[Paper Review] Uncovering the Underlying Physics of Degrading System Behavior Through a Deep Neural Network Framework: The Case of Remaining Useful Life Prognosis

Sergio Cofre-Martel, Enrique López Droguett+1 more|arXiv (Cornell University)|Jun 10, 2020

Fault Detection and Control Systems32 references4 citations

TL;DR

This paper proposes a physics-informed deep learning framework that uncovers degradation mechanisms in engineering systems by discovering latent variables and underlying partial differential equations (PDEs) from sensor data. By integrating physical constraints into a three-stage deep neural network, the model predicts remaining useful life (RUL) while producing interpretable health indicators, enabling transparent, physically consistent prognosis beyond black-box regression.

ABSTRACT

Deep learning (DL) has become an essential tool in prognosis and health management (PHM), commonly used as a regression algorithm for the prognosis of a system's behavior. One particular metric of interest is the remaining useful life (RUL) estimated using monitoring sensor data. Most of these deep learning applications treat the algorithms as black-box functions, giving little to no control of the data interpretation. This becomes an issue if the models break the governing laws of physics or other natural sciences when no constraints are imposed. The latest research efforts have focused on applying complex DL models to achieve a low prediction error rather than studying how the models interpret the behavior of the data and the system itself. In this paper, we propose an open-box approach using a deep neural network framework to explore the physics of degradation through partial differential equations (PDEs). The framework has three stages, and it aims to discover a latent variable and corresponding PDE to represent the health state of the system. Models are trained as a supervised regression and designed to output the RUL as well as a latent variable map that can be used and interpreted as the system's health indicator.

Motivation & Objective

• To address the lack of interpretability in deep learning models used for remaining useful life (RUL) prognosis in prognostics and health management (PHM).
• To develop a framework that uncovers the underlying physical laws governing system degradation, rather than treating deep learning as a black box.
• To enable the discovery of a latent variable and corresponding partial differential equation (PDE) that represent the system's health state.
• To ensure model predictions remain consistent with physical laws by embedding constraints into the deep learning architecture.
• To produce interpretable health indicators alongside RUL predictions for improved system diagnostics and decision-making.

Proposed method

• The framework employs a three-stage deep neural network (DNN) architecture to learn a latent representation of system degradation from sensor data.
• The first stage encodes raw sensor data into a low-dimensional latent space representing the system's health state.
• The second stage learns a partial differential equation (PDE) that governs the evolution of the latent variable over time and space.
• The third stage performs supervised regression to predict the remaining useful life (RUL) using the learned latent variable and PDE.
• The model is trained with a loss function that enforces physical consistency, ensuring the PDE aligns with known degradation dynamics.
• The output includes both RUL predictions and a physically interpretable health indicator derived from the latent variable map.

Experimental results

Research questions

• RQ1 Can a deep neural network framework discover an interpretable latent variable that represents the health state of a degrading system?
• RQ2 Can the framework uncover an underlying partial differential equation (PDE) that governs the degradation process from sensor data alone?
• RQ3 How well can the model predict remaining useful life (RUL) while maintaining physical consistency with known degradation laws?
• RQ4 To what extent does the inclusion of physical constraints improve model interpretability and reliability compared to standard black-box deep learning?
• RQ5 Can the learned latent variable serve as a meaningful health indicator for system diagnostics and prognosis?

Key findings

• The proposed framework successfully discovers a latent variable that correlates with the system's health state and can be interpreted as a health indicator.
• The model identifies a partial differential equation (PDE) that governs the degradation process, revealing underlying physical mechanisms from data.
• The framework achieves accurate RUL predictions while ensuring physical consistency, avoiding violations of known physical laws.
• The latent variable map provides interpretability, allowing engineers to understand degradation progression beyond raw sensor outputs.
• The integration of physical constraints into the deep learning architecture enhances model reliability and reduces the risk of unphysical predictions.
• The method demonstrates the feasibility of using deep learning not just for prediction, but for uncovering hidden physical dynamics in degrading systems.