Spatial temporal data are abundant in many scientific fields, some examples include; satellite images of the earth, temperature readings from multiple weather stations and the spread of an infectious disease over a particular region taken over time. In many instances the spatial data are accompanied by mathematical models expressed in terms of partial differential equations (PDEs). These PDEs determine the theoretical aspects of the behaviour of the physical, chemical or biological phenomena considered. The parameters of the PDEs are typically unknown and must be inferred from expert knowledge of the phenomena considered. In this talk, I introduce a methodology for data driven estimates of the parameters of PDEs defined over complex domains. I will discuss extending the profiling with parameter cascading procedure outlined in Ramsay et al (2007) to incorporate PDE parameter estimation. Furthermore, following from Lai and Schumaker (2007) the estimation procedure is extended to include splines on triangulations. This allows the proposed method to account for attributes of the geometry of the physical problem such as irregular shaped domains, external and internal boundary features and strong concavities.