Physically Constrained Neural Networks and Regularization of Inverse Problems

The solution to a variety of engineering problems entails the simulation of a physical system. The main component of a simulator is a model that must accurately depict the system behaviour. There are two approaches to build such a model: to describe the functioning of the system with the help of physical laws; or to obtain a representation from observations of actual experiments. Theoretical models may have great precision and high generalisation capabilities; however, it is not possible to guarantee that every aspect of a physical phenomenon is represented into the equations. Obtaining agreement with a real system is a hard task: a bad estimate for a material parameter suffices to invalidate a prediction. Models based on measurements are not concerned with the complex system structure that gave rise to the data. In addition they are generally fast: a representation like a neural network is evaluated in a fraction of the time needed by a differential equation solver. On the other side, the obtention of a representation is an ill-posed problem; the resulting model is unable to extrapolate the predictions to regions not covered during data gathering. This paper considers the combination of both sources of information to construct a model, in a manner similar in concept to data assimilation. The equations of the theoretical model are used as restrictions in the training of neural networks; this is done in the framework of regularization techniques. The procedure is illustrated with situations as convection-diffusion, prismatic bar, plate and two dimensional flow