Speaker
Description
Artificial Neural Networks (ANNs) are algorithmic structures used in Machine Learning (ML), which is the ability to learn without being explicitly programmed. Recently, ANNs have been used to solve boundary value problems as an alternative to traditional numerical methods in computational mechanics. Physics-informed Neural Networks (PINNs) are the most promising type of ANNs in this context. PINNs are guided by loss functions that incorporate the laws of physics relevant to the given boundary value problem, defining the error to be minimized. Thus, this is referred to as unsupervised training, i.e., no training data with labels is required. Different approaches are utilised to construct the loss function of PINNs such as using the strong form of the governing partial differential equation or evaluating the variational energy formulation of the boundary value problem, thereby minimizing the energy.
In this regard, it is an open question for PINNs how to optimally select the number of collocation points. Moreover, it is unclear how the smoothness of the solution influences their performance. Therefore, we perform a systematic study to evaluate the performance of PINNs when dealing with different combinations of inputs, viz., the location and number of collocation points, and the smoothness of the body load.
In this study, we keep the geometry as simple as a 3D unit cube but vary the body force by changing the respective polynomial degree. The performance is evaluated using error norms established in computational mechanics, such as the normalized L2 norm. This norm is calculated using the predictions from the PINN and the analytical solution for various combinations of surface and interior points of the unit cube. We observe the minimum number of collocation points required to fit the solution and discuss the associated computational efforts.
