Surrogate-accelerated parameterisation of physics-based Li-ion battery models

Technical Summary: Surrogate-Accelerated Parameterisation of Physics-Based Li-ion Battery Models

Problem Statement
Physics-based continuum models, such as the Doyle–Fuller–Newman (DFN) model and its reduced counterparts like the Single-Particle Model with electrolyte dynamics (SPMe), offer access to physically meaningful internal electrochemical states (e.g., stoichiometries, concentration gradients) that are not directly measurable. However, inferring cell-specific parameters for these models from terminal current–voltage data is computationally expensive and limited by issues of identifiability. Existing approaches often rely on sensitivity-based parameter selection or staged estimation, and many studies fail to recover unique or physically meaningful parameters despite achieving accurate voltage fits. Furthermore, high-dimensional simultaneous fits involving state-dependent transport functions (diffusivities) remain uncommon. A significant challenge is the "inverse crime," where inference models are identical to data-generating models, failing to account for structured model discrepancies.

Methodology
The authors present a surrogate-accelerated inverse framework based on the SPMe to address these challenges. The methodology proceeds through several key stages:

1. Model Reformulation and Redundancy Removal: The SPMe is rescaled to remove exact structural redundancies. By normalizing spatial coordinates, concentrations, and fluxes, the authors derive a non-redundant set of parameter groups. This includes scalar groups for diffusion time scales ($\tau_k$), maximum lithium inventory ($Q_k$), exchange current scales ($i^{\text{ref}}_{k,0}$), and electrolyte properties, as well as functional parameters for state-dependent solid and electrolyte diffusivities.
2. Surrogate Acceleration: The forward map of the SPMe is accelerated using the Artiphy surrogate framework. This differentiable surrogate enables rapid evaluation of voltage and internal states, making gradient-based optimization of a high-dimensional parameter space computationally feasible.
3. Synthetic Benchmark: To avoid the inverse crime, the benchmark data is generated using a more detailed DFN model (via the Dandeliion simulator) under a WLTP-like dynamic current protocol. The inference is performed using the SPMe, introducing controlled model discrepancy while retaining known reference parameters.
4. Inference Formulation: The study employs a Maximum A Posteriori (MAP) estimation approach. The parameter vector includes 27 scalar parameters and three functional parameters (solid diffusivities represented by truncated Fourier expansions, and electrolyte diffusivity as an exponential shape function). The objective function combines a Gaussian likelihood (based on voltage residuals), literature-derived priors (from LiionDB), and soft physical-consistency penalties (ensuring non-negative concentrations and stoichiometries within $[0,1]$).
5. Sensitivity and Identifiability Analysis: Following the inference, the authors conduct a local sensitivity and Fisher Information Matrix (FIM) analysis. This involves transforming parameters into non-dimensional coordinates to compute the Jacobian and FIM, assessing which parameter combinations are constrained by the voltage data and identifying correlated directions.

Key Results

• Voltage Reconstruction: The inferred SPMe reproduces the DFN benchmark voltage with an error of approximately 1 mV (RMSE of 1.09 mV when evaluated with a conventional PyBaMM solver).
• Parameter Recovery:
  • Capacities: Electrode capacities ($Q_n, Q_p$) are recovered with high fidelity (relative differences < 2.5%).
  • Positive Electrode: The positive-electrode diffusivity is accurately recovered over much of the probed stoichiometric range.
  • Negative Electrode: Despite the excellent voltage fit, the negative-electrode diffusivity differs substantially from the benchmark. This is attributed to the flat Open-Circuit Potential (OCP) profile of the graphite-like anode, which renders the terminal voltage insensitive to anode concentration variations over large portions of the drive cycle.
  • Electrolyte: The electrolyte diffusivity shape parameter ($\beta_e$) is poorly informed, showing a large deviation from the benchmark, consistent with its extremely small voltage footprint (~1 µV).
• Identifiability and Correlations:
  • Correlated Groups: The analysis reveals strong correlations between kinetic and Ohmic parameters, as well as within the electrolyte transport block ($Q_e, \tau_e, t_+$).
  • Information Localization: Information regarding diffusivity is not uniformly distributed. For the negative electrode, information is concentrated in a single dominant mode (82% of total information), acquired primarily when the anode OCP gradient is steep. For the positive electrode, information is more distributed but still concentrated in the leading modes.
  • Fisher Curvature vs. Physical Accuracy: The study highlights a critical distinction: narrow Fisher-curvature envelopes (indicating high local precision within the model) do not guarantee accurate recovery of the true physical diffusivity function, especially when the parameterization (e.g., global Fourier basis) propagates local constraints across the entire domain.

Significance and Claims
The paper claims to represent a step towards rapid, physics-based in-silico parameterisation and a reduced reliance on destructive cell characterisation. Its primary contributions are:

1. Demonstration of Feasibility: It shows that a fast, differentiable SPMe forward map can make high-dimensional physics-based parameter inference from dynamic current–voltage data computationally practical.
2. Handling Model Discrepancy: By using a DFN-generated benchmark for SPMe inference, the work provides a controlled testbed for understanding how model discrepancy affects parameter recovery, moving beyond the "inverse crime."
3. Nuanced Identifiability: The study clarifies that accurate voltage reproduction does not imply unique or physically meaningful parameter recovery. It demonstrates how weak voltage sensitivity (e.g., in flat OCP regions) and parameter correlations can mask non-uniqueness, even when the fitted model fits the data perfectly.
4. Role of Surrogates and Priors: The authors emphasize that the surrogate is a tool for acceleration, not a replacement for the physical model, and that priors play a crucial role in regularizing weak directions where voltage data alone provides insufficient information.

The authors maintain a modest stance, noting that while the surrogate accelerates inference, the resolution of parameters is ultimately determined by the model structure and the information content of the experimental protocols. They caution that the reported Fisher-curvature envelopes are conditional on the specific SPMe structure and independent noise assumptions, and may be optimistic proxies for physical uncertainty in the presence of temporally correlated model discrepancies.