MoPINNEnKF: Iterative Model Inference using generic-PINN-based ensemble Kalman filter

2025-05-31

AI Summary

Solving forward and inverse partial differential equation (PDE) problems under noisy observations and incomplete physical knowledge remains challenging for physics-informed neural networks (PINNs). Method: This paper proposes an iterative multi-objective PINN–EnKF fusion framework that uniquely couples the NSGA-III multi-objective optimizer with ensemble Kalman filtering (EnKF). It constructs a PINN ensemble on the Pareto front, explicitly quantifying model uncertainty, and dynamically updates the data-misfit loss to accommodate observation noise and missing physics. Results: Experiments on the Burgers equation and time-fractional hybrid diffusion-wave equation demonstrate substantial improvements over standard PINNs: significantly enhanced noise robustness, over 30% higher inversion accuracy, and strong resilience under Gaussian noise exceeding 10% or critical physical terms omission. The framework establishes a new interpretable, uncertainty-quantified paradigm for data–physics joint modeling.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 (MoPINNEnKF) framework that improves the robustness and accuracy of PINNs in both forward and inverse problems by using the extit{ensemble Kalman filter} and the extit{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.

Problem

• Improves PINNs robustness with noisy data
• Enhances accuracy in forward and inverse PDE problems
• Combines ensemble Kalman filter and NSGA-III optimization

Innovation

• Uses ensemble Kalman filter for data assimilation
• Applies NSGA-III for multi-objective optimization
• Iteratively updates PINNs parameters for accuracy