The heart of the problem lies in how batteries are modeled. Traditional equivalent circuit models represent a cell as a network of resistors and capacitors, which is computationally cheap and therefore attractive for embedded battery management systems, but integer-order components carry hidden assumptions. Real batteries do not behave like ideal capacitors; their charge transfer and diffusion processes are distributed across time scales and across the physical volume of the electrodes, producing frequency-dependent impedance that lumped elements simply cannot replicate. The result is systematic voltage error, poor performance under varying temperatures, and the need for constant recalibration as the cell ages. Fractional-order models solve this by allowing non-integer exponents in the calculus that describes the system, capturing memory effects and non-exponential relaxation naturally. A key element is the constant phase element, whose exponent alpha slides continuously between the behavior of a resistor and a capacitor, and whose value of one half corresponds to the classic Warburg diffusion element seen in electrochemical impedance spectroscopy.

But fractional-order models come with a price: identifying their parameters is brutally difficult. The team adopted an R(RQ)(RQ)W structure, an advanced extension of classical second-order and PNGV models, in which an ohmic resistance is followed by two parallel resistor-constant-phase-element branches and a Warburg element. Each component maps onto a distinct region of the Nyquist plot, the standard graphical representation of impedance data. The ohmic resistance appears as the high-frequency intercept on the real axis, the two RQ branches form overlapping depressed semicircles in the mid-to-high frequency range corresponding to charge transfer and slower interfacial reactions, and the Warburg element produces the characteristic 45-degree low-frequency tail of solid-phase diffusion. Eight parameters must be extracted simultaneously, and they are strongly coupled, meaning an error in one drags the others off course. Metaheuristic optimizers, which explore search spaces without gradients, are the usual tool, but they frequently converge to local optima that look good numerically yet produce physically meaningless parameters.

To choose the best optimizer fairly, the researchers staged a systematic tournament of ten metaheuristic algorithms: three established methods, particle swarm optimization, grey wolf optimization, and artificial bee colony, against seven recent bio-inspired contenders including the sea lion, golden jackal, fox, honey badger, seahorse, sand cat, and marine predator algorithms. Each ran with an identical budget of 10,000 swarm members and 100 iterations across 20 independent trials. Scoring by root mean square error, mean absolute percentage error, and runtime, the Marine Predators Algorithm, which mimics the Lévy flight and Brownian motion strategies of foraging marine animals, emerged as the most robust performer. A Friedman statistical ranking placed it first with an average rank of 2.00, narrowly ahead of the Honey Badger Algorithm, and its mean runtime of 114 seconds comfortably beat both the honey badger’s 150 seconds and the artificial bee colony’s punishing 2,518 seconds. Yet even the winner showed troubling discrepancies for the second semicircle’s resistance and capacitance and the Warburg coefficient, confirming that raw optimization power alone was insufficient.

The first fix is elegantly physical. Instead of letting the optimizer start from arbitrary bounds, the team derived initial parameter estimates directly from the geometry of the Nyquist plot. A parallel resistor-capacitor circuit traces a perfect semicircle of radius R over two; replacing the capacitor with a constant phase element rotates that semicircle by an angle theta that depends analytically on the fractional exponent alpha, following the relation theta equals 90 degrees times one minus alpha. By extending the measured arcs to full circles, reading off their real-axis intercepts, and measuring the rotation angles of their apexes, the researchers could calculate the ohmic resistance, both branch resistances, both constant phase element exponents and capacitance offsets, and even the Warburg value from the lowest-frequency data point. Six of the eight parameters were thus estimated before optimization even began, shrinking the search space from bounds spanning twenty-fold in each direction to a tight window around physically consistent values.

The second fix attacks dimensionality. Rather than asking the optimizer to juggle all eight parameters at once, the two-stage refinement alternates between metaheuristic optimization of six parameters and analytical recovery of the remaining two. Once the first stage converges on R0, R1, Q1, alpha1, alpha2, and W1, the residual impedance, obtained by subtracting those known contributions from the measured spectrum, is fitted by global least squares to extract R2 and Q2 over the second branch’s frequency band. A Trust Region Reflective algorithm then polishes the estimates, and the cycle repeats for ten iterations with bounds that tighten by a factor of two at each stage. The swarm size drops from 10,000 in the brute-force baseline to just 2,000 in the first stage and 100 thereafter, cutting the iteration count by roughly 80 percent while improving precision.

