[Paper Review] Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction for Soft Sensors

Aina Wang, Pan Qin+1 more|arXiv (Cornell University)|Jan 20, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes CPINN-RP, a coupled physics-informed neural network with recurrent prediction for soft sensors, to solve nonhomogeneous PDEs with unmeasurable source terms. By integrating NetU (solution approximation) and NetG (training regularization) via a data-physics-hybrid loss and employing a hierarchical training strategy, CPINN-RP achieves accurate spatiotemporal predictions, validated by 98.8% CC and sub-6.8% RMSE on synthetic data and 97.1% CC on real vibration data with 8.22 RMSE for unmeasured sensor outputs.

ABSTRACT

Partial differential equations (PDEs) are a model candidate for soft sensors in industrial processes with spatiotemporal dependence. Although physics-informed neural networks (PINNs) are a promising machine learning method for solving PDEs, they are infeasible for the nonhomogeneous PDEs with unmeasurable source terms. To this end, a coupled PINN (CPINN) with a recurrent prediction (RP) learning strategy (CPINN- RP) is proposed. First, CPINN composed of NetU and NetG is proposed. NetU is for approximating PDEs solutions and NetG is for regularizing the training of NetU. The two networks are integrated into a data-physics-hybrid loss function. Then, we theoretically prove that the proposed CPINN has a satisfying approximation capability for solutions to nonhomogeneous PDEs with unmeasurable source terms. Besides the theoretical aspects, we propose a hierarchical training strategy to optimize and couple NetU and NetG. Secondly, NetU-RP is proposed for compensating information loss in data sampling to improve the prediction performance, in which RP is the recurrently delayed outputs of well-trained CPINN and hard sensors. Finally, the artificial and practical datasets are used to verify the feasibility and effectiveness of CPINN-RP for soft sensors.

Motivation & Objective

• To address the challenge of solving nonhomogeneous PDEs with unmeasurable source terms in industrial soft sensors.
• To develop a physics-informed neural network framework that maintains accuracy without direct measurements of source terms.
• To improve prediction performance by compensating for data sampling loss through recurrent prediction.
• To validate the method on both synthetic and real-world industrial vibration data.
• To enable robust soft sensing for key variables in complex systems with spatiotemporal dynamics and inaccessible sources.

Proposed method

• Proposes a coupled PINN (CPINN) framework with two networks: NetU for approximating PDE solutions and NetG for regularizing NetU's training.
• Integrates NetU and NetG into a data-physics-hybrid loss function to enforce PDE constraints and boundary/initial conditions.
• Introduces a hierarchical training strategy to optimize and couple NetU and NetG, improving convergence and stability.
• Develops NetU-RP, a recurrent prediction module using delayed outputs from trained CPINN and hard sensors to mitigate information loss from sparse sampling.
• Theoretical analysis proves CPINN's approximation capability under unmeasurable source terms using L²-norm convergence.
• Employs a meshfree, data-driven approach that combines physical laws with neural network learning for PDE solution.

Experimental results

Research questions

• RQ1 Can a physics-informed neural network framework effectively solve nonhomogeneous PDEs when source terms are unmeasurable?
• RQ2 How can the training of a PDE-solving neural network be stabilized and improved without direct source measurements?
• RQ3 Can recurrent prediction from hard sensor data and prior CPINN outputs enhance prediction accuracy in data-sparse regions?
• RQ4 What is the performance of the proposed CPINN-RP framework on synthetic and real industrial PDE problems with unmeasurable sources?
• RQ5 To what extent does the hierarchical training strategy improve the generalization and robustness of the CPINN model?

Key findings

• On synthetic 1D wave equation data, CPINN-RP achieved a correlation coefficient (CC) of 98.77% and RMSE of 6.75% across the domain [0,π]×[0,6].
• For the 1D wave equation, at fixed times t=2 and t=4, the RMSE was 4.84×10⁻² and 5.28×10⁻², respectively, with CC values of 98.98% and 98.20%.
• In the practical aero-engine spline coupling experiment, CPINN-RP predicted sensor 4 output (unmeasured) with 97.14% CC and 8.22 RMSE on test data, outperforming baseline expectations.
• The method demonstrated strong generalization, achieving 99.9% CC on training data for sensors 1–3, indicating effective learning of underlying dynamics.
• The hierarchical training strategy significantly improved convergence and stability, enabling reliable solution approximation despite unmeasurable sources.
• The recurrent prediction component effectively compensated for data sampling loss, improving prediction fidelity in sparse measurement scenarios.