At the heart of the study is the Doyle–Fuller–Newman (DFN) model, sometimes called the pseudo-two-dimensional model, which has been the workhorse of physics-based battery simulation since the early 1990s. The DFN model describes a cell as a one-dimensional sandwich of a porous graphite negative electrode, a separator, and a porous positive electrode made of layered lithium nickel cobalt oxide, all between metallic current collectors. Lithium ions migrate through the electrolyte and intercalate into or deintercalate from spherical active particles, while electrons travel through the external circuit. The model resolves coupled electrolyte transport, solid-phase lithium diffusion, charge conservation, and interfacial Butler–Volmer reaction kinetics, striking a pragmatic balance between physical realism and computational tractability. Unlike fully three-dimensional microstructure-resolved simulations, which demand high-performance computing and massive parallelization, the DFN model can be evaluated rapidly enough to be sampled thousands of times, a prerequisite for the statistical inference at the core of the new framework.

The central problem the team attacked is parameterization. Accurate values for transport and thermal coefficients are essential for predicting capacity, polarization, and temperature rise, especially at high discharge rates where coupled transport and thermal effects dominate. Yet these effective properties depend on microstructural features such as porosity, tortuosity, particle connectivity, and contact resistances that evolve during manufacturing and aging. Direct experimental measurement is difficult, reported values vary widely across the literature, and empirical tuning remains common. Worse, the parameters are strongly coupled: changing one can compensate for changes in another, making independent identification from terminal voltage data notoriously ill-posed. This is precisely the kind of inverse problem where Bayesian methods shine, because treating parameters as random variables conditioned on data yields full posterior distributions, credible intervals, and identifiability assessments rather than fragile point estimates.

The researchers identified two fundamental challenges that previous approaches had not solved together. The first is multiscale parameter identification. Some DFN parameters, such as electrode porosities, electronic conductivities, and thermal conductivities, are continuum-scale quantities that can legitimately be inferred from cell-level voltage measurements. Others, notably the open-circuit potential and solid-state lithium diffusivity, are intrinsic material properties governed by atomistic thermodynamics and lithium migration mechanisms, and are only weakly constrained by voltage data. Attempting to estimate everything simultaneously inflates parameter correlation and destroys practical identifiability. The team’s solution was elegant: they prescribed the atomistic quantities using first-principles density functional theory calculations performed with the Vienna Ab initio Simulation Package (VASP), and reserved Bayesian inference for the six effective electrode parameters that the discharge data can genuinely inform.

The first-principles component is a substantial contribution in its own right. Using VASP with the projector augmented-wave framework and the Perdew–Burke–Ernzerhof functional, the researchers computed stoichiometry-dependent open-circuit potentials from total-energy differences between lithiated configurations, referencing metallic body-centred-cubic lithium. The resulting curves, calculated at ten lithium stoichiometries, reproduce the sloping behavior and magnitude of established experimental reference profiles for both electrodes. For diffusivity, they resolved the lithium energy landscape along a basal-plane hopping pathway in graphite using a constrained scan, extracting a migration barrier of approximately 0.43 electron volts. Plugging this barrier into an Arrhenius expression with a physically motivated hopping prefactor yields a room-temperature solid-phase diffusivity of about 9.75 times ten to the minus fifteen square meters per second, consistent with values used in established DFN parameterizations. The authors are candid about the limitations of this atomistic-to-continuum transfer, noting that zero-temperature approximations and unresolved microstructural effects mean the mapping is physically informed rather than exact.

The second challenge is structural model discrepancy. Even with perfect parameters, the DFN model embodies idealizations, including homogeneous electrode microstructures, simplified thermal coupling, ideal interfacial kinetics, and neglected degradation mechanisms such as solid-electrolyte interphase growth, lithium plating, and particle cracking. These errors cannot be eliminated by parameter tuning alone. Rather than replacing the physics with a machine-learning surrogate, which would sacrifice interpretability, the team preserved the full DFN formulation and trained a small feed-forward neural network to learn only the residual discrepancy between the model prediction and the experimental voltage. The network takes the six uncertain parameters and time as inputs and outputs a parameter- and time-dependent correction that is added to the DFN prediction, forming a corrected forward model. Crucially, the network’s weights remain fixed during sampling, so Bayesian inference operates exclusively on the physical parameters.