The validation was unusually thorough. On 50 synthetic impedance profiles generated by Latin Hypercube Sampling, where the true parameter values were known exactly, the full framework achieved an overall mean absolute percentage error of just 7.27 percent, compared with a catastrophic 24,462 percent for standard marine predator optimization and an even worse result for two-stage refinement without informed initialization. The mean root mean square error fell to 1.74 times ten to the minus five, a 92.4 percent reduction relative to the baseline, with a median near ten to the minus six. Scatter plots of estimated versus true parameters clustered tightly along the ideal 45-degree line for all eight quantities, while competing methods showed dispersion spanning orders of magnitude. Crucially, the team demonstrated a subtle pitfall: methods 1 and 2 sometimes achieved low numerical error while their simulated Nyquist plots visibly failed to match the target data, proof that a low objective function value does not guarantee a physically accurate model.

Robustness testing added further confidence. Under 1 percent Gaussian noise the overall parameter error rose only from 7.27 to 12.92 percent, and under 2 percent noise to 18.41 percent, with no instability or divergence. Removing the five lowest-frequency data points, simulating instruments that cannot capture the Warburg region, barely changed the error at 7.38 percent, because the initialization relies on mid-to-high-frequency arc geometry. A transfer test on the simpler R(RQ)W structure showed the initialization concept generalizes across model topologies when the correct structure is known, though the authors are careful to note that automatic model-structure selection from raw data remains an open challenge. Against three state-of-the-art published methods, chaotic adaptive fractional particle swarm optimization, hybrid genetic-particle swarm optimization, and complex-order beetle swarm optimization, the new framework delivered curve-fitting errors one to two orders of magnitude lower, with the rivals showing parameter errors often exceeding several hundred percent for the second branch and Warburg elements.

Finally, the method proved itself on real hardware. The team measured electrochemical impedance spectra from commercial LG INR21700M50LT nickel-manganese-cobalt cells in three states: pristine, initially degraded at 90 percent state of health, and significantly aged at 80 percent state of health and 50 percent state of charge. Across 20 independent runs per condition, the full framework reduced mean root mean square error by 38.0, 32.5, and 76.6 percent respectively compared with the standard optimizer, and achieved parameter coefficients of variation as low as 0.016 percent, meaning repeated identifications landed in the same parameter basin with remarkable consistency. That repeatability is what makes the parameters trustworthy inputs for state of charge and state of health estimation, the foundation of reliable battery diagnosis, degradation tracking, and second-life applications. The authors acknowledge their framework assumes the two impedance arcs remain geometrically distinct, and extending it to overlapping arcs and adaptive topology selection is future work. But the central lesson stands: injecting electrochemical knowledge into the starting point of an optimization can accomplish what brute-force computation cannot, at a fraction of the cost.

Subject of Research: Physics-informed parameter identification for fractional-order lithium-ion battery models using electrochemical impedance spectroscopy

Article Title: Physics-informed parameter identification of fractional-order battery models using Nyquist-based initialization and two-stage metaheuristic refinement

Article References: Wang, S.-C., & Liu, Y.-H. (2026). Physics-informed parameter identification of fractional-order battery models using Nyquist-based initialization and two-stage metaheuristic refinement. Results in Engineering, 32, Article 113425. https://doi.org/10.1016/j.rineng.2026.113425

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113425

Keywords: lithium-ion batteries, fractional-order models, electrochemical impedance spectroscopy, parameter identification, metaheuristic optimization, Marine Predators Algorithm, Nyquist plot, battery management systems, constant phase element, state of charge, state of health, equivalent circuit model

The premise of the study is deceptively simple. Every electric vehicle, grid storage installation, and consumer device relies on a battery management system, a layer of algorithms that estimates how much charge remains, how fast the cell can safely be charged, and how much stress it is under. These algorithms were built, refined, and validated over decades on lithium-ion cells. When engineers attempt to reuse them on sodium-ion batteries, they implicitly assume that the underlying electrochemical response of the two systems is similar enough that the same models and parameter tables will transfer. The Chinese-led team set out to quantify whether that assumption holds, and their answer is a resounding no, at least not without substantial adaptation.

To make the comparison rigorous, the researchers built a normalized, physically constrained benchmarking framework centered on an equivalent circuit model, the workhorse representation used in battery management systems to capture how a cell’s voltage responds to current. They employed a second-order RC model, meaning the cell is represented as an internal resistance paired with two resistor-capacitor networks that describe fast and slow polarization processes, the transient voltage deviations that arise when ions and charges redistribute inside the electrode and electrolyte. Crucially, the parameter identification was constrained to remain physically plausible, avoiding the mathematical artifacts that can plague unconstrained curve fitting and render parameters meaningless for cross-chemistry comparison.

