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Symbolic regression via genetic programming (hereafter, referred to simply as symbolic regression) has proven to be a very important tool for industrial empirical modeling (Kotanchek et al., 2003). Two of the primary problems with industrial use of symbolic regression are (1) the relatively large computational demands in comparison with other nonlinear empirical modeling techniques such as neural networks and (2) the difficulty in making the trade-off between expression accuracy and complexity. The latter issue is significant since, in general, we prefer parsimonious (simple) expressions with the expectation that they are more robust with respect to changes over time in the underlying system or extrapolation outside the range of the data used as the reference in evolving the symbolic regression.In this chapter, we present a genetic programming variant, ParetoGP, which exploits the Pareto front to dramatically speed the symbolic regression solution evolution as well as explicitly exploit the complexity-performance trade-off. In addition to the improvement in evolution efficiency, the Pareto front perspective allows the user to choose appropriate models for further analysis or deployment. The Pareto front avoids the need to apriori specify a trade-off between competing objectives (e.g. complexity and performance) by identifying the curve (or surface or hyper-surface) which characterizes, for example, the best performance for a given expression complexity.

Polymer distribution in rigid PU foams is a complex, position de pendent parameter which can be described in terms of: 1) the overall density and density distribution, 2) the average cellsize and cellsize distribution, 3) cell elonga tion and orientation and 4) the material distribution between struts and windows within the individual cells. Considering that a typical foam contains only 2-3 volume % of polymer, it is not surprising that any variation in the way this material is distributed will have a large impact on most of the physical properties of the foam. In the present paper, we evaluate what effects can be expected from changes in cellsize and/or density on the polymer distribution in foams and consequently on foam physical properties like compressive strength, initial thermal conductivity, and thermal conductivity aging. These results are obtained using a blend of modelling techniques and experimental data. Generally speaking, in PU foams a reduction in the average cellsize will also cause an increase in the amount of material which is present in the cell windows versus the cell struts. It is observed that in most systems, the window thickness remains relatively unaffected by cell size reduction. Also, for a given density, there exists a cell size at which the cells will no longer be closed simply because there is not enough material available to ensure film stability. Since the cell windows represent most of the barrier to diffusion, the rate of aging will be reduced dramatically by this shift in the material distribution as long as the cells are still closed. A reduction in cellsize will, in general, also cause a reduction in the initial thermal conductivity, mostly because of the reduction in the thermal radiation component to foam thermal conductivity. There exists a region, however, where this effect is reversed. Further reductions in cellsize will cause an increase in thermal conductivity rather than a decrease. The impact of cellsize reduction on foam compressive strength is usually negative, again because of the concomitant change in polymer distribution.

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