My research expertise and interest are described in the figure above. They consist of three steps and their relations between them. It can be summarized as follows.

1.  What can I expect from my data?
   Depending on the sparsity in the measurements, the quality of the signals, and the fidelity of the model, one can decide on the problem type. In the more restrictive case, one can simulate the system and infer its state relatively far from measurements, this is a digital twin framework. In case of more data, one can start estimating uncertain parameters in the model or propose a richer dynamic. Long-term inference can also be made possible.

2. How can I train a PINN?
   Training a PINN is like solving a learning problem under constraint. There are then different methodologies coming from constrained optimization, mainly Lagrangian relaxations. This choice is highly correlated to the type of problem discussed previously and the limited training time. The architecture of the neural network also reflects the a-priori knowledge of the system and affects the computational/memory burden.

3. Which tool do I need to attain the desired level of certification?
   Finally, the last step is to evaluate the quality of the training. The general approach tends to consider the inherent stochasticity of the training process and gives confidence intervals on the inference. A more formal approach evaluates the solution using IQCs to ensure that the signal has the desired level of performance. Once the PINN is trained, this can also be used as an initial guess to standard algorithms already certified. This last step can also be tackled first depending on the problem.

These three questions shape the methodology that I will apply in the research program described below. My research is focusing on re-interpretation and coupling between system theory and machine learning. This is conducted in an international environment, mainly with researchers from all over Europe and the US.