[Paper Review] A Physics-Regularized Neural Network and Kirchhoff Markov Random Field Framework for Inferring Internal Electrochemical States from Operando Spectromicroscopy

Naoki Wada, Yuta Kimura+4 more|arXiv (Cornell University)|Feb 23, 2026

Machine Learning in Materials Science0 citations

TL;DR

This paper presents a physics-regularized neural network combined with a Kirchhoff-based Markov random field to infer internal electrochemical states from operando X-ray spectromicroscopy data in Li-ion battery cathodes, enabling quantitative visualization of internal transport phenomena.

ABSTRACT

Quantitative understanding of coupled reaction and transport processes in lithium-ion battery (LIB) composite electrodes remains challenging because key internal states cannot be measured directly. In this study, we develop a physics-integrated, data-driven analysis pipeline to estimate internal electrochemical states from operando microscopic X-ray absorption fine structure ($μ$-XAFS) hyperspectral data of LIB cathodes with LiPF$_6$ electrolyte. State-of-charge (SOC) maps are first constructed from Co K-edge spectra. To resolve ambiguities in the two-phase reaction region, a physics-regularized three-layer neural network is introduced, enforcing spatial continuity of SOC and current conservation. The inferred SOC dynamics are then incorporated into a Kirchhoff-based Markov random field framework that integrates Kirchhoff's current and voltage laws, Ohm's law, and a symmetric Butler-Volmer relation to estimate interfacial current, ionic current, electrolyte potential, and effective ionic conductivity. Application to composite electrodes with different initial electrolyte concentrations (0.3, 1, and 2M LiPF$_6$) reveals distinct reaction propagation behaviors governed by electrolyte concentration-dependent conductivity. The inferred electrolyte concentration distributions show qualitative agreement with independent operando X-ray transmission imaging performed on LIB composite cathodes employing a LiAsF$_6$ electrolyte. This framework enables quantitative visualization of otherwise inaccessible internal transport phenomena.

Motivation & Objective

• Develop a data-driven pipeline that can estimate internal electrochemical states from operando micrometer-scale X-ray spectromicroscopy data of LIB cathodes.
• Resolve ambiguities in the two-phase reaction region through physics-regularization enforcing spatial SOC continuity and current conservation.
• Integrate SOC dynamics into a Kirchhoff-based Markov random field to estimate interfacial current, ionic current, electrolyte potential, and conductivity.
• Demonstrate the framework on composite electrodes with varying LiPF6 electrolyte concentrations and validate qualitatively against independent imaging.

Proposed method

• Construct SOC maps from Co K-edge spectra to initialize the analysis.
• Introduce a three-layer neural network with physics regularization enforcing spatial SOC continuity and current conservation in the two-phase region.
• Incorporate inferred SOC dynamics into a Kirchhoff-based Markov random field framework that combines Kirchhoff’s laws, Ohm’s law, and a symmetric Butler–Volmer relation.
• Estimate interfacial current, ionic current, electrolyte potential, and effective ionic conductivity from the integrated model.
• Apply the framework to electrodes with different initial LiPF6 concentrations (0.3, 1, 2 M) to study conductivity-driven reaction propagation.

Experimental results

Research questions

• RQ1 Can operando X-ray spectromicroscopy data reliably infer internal SOC distributions in LIB cathodes and resolve two-phase region ambiguities?
• RQ2 How can a physics-regularized neural network enforce physical constraints (SOC continuity, current conservation) in the interior of a battery electrode?
• RQ3 Can a Kirchhoff-based Markov random field framework accurately estimate interfacial current, ionic current, electrolyte potential, and conductivity from inferred SOC dynamics?
• RQ4 Do electrolyte concentration variations (0.3, 1, 2 M LiPF6) lead to distinct propagation behaviors that the framework can capture?
• RQ5 How well do the inferred electrolyte concentration distributions align qualitatively with independent operando X-ray imaging?

Key findings

• The framework enables quantitative visualization of internal transport phenomena that are otherwise inaccessible.
• A physics-regularized three-layer neural network resolves ambiguities in the two-phase region by enforcing spatial continuity of SOC and current conservation.
• Kirchhoff-based Markov random field integration yields estimates of interfacial current, ionic current, electrolyte potential, and effective ionic conductivity consistent with physical laws.
• Electrolyte concentration variations (0.3, 1, 2 M LiPF6) show distinct reaction propagation behaviors governed by conductivity.
• Inferred electrolyte concentration distributions qualitatively agree with independent operando X-ray transmission imaging using LiAsF6 electrolyte.