Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks
This work addresses uncertainty quantification for researchers using PINNs to solve differential equations, but it is incremental as it builds on existing error bounds and focuses on linear systems.
The paper tackles the lack of credibility in Physics-Informed Neural Networks (PINNs) by proposing a framework for Uncertainty Quantification (UQ) in Bayesian PINNs (B-PINNs) that incorporates the discrepancy between the B-PINN solution and the true solution, demonstrating predictive uncertainty on linear ODEs.
Physics-Informed Neural Networks (PINNs) are gaining popularity as a method for solving differential equations. While being more feasible in some contexts than the classical numerical techniques, PINNs still lack credibility. A remedy for that can be found in Uncertainty Quantification (UQ) which is just beginning to emerge in the context of PINNs. Assessing how well the trained PINN complies with imposed differential equation is the key to tackling uncertainty, yet there is lack of comprehensive methodology for this task. We propose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates the discrepancy between the B-PINN solution and the unknown true solution. We exploit recent results on error bounds for PINNs on linear dynamical systems and demonstrate the predictive uncertainty on a class of linear ODEs.