Pre-Conditioned Physics-Informed Neural Network for Inverse Problems

Physics-informed neural networks (PINNs) can solve inverse problems exclusively, circumventing the dependence of other existing computational inverse methods on the results of the forward problem numerical solvers. However, it has been found that when the magnitudes of different physical parameters in the inverse problem vary significantly from each other, PINNs may either fail to converge or converge to incorrect solutions, resulting in poor inverse accuracy. Our previous study has revealed that the standard preprocessing in neural networks normalizes only the input and output data, neglecting the newly introduced trainable inverse parameters within the PINN framework. Thereby, it fails to ensure an appropriate scale for the overall trainable parameter space, potentially leading to an inappropriately “flattened” elliptical distribution. In addition, the scale differences of physical parameters affect the loss terms in PINNs, exacerbating the loss imbalance and leading to notable gradient biases. To address these issues, two crucial pre-constraint strategies are proposed in this study: (1) preprocessing is enhanced through nondimensionalization, scaling highest-order derivative coefficients, and selecting PINN inverse parameters to ensure that the magnitude scales of the trainable parameter space are appropriate and the loss terms are in unit scale to mitigate the loss imbalance; (2) the network architecture is constrained by the initial conditions, boundary conditions, and inverse data within the neural network through a constructive function, to eliminate the loss imbalance completely. The proposed method is thus termed as pre-conditioned PINN (PC-PINN), and the effectiveness of the currently proposed PC-PINN method is validated against vanilla PINN and hPINN methods through numerical examples. The research results indicate that the current PC-PINN method can effectively address the preprocessing and loss imbalance problems of PINNs, and its inverse accuracy is an order of magnitude higher than those of the adjusted PINN and hPINN methods.