*1.3. Towards Real-Time Decision Making*

In summary, on one side models based on physics can be solved fast but, as discussed, in many engineering areas they remain poor approximations of the real components and systems. On the other hand, when approaching the problem from the data perspective, impressive amounts of data are sometimes needed (with the associated cost and technological difficulty of collecting them), to be processed in real time and then explained in order to certify both the designs and the decisions.

A possible winning option consists of merging both concepts and methodologies. The *hybrid paradigm* was born [19,20], associating in it two type of models: the first based on physics; the second being a completely new type of model, more pragmatic and phenomenological, based on data.

Real-time decision making in engineering design, manufacturing and predictive and operational maintenance, needs the evaluation of quantities of interest in almost real-time. The present work aims at proposing a technique able to determine under the stringent real-time constraints, effective properties of a complex microstructure by assimilating an image of it.

To conciliate accuracy and real-time constraints, the hybrid paradigm is retained: (i) the prediction engine will be trained offline from data coming form physics; then (ii) a non-linear regression, acting on some topological descriptors extracted from those images, will ensure a real-time evaluation of the effective properties (in the present case the homogenized thermal conductivity).

As previously discussed, dealing with shapes and topologies, the parametric description requires performant techniques able to express them in a compact and concise way. In this paper, we propose using Topological Data Analysis, TDA [21], for representing these rich topologies and morphologies, and then using the associated topological descriptors for generating accurate supervised classification and nonlinear regressions, enabling an almost real-time evaluation of the quantities of interest.

In the next section we will present the main methodologies used in the present study, that will be considered later for the training and then for the real-time evaluation of effective properties (homogenized thermal conductivity) of rich microstructures.
