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.

翻译：物理成形神经网络(PINNs)作为解决差异方程式的一种方法越来越受欢迎。 虽然在某些情况下比传统数字技术更可行,但PINNs仍然缺乏可信度。 在不确定性量化(UQ)中可以找到的补救方法刚刚开始出现。 评估经过培训的PINN在多大程度上符合强制规定的差异方程式是解决不确定性的关键,但缺乏应对这一任务的全面方法。 我们提出了巴伊西亚的UQ PINNs(B-PINNs)框架,其中纳入了B-PINN的解决方案与未知的真实解决方案之间的差异。 我们利用了近期在线性动态系统中PINNs错误界限的结果,并展示了某类线性ODs的预测不确定性。