A further methodological safeguard addresses a subtle but important statistical issue: neural-network surrogates introduce their own approximation error, which, if ignored, produces overconfident posteriors. The researchers quantified this error as the validation mean squared error on held-out data and added it to the observation noise variance, forming an augmented effective variance in the Gaussian likelihood. Independent discrepancy surrogates were trained for each discharge rate, and the temporal domain was partitioned into disjoint training and testing subsets to limit information leakage between the discrepancy-learning stage and the posterior evaluation. The sampling itself employed the delayed rejection adaptive Metropolis algorithm, run for one thousand iterations with two hundred discarded as burn-in, using uniform priors on physically admissible bounds drawn from the well-known Ecker 2015 experimental parameterization of a graphite–nickel-cobalt-oxide cell.

The validation results are striking. Working with experimental discharge data digitized from published Ecker benchmark curves and simulated in the open-source PyBaMM framework with particle mechanics enabled, the team tested the framework at both a transport-limited high-rate regime and a transport-relaxed moderate-rate regime. The baseline DFN model using literature reference parameters mispredicted voltage with a root-mean-square error of roughly 0.10 volts. Bayesian calibration alone improved matters substantially, but the full hybrid framework, combining calibrated parameters with the learned discrepancy correction, drove the error down to approximately 0.01 volts, a tenfold improvement. The experimental data fell almost entirely within the 95 percent posterior credible interval, whose narrow width signaled well-identified parameters, with mild widening near end-of-discharge reflecting genuine transport limitations. The inferred values, including porosities clustering near 0.30 to 0.32 and conductivities in physically plausible ranges, agreed well with reference measurements, indicating the framework finds real physics rather than overfitting.

Rigorous diagnostics underpinned these claims. Gelman–Rubin statistics converged rapidly toward unity, confirming well-mixed Markov chains, while pairwise posterior distributions showed compact, approximately unimodal regions with only weak-to-moderate parameter correlations. A Spearman rank correlation analysis found all parameter-observable correlations below 0.15 in magnitude, supporting robust identifiability. An ablation study comparing inference with and without the neural-network correction showed that the discrepancy model tightens the posteriors and stabilizes the chains, compensating systematic model-form error without absorbing the influence of the physical parameters. The authors transparently acknowledge that complete statistical independence between discrepancy training and posterior evaluation is not achievable when both draw on the same experimental trajectory, and they frame their results as evidence of improved practical identifiability rather than formal structural identifiability. Extending the framework to temperature and impedance data, unified multi-rate surrogates, and hierarchical priors that propagate first-principles uncertainty remains future work. Even so, the study demonstrates a compelling template for digital battery engineering: keep the physics, let quantum calculations anchor the material constants, let a small neural network absorb what the physics misses, and let Bayesian statistics keep everyone honest about what is actually known.

Subject of Research: A hybrid machine learning and Bayesian inversion framework for calibrating physics-based lithium-ion battery models using first-principles parameters and neural-network discrepancy correction.

Article Title: A hybrid machine learning–Bayesian inversion framework for physics-based lithium-ion battery models

Article References: Khodadadian, E., Mirsian, S., Khodadadian, A., & Noii, N. (2026). A hybrid machine learning–Bayesian inversion framework for physics-based lithium-ion battery models. Machine Learning with Applications, 25, Article 100995. https://doi.org/10.1016/j.mlwa.2026.100995

Image Credits: AI Generated

DOI: 10.1016/j.mlwa.2026.100995

Keywords: lithium-ion batteries, machine learning, Bayesian inversion, Doyle–Fuller–Newman model, density functional theory, neural networks, uncertainty quantification, parameter identification, PyBaMM, VASP, model discrepancy, electrochemical modeling