Differentiable Electrochemistry: A paradigm for uncovering hidden physical phenomena in electrochemical systems

TL;DR

Differentiable Electrochemistry addresses a long-standing gap in electrochemical research by unifying physics-based models with differentiable computation to enable end-to-end gradient-based inference from experimental data. The approach delivers differentiable simulators that couple thermodynamics, kinetics, and mass transport, allowing direct parameter identification without gradient-free search or region selection. The authors demonstrate three mechanistic applications beyond traditional methods—refining Tafel/Nicholson analysis, identifying Li metal electrode kinetics with full Marcus-Hush-Chidsey formalism, and interpreting Operando X-ray transport measurements—along with an open-source software stack. This framework offers a data-efficient, physics-consistent pathway to mechanistic discovery and predictive modeling in complex electrochemical systems.

Abstract

Despite the long history of electrochemistry, there is a lack of quantitative algorithms that rigorously correlate experiment with theory. Electrochemical modeling has had advanced across empirical, analytical, numerical, and data-driven paradigms. Data-driven machine learning and physics based electrochemical modeling, however, have not been explicitly linked. Here we introduce Differentiable Electrochemistry, a mew paradigm in electrochemical modeling that integrates thermodynamics, kinetics and mass transport with differentiable programming enabled by automatic differentiation. By making the entire electrochemical simulation end-to-end differentiable, this framework enables gradient-based optimization for mechanistic discovery from experimental and simulation data, achieving approximately one to two orders of improvement over gradient-free methods. We develop a rich repository of differentiable simulators across diverse mechanisms, and apply Differentiable Electrochemistry to bottleneck problems in kinetic analysis. Specifically, Differentiable Electrochemistry advances beyond Tafel and Nicholson method by removing several limitations including Tafel region selection, and identifies the electron transfer mechanism in Li metal electrodeposition/stripping by parameterizing the full Marcus-Hush-Chidsey formalism. In addition, Differentiable Electrochemistry interprets Operando X-ray measurements in concentrated electrolyte by coupling concentration and velocity theories. This framework resolves ambiguity when multiple electrochemical theories intertwine, and establishes a physics-consistent and data-efficient foundation for predictive electrochemical modeling.

Figures (6)

• Figure 1: Differentiable Electrochemistry is proposed as the new paradigm of electrochemical modeling. (a) The four paradigms of scientific modeling: observations for correlation and inspiration, linearization of analytical theory (e.g. Nicholson method), numerical simulation with gradient-free optimization, and data-driven ML models. (b) The principle of Differentiable Electrochemistry simulations in comparison with conventional black-box simulators. The parameter set $\Theta$ populates initial conditions, boundary conditions, and governing equations of simulations for time stepping (forward run). The simulated responses are then compared with experimental responses to obtain loss values $\mathcal{L}\left(\Theta\right)$. In a conventional simulator, the losses are not differentiable with respect to $\Theta$, and thus requires parameter sweep or gradient-free optimization for parameter discovery. On the contrary, Differentiable Electrochemistry simulation utilizes differentiable programming and automatic differentiation to obtain the algorithmic gradients of loss with respect to parameters $\left(\frac{\partial \mathcal{L}\left(\Theta\right)}{\partial \Theta}\right)$ for gradient-based optimization.
• Figure 2: Three applications of Differentiable Electrochemistry. (a) Differentiable Electrochemistry offers a better alternative to traditional Tafel analysis and Nicholson method (not shown in this figure). Compared with conventional Tafel analysis that selects an arbitrary Tafel region, considers only cathodic or anodic reaction, and neglects mass transport, Differentiable Electrochemistry simulation enables a differentiable continuum modeling of the entire reaction domain to consider the entire current-potential data, with full electrokinetics and mass transport. Thus, Differentiable Electrochemistry removes the limitations of conventional Tafel analysis, offering a more accurate, objective, and standardized alternative. (b) Differentiable Electrochemistry enables mechanistic identification of Li metal anode electrodeposition and stripping. Specifically, it allows direct parameterization of Marcus-Hush-Chidsey (MHC) kinetics without closed-form approximations. (c) Operando Heterodyne XPCS and XAM on symmetric Li cell generates dynamic velocity and concentration profiles. Differentiable Electrochemistry couples concentrated transport theory with a hydrodynamic continuity constraint to interpret the two fields to identify and explain abnormal Li transference number $t_+^0$.
• Figure 3: Differentiable Electrochemistry simulation and optimization of $\ch {Fe^{3+}}$/$\ch{Fe^{2+}}$ redox couple as an example to advance Tafel analysis and Nicholson method. a) The experimental voltammograms (dashed lines) are overlaid with simulated voltammograms (solid lines) using parameters estimated by Differentiable Electrochemistry. Subplots b-e are the optimization trajectories of Differentiable Electrochemistry. The mean and standard deviation of 30 runs are shown in solid lines and shadowed regions, respectively. The trajectories of (b) mean squared error of currents, (c) electrochemical rate constant, (d) transfer coefficients, and (e) average diffusion coefficient are shown.
• Figure 4: Differentiable Electrochemistry simulation for acidic HER on a rotating disk electrode. (a) The experimental linear sweep voltammogram (black dashed line) of acidic HER on a rotating Pt electrode in $1\ \mathrm{M}\ \ch{HClO4}$ at $2\ \mathrm{mV/s}$ and 2500 rpm reported by Koper et al.AP1, and the simulated voltammogram (blue solid line) using parameters provided by Differentiable Electrochemistry. The Tafel region suggested by Duan et al. and the Tafel region used by Koper et al. are shown in orange and red dashed boxes.wan2024unravelingAP1 The Differentiable Electrochemistry learning trajectories for (b) MSE of current density, (c) electrochemical rate constant, and (d) cathodic transfer coefficient. The mean trajectories and standard deviations are shown in solid lines and dashed trajectories, respectively.
• Figure 5: The predictions of experimental current densities (red scatters) reported by Boyel et al.boyle2020transient using Differentiable Electrochemistry with MH (red lines) and MHC formalisms (blue lines) for four different solvents (a) PC, (b) DEC, (c) EC:DEC, (d) EC:DC w. 10% FEC. The reorganization energies of MH ($\lambda_{MH}$) and MHC ($\lambda_{MHC}$) formalisms are reported. The root mean square errors of MH ($\mathrm{RMSE_{MH}}$), closed form approximation of MHC ($\mathrm{RMSE_{MHC,approx}}$), and MHC ($\mathrm{RMSE_{MHC}}$) are also shown in each subplot.