Data-driven reduced bond graph for nonlinear multiphysics dynamic systems

We propose a data-driven approach aiming at reducing Bond Graphs by using simulation data. The output of the approach is a hybrid model that contains a Reduced Bond Graph coupled to a simple artificial neural network. In this paper, the neural network is obtained by a linear calibration procedure. We propose a method that contains two training steps. First, the method selects the components of the original Bond Graph that are kept in the reduced Bond Graph. Secondly, the method builds an artificial neural network that supplements the reduced Bond Graph. Because the output of the proposed approach is a hybrid model, not solely data, it becomes difficult to use a usual Backpropagation Through Time to calibrate the weights of the neural network. So, in a first attempt, a very simple neural network is proposed by following a model reduction approach. As an industrial application, we consider the modeling of the automotive cabins thermal behavior. The data used for the training step are obtained via solutions of differential algebraic equations by means of a design of experiment. Simple cooling simulations are run during the training step. We show a simulation speed-up when the reduced bond graph is used to simulate the driving cycle of the WLTP vehicles homologation procedure, while preserving accuracy on output variables. The variables of the original bond graph are split into a set of primary variables, a set of secondary variables and a set of tertiary variables. The reduced bond graph contains all the primary variables, but none of the tertiary variables. While secondary variables are coupled to primary ones via an artificial neural network. Besides, the mathematical modeling used to describe the proposed method allows generalizing it to any set of differential algebraic equations, and not only to bond graphs. Finally, we discuss the extension of the coupling approach to more complex artificial neural networks.