Robust Physics Informed Neural Networks

We introduce a Robust version of the Physics-Informed Neural Networks (RPINNs) to approximate the Partial Differential Equations (PDEs) solution. Standard Physics Informed Neural Networks (PINN) takes into account the governing physical laws described by PDE during the learning process. The network is trained on a data set that consists of randomly selected points in the physical domain and its boundary. PINNs have been successfully applied to solve various problems described by PDEs with boundary conditions. The loss function in traditional PINNs is based on the strong residuals of the PDEs. This loss function in PINNs is generally not robust with respect to the true error. The loss function in PINNs can be far from the true error, which makes the training process more difficult. In particular, we do not know if the training process has already converged to the solution with the required accuracy. This is especially true if we do not know the exact solution, so we cannot estimate the true error during the training. This paper introduces a different way of defining the loss function. It incorporates the residual and the inverse of the Gram matrix, computed using the energy norm. We test our RPINN algorithm on two Laplace problems and one advection-diffusion problem in two spatial dimensions. We conclude that RPINN is a robust method. The proposed loss coincides well with the true error of the solution, as measured in the energy norm. Thus, we know if our training process goes well, and we know when to stop the training to obtain the neural network approximation of the solution of the PDE with the true error of required accuracy.

Approximation of the objective insensitivity regions using Hierarchic Memetic Strategy coupled with Covariance Matrix Adaptation Evolutionary Strategy

One of the most challenging types of ill-posedness in global optimization is the presence of insensitivity regions in design parameter space, so the identification of their shape will be crucial, if ill-posedness is irrecoverable. Such problems may be solved using global stochastic search followed by post-processing of a local sample and a local objective approximation. We propose a new approach of this type composed of Hierarchic Memetic Strategy (HMS) powered by the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) well-known as an effective, self-adaptable stochastic optimization algorithm and we leverage the distribution density knowledge it accumulates to better identify and separate insensitivity regions. The results of benchmarks prove that the improved HMS-CMA-ES strategy is effective in both the total computational cost and the accuracy of insensitivity region approximation. The reference data for the tests was obtained by means of a well-known effective strategy of multimodal stochastic optimization called the Niching Evolutionary Algorithm 2 (NEA2), that also uses CMA-ES as a component.