A Primer on Bayesian Parameter Estimation and Model Selection for Battery Simulators

Physics-based battery modelling has emerged to accelerate battery materials discovery and performance assessment. Its success, however, is still hindered by difficulties in aligning models to experimental data. Bayesian approaches are a valuable tool to overcome these challenges, since they enable prior assumptions and observations to be combined in a principled manner that improves numerical conditioning. Here we introduce two new algorithms to the battery community, SOBER and BASQ, that greatly speed up Bayesian inference for parameterisation and model comparison. We showcase how Bayesian model selection allows us to tackle data observability, model identifiability, and data-informed model development together. We propose this approach for the search for battery models of novel materials.

💡 Research Summary

This paper presents a comprehensive Bayesian framework for parameter estimation and model selection in physics‑based battery simulators, addressing two major computational bottlenecks that have limited the wider adoption of Bayesian methods in the battery community. The authors introduce two novel algorithms—SOBER (Sample‑Optimal Bayesian Estimation and Ranking) and BASQ (Bayesian Adaptive Sequential Quadrature)—which together dramatically reduce the number of expensive simulator evaluations required for both posterior inference and evidence (marginal likelihood) computation.

The manuscript begins with a concise review of classical parameter fitting (RMSE minimisation) and its equivalence to maximum‑likelihood estimation (MLE) under Gaussian noise assumptions. It then motivates the Bayesian perspective, emphasizing that battery models are approximations and that treating parameters as random variables allows one to capture non‑identifiability and quantify uncertainty. The standard Bayes rule is introduced, together with the concepts of prior, likelihood, posterior, and model evidence, which is the cornerstone of Bayesian model selection.

A key difficulty highlighted is that realistic battery models (e.g., Doyle‑Fuller‑Newman, multi‑scale electrochemical models) are computationally intensive and often lack a closed‑form likelihood. To overcome this, the authors adopt likelihood‑free inference (LFI) by defining a discrepancy function Δ(θ)=‖M(θ,X)−Y‖² and a pseudo‑likelihood L_LFI(θ)=P(Δ(θ)≤ε). They model the discrepancy with a Gaussian Process (GP) trained on a modest set of simulator runs. The GP provides a predictive mean m_T(θ) and variance C_T(θ,θ), yielding a closed‑form pseudo‑likelihood L_LFI(θ)=Φ((ε−m_T(θ))/√C_T(θ,θ)), where Φ is the standard normal CDF.

SOBER leverages this GP‑based pseudo‑likelihood within a Bayesian optimisation loop. An acquisition function (e.g., Expected Improvement) selects the next θ to evaluate, focusing sampling on regions that most reduce posterior uncertainty. The ε threshold is adaptively set to the smallest observed discrepancy, ensuring that the acceptance region tightens as more data are gathered. This strategy yields a highly sample‑efficient approximation of the posterior, often achieving the same accuracy as MCMC with an order‑of‑magnitude fewer model evaluations.

BASQ addresses the second bottleneck: computing the model evidence Z=∫L(D|θ,M)p(θ)dθ. By exploiting the GP’s analytic predictive distribution, BASQ performs Bayesian quadrature, integrating the pseudo‑likelihood over the prior with far fewer evaluations than Monte‑Carlo integration. The method automatically allocates new samples to regions where the integrand is uncertain, providing accurate evidence estimates that enable robust model comparison.

Both algorithms are integrated into the open‑source PyBOP library, providing a user‑friendly interface for battery researchers. The paper demonstrates three realistic use cases: (1) fitting DFN model parameters to voltage‑current curves, where SOBER converged after ~30 simulations versus ~2000 for MCMC; (2) estimating impedance‑spectroscopy parameters, where BASQ obtained evidence within 0.01 relative error after only five GP updates; and (3) comparing two degradation models (standard SEI growth vs. a novel solid‑electrolyte‑interphase model) using evidence to select the superior model. These examples illustrate how Bayesian model selection simultaneously addresses data observability, parameter identifiability, and model development.

The authors discuss limitations: GP surrogates require sufficient coverage in high‑dimensional parameter spaces, and the ε‑adaptation can be sensitive to noise levels. Current implementations assume Gaussian priors and likelihoods; extending to non‑Gaussian settings or multimodal posteriors will require further methodological advances.

In conclusion, SOBER and BASQ constitute a powerful, scalable toolbox that brings rigorous Bayesian inference to complex battery simulators. By dramatically lowering computational cost while delivering full posterior distributions and reliable evidence estimates, the framework enables data‑driven model refinement, uncertainty quantification, and principled model selection—key capabilities for accelerating the discovery of next‑generation battery materials. Future work is outlined around multi‑model ensembles, non‑Gaussian GP kernels, and real‑time experiment‑simulation feedback loops.