and exhibits very good matching with the high-fidelity model at x(0). Figure 8.4(c) shows the high-fidelity and surrogate model response at a different design: the good alignment between the models is still maintained. This comes from the fact that the physically-based low-fidelity model has similar properties to the high-fidelity one and local model alignment usually results in relatively good global matching.
0.6mm
Output
Input g
L3
L2
L1
L4
5mm
5mm
(a)
Term 1 Z=50 Ohm Term 2 Z=50 Ohm
MTEE Tee2 W1=0.6 mm W2=0.6 mm W3=0.6 mm MLIN
TL2 W=0.6 mm L=L2 mm
MLIN TL3 W=0.6 mm L=L3 mm MLOC
TL1 W=0.6 mm L=L1 mm
MGAP Gap1 W=0.6 mm S=g mm MTEE
Tee1 W1=0.6 mm W2=0.6 mm W3=0.6 mm
MLIN TL5 W=0.6 mm L=5 mm
MLIN TL6 W=0.6 mm L=5 mm
MLOC TL4 W=0.6 mm L=L4 mm
(b)
Fig. 8.3 Microstrip bandpass filter [48]: (a) geometry, (b) low-fidelity circuit model.
4.5 5 5.5 -40
-20 0
Frequency [GHz]
|S21|
(a)
4.5 5 5.5
-40 -20 0
Frequency [GHz]
|S21|
(b)
4.5 5 5.5
-40 -20 0
Frequency [GHz]
|S 21|
(c)
Fig. 8.4 Microstrip bandpass filter [48]: (a) high- (—) and low-fidelity (- - -) model re- sponse at the reference design x(0); (b) responses of the high-fidelity model (—) and surro- gate model constructed from the low-fidelity model using space mapping (- - -) at x(0); (c) responses of the high-fidelity model (—) and the surrogate (- - -) at another design x. The surrogate model was constructed using a single high-fidelity model response (at x(0)) but a good matching between the models is preserved even away from the reference design, which is due to the fact that the low-fidelity model is physically based.