Learning to Estimate Regions of Attraction of Autonomous Dynamical Systems Using Bayesian Physics Informed Neural Networks

The emergence of neural networks as effective and adaptable computational tools has effectuated a congruent rise in research pertaining to how these techniques may be leveraged to model complex physical phenomena across a myriad of scientific disciplines. One application of particular interest to modern control theory is that of region of attraction (ROA) estimation for nonlinear dynamical systems. While ROAs reveal important stability, robustness, and safety information, their precise determination can require the solution of a Hamilton-Jacobi partial differential equation (PDE) that is often impractical for standard analytical and numerical techniques. As a result, neural PDE solution methodologies, such as the physics-informed neural network (PINN) framework, promise to be potent strategies for ROA estimation so long as well suited hyperparameters can be determined efficiently. Toward this end, this work applies Bayesian optimization (BO) to select hyperparameters for a modified PINN designed to estimate the ROAs of autonomous dynamical systems. Since the BO must determine the training loss weights themselves, ground truth solutions are typically required to evaluate the merits of any particular set of hyperparameters. However, by exploiting domain specific knowledge concerning ROA boundary crossings, the need for a prior ground truth solution to direct the BO can be obviated in favor of a unique classification loss metric. The application of this classification guided BO method to two different ROA estimation benchmark problems is shown to significantly improve the rate of hyperparameter convergence. These results support the idea that PINN variations can be efficiently augmented with Bayesian hyperparameter tuning without the need for pre-existing solutions if sufficient domain knowledge is available.