iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filter

TL;DR

This paper introduces iPINNER, an iterative physics-informed neural network framework enhanced with ensemble Kalman filter and multi-objective optimization to improve robustness and accuracy in solving PDEs with noisy data and missing physics.

Contribution

The paper proposes a novel iterative multi-objective PINN ensemble Kalman filter framework combining NSGA-III and EnKF for better PDE solutions under noisy and incomplete data conditions.

Findings

Outperforms standard PINNs in noisy data scenarios.

Effectively handles missing physics in inverse problems.

Demonstrates improved accuracy on benchmark PDE problems.

Abstract

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving forward and inverse problems involving partial differential equations (PDEs) by incorporating physical laws into the training process. However, the performance of PINNs is often hindered in real-world scenarios involving noisy observational data and missing physics, particularly in inverse problems. In this work, we propose an iterative multi-objective PINN ensemble Kalman filter (iPINNER) framework that improves the robustness and accuracy of PINNs in both forward and inverse problems by using the \textit{ensemble Kalman filter} and the \textit{non-dominated sorting genetic algorithm} III (NSGA-III). Specifically, NSGA-III is used as a multi-objective optimizer that can generate various ensemble members of PINNs along the optimal Pareto front, while accounting the model uncertainty in the solution space. These ensemble members are then utilized within the EnKF to assimilate noisy observational data. The EnKF's analysis is subsequently used to refine the data loss component for retraining the PINNs, thereby iteratively updating their parameters. The iterative procedure generates improved solutions to the PDEs. The proposed method is tested on two benchmark problems: the one-dimensional viscous Burgers equation and the time-fractional mixed diffusion-wave equation (TFMDWE). The numerical results show it outperforms standard PINNs in handling noisy data and missing physics.