  How to solve inverse problems is the challenge of many engineering and
industrial applications. Recently, physics-informed neural networks (PINNs)
have emerged as a powerful approach to solve inverse problems efficiently.
However, it is difficult for PINNs to quantify the uncertainty of results.
Therefore, this study proposed ensemble PINNs (E-PINNs) to handle this issue.
The E-PINN uses ensemble statistics of several basic models to provide
uncertainty quantifications for the inverse solution based on the PINN
framework, and it is employed to solve the inverse problems in which the
unknown quantity is propagated through partial differential equations (PDEs),
especially the identification of the unknown field (e.g., space function) of a
given physical system. Compared with other data-driven approaches, the
suggested method is more than straightforward to implement, and also obtains
high-quality uncertainty estimates of the quantity of interest (QoI) without
significantly increasing the complexity of the algorithm. This work discusses
the good properties of ensemble learning in field inversion and uncertainty
quantification. The effectiveness of the proposed method is demonstrated
through several numerical experiments. To enhance the robustness of models,
adversarial training (AT) is applied. Furthermore, an adaptive active sampling
(AS) strategy based on the uncertainty estimates from E-PINNs is also proposed
to improve the accuracy of material field inversion problems.
