Effect of Loss Functions on the Learning Capabilities of Physics-Informed Neural Networks in Mechanical Systems

In the field of aerospace and mechanical engineering, the identification of mathematical models capable of accurately representing the dynamics of mechanical structures is extremely important to improve the design and accelerate the certification of new systems and structures. Recent developments in the field of machine learning have demonstrated that Neural Networks (NNs) can accurately model the dynamics of linear and nonlinear systems in the time domain. Physics-Informed Neural Networks (PINNs) exploit this property and utilise physics-informed loss functions to identify the parameters of systems. Nonetheless, it is still not completely clear how the loss functions affect the learning process of the NNs and which type of function improves/deteriorates the identification process of the parameters associated with mechanical systems. In this chapter, we investigate the effect of three different loss functions on the learning and identification capabilities of PINNs when mechanical problems are considered. To this end, classic Forward Neural Networks (FNNs) are embedded in a parameter identification scheme based on physics-informed loss functions, and the parameters (natural frequency and damping) of a single-degree-of-freedom (SDOF) mechanical oscillator are identified via numerical experiments. In order to minimise the required training data, the loss functions are built considering the governing equations of motion and a single dynamics response of the oscillator in the form of either acceleration, velocity, or displacement. Their effect is then evaluated in terms of the accuracy of the identified unknown parameters and the capacity of the NN to predict the unknown physical dynamic responses. We demonstrate that loss functions based on the acceleration time series allow the NN to correctly learn the unknown physical dynamic responses, i.e., the velocity and the displacement, with great accuracy; this results in faster and more efficient learning of the dynamic behaviour of the system which, in turn, allows to identify the correct mechanical parameters.