Enhancing Dynamical System Modeling through Interpretable Machine Learning Augmentations: A Case Study in Cathodic Electrophoretic Deposition

TL;DR

The paper addresses the challenge of accurately modeling cathodic electrophoretic deposition (EPD) under variable driving conditions by combining uncertainty quantification, identifiability analysis, and data-driven augmentations. It first analyzes a baseline 1D EPD model, reveals parameter identifiability limitations with onset criteria $j_{min}$ and $Q_{min}$, and then introduces an inference-informed modification to yield a continuous, more generalizable formulation. When the baseline and inference-informed models still fail to capture all experimental dynamics, the authors implement interpretable machine-learning augmentations within a NeuralODE framework to learn additional dynamics while preserving physical meaning. The resulting framework improves predictive accuracy for film thickness and current dynamics, demonstrates trade-offs between model complexity and offline computation, and offers a principled approach for integrating data-driven methods with physics-based models in other dynamical systems.

Abstract

We introduce a comprehensive data-driven framework aimed at enhancing the modeling of physical systems, employing inference techniques and machine learning enhancements. As a demonstrative application, we pursue the modeling of cathodic electrophoretic deposition (EPD), commonly known as e-coating. Our approach illustrates a systematic procedure for enhancing physical models by identifying their limitations through inference on experimental data and introducing adaptable model enhancements to address these shortcomings. We begin by tackling the issue of model parameter identifiability, which reveals aspects of the model that require improvement. To address generalizability , we introduce modifications which also enhance identifiability. However, these modifications do not fully capture essential experimental behaviors. To overcome this limitation, we incorporate interpretable yet flexible augmentations into the baseline model. These augmentations are parameterized by simple fully-connected neural networks (FNNs), and we leverage machine learning tools, particularly Neural Ordinary Differential Equations (Neural ODEs), to learn these augmentations. Our simulations demonstrate that the machine learning-augmented model more accurately captures observed behaviors and improves predictive accuracy. Nevertheless, we contend that while the model updates offer superior performance and capture the relevant physics, we can reduce off-line computational costs by eliminating certain dynamics without compromising accuracy or interpretability in downstream predictions of quantities of interest, particularly film thickness predictions. The entire process outlined here provides a structured approach to leverage data-driven methods. Firstly, it helps us comprehend the root causes of model inaccuracies, and secondly, it offers a principled method for enhancing model performance.

Figures (19)

• Figure 1: Initial setup for the 1D case.
• Figure 2: Experimental setup
• Figure 3: Visualization of the $\{j\}_1$ experimental data (all 13 trials) for configuration VR = 1.0. Each trial ends at a different time, and data is sampled at a rate of 10 Hz.
• Figure 4: Negative log-likelihoods computed from simulated data on a voltage ramp experiment using the baseline model with experimental conditions $V_R = 0.125$, $\sigma=0.14$, $-\log C_v=8.5$, $Q_{min} = 100.0$, and $j_{min} = 1.5$.
• Figure 5: Identifiability regions of the baseline model for two different experimental conditions. The log-likelihood on the simulated experimental data will be constant in the purple and cyan regions, indicating that little information is gained about $j_{min}$ if the true value lies in the purple region or $Q_{min}$ if the true value lies in the cyan region. Note: the 'stepping' behavior observed in the identifiability boundaries here are a product of discretizing the $j_{min}$ and $Q_{min}$ domains, but the boundaries are in fact smooth.