Failure-Informed Adaptive Sampling for PINNs, Part III: Applications to Inverse Problems (CSIAM Transactions on Applied Mathematics, 2024)
In this paper, we present a novel adaptive sampling strategy for enhancing the performance of physics-informed neural networks (PINNs) in addressing inverse problems with low regularity and high dimensionality. The framework is basedon failure-informed PINNs, which was recently developed in [Gao et al., SIAM J. Sci.Comput., 45(4), 2023]. Specifically, we employ a truncated Gaussian mixture model toestimate the failure probability; this model additionally serves as an error indicator inour adaptive strategy. New samples for further computation are also produced usingthe truncated Gaussian mixture model. To describe the new framework, we considertwo important classes of inverse problems: the inverse conductivity problem in electrical impedance tomography and the inverse source problem in a parabolic system. Theeffectiveness of our method is demonstrated through a series of numerical examples.
Keywords: Inverse problem; FI-PINNs; Gaussian mixture model.