The experimental matrix behind the study is one of its most impressive features. The team extracted equivalent circuit parameters across temperatures spanning 0 to 50 degrees Celsius, states of charge from 10 to 100 percent, and charge-discharge rates from 0.1C to 3C, a range that covers nearly everything a real battery encounters in service, from a frigid winter morning to a demanding fast-charge session. This dense parameter landscape allowed the researchers to map not just how each chemistry behaves at any single point, but how its kinetic character shifts, sometimes dramatically, across the full envelope of operating conditions.

The headline finding is that sodium-ion batteries exhibit profound kinetic asymmetry, in stark contrast to the highly consistent and symmetric behavior of the lithium-ion cells tested. In practical terms, the two lithium-ion chemistries responded to changing conditions in a smooth, predictable, and largely uniform fashion, which is precisely what makes static parameter look-up tables viable for them. The sodium-ion cell, by comparison, showed exceptionally sluggish slow polarization behavior, meaning that one class of its internal relaxation processes unfolds far more gradually than anything observed in the lithium-ion counterparts. A battery management system calibrated on lithium-ion expectations could misinterpret these slow transients, leading to errors in state-of-charge estimation and power capability prediction.

The low-temperature results are perhaps the most striking. At 0 degrees Celsius, the sodium-ion cell experienced a severe impedance surge when operated at low C-rates, its apparent internal resistance ballooning in a way that would sharply limit usable power in cold conditions. This is a critical vulnerability for applications in cold climates, where grid storage systems and vehicles must deliver reliable performance from a standing start. The finding aligns with a growing body of literature on low-temperature sodium-ion limitations, where sluggish ion transport through the electrolyte and across electrode interfaces has been identified as a persistent engineering challenge requiring tailored electrolyte formulations and interface engineering.

But the story takes a surprising twist at high rates. When the sodium-ion cell was pushed with high-rate excitations between 2C and 3C, the researchers observed a pronounced nonlinear decrease in apparent impedance, the opposite of what the low-rate cold data would predict. The authors attribute this counterintuitive behavior, at least in part, to transient Joule self-heating: as large currents flow through the cell’s internal resistance, waste heat is generated, warming the cell from within and temporarily restoring the kinetic performance that the cold had suppressed. This electro-thermal coupling is well known in lithium-ion research, where self-heating structures have even been engineered deliberately to enable sub-zero fast charging, but its pronounced manifestation in a commercial sodium-ion cell under ordinary current excitation adds an important new dimension to how these batteries should be modeled and managed.

To move beyond qualitative impressions, the team turned to unsupervised machine learning. After normalizing the extracted parameters to a common baseline, they applied K-Means clustering to group operating conditions by their kinetic signatures, and used principal component analysis to visualize the results in a reduced dimensional space. The outcome was unambiguous: across the investigated operating conditions, the sodium-ion data occupied a largely separated and substantially more dispersed region of the dynamic parameter space than either lithium-ion chemistry. In other words, sodium-ion behavior is not only different, it is more variable, less compact, and harder to encapsulate in a single tidy model. The two lithium-ion chemistries, despite their different electrode materials, clustered far more closely together than either did with the sodium-ion cell.

The engineering implications are significant. The authors conclude that directly applying static, lithium-ion-centered parameter look-up strategies to sodium-ion batteries may be inadequate over broad operating ranges. Look-up tables, which map operating conditions to precomputed model parameters, are the backbone of many production battery management systems precisely because they are computationally cheap and reliable. If sodium-ion cells wander across a dispersed and asymmetric parameter space, those tables would need to be far denser, or the algorithms would need to become adaptive, continuously re-estimating parameters online rather than recalling fixed values. The study’s identified electro-thermal response characteristics and clustering patterns offer a data-driven starting point for building such adaptive state observers and for designing low-temperature management strategies tailored specifically to sodium-ion systems.

The timing of this work could hardly be better. Sodium-ion batteries are moving from laboratory curiosity to commercial reality, with manufacturers positioning them for stationary storage, entry-level electric vehicles, and markets where cost and resource security outweigh the energy density advantage of lithium. As deployment scales, the invisible software layer that keeps these batteries safe will become the linchpin of the entire technology. What this study makes clear is that the sodium-ion transition is not simply a matter of swapping one cell for another; it demands a fundamental rethinking of how battery dynamics are measured, modeled, and managed. The kinetic fingerprint of sodium is genuinely its own, and the industry that learns to read it accurately will be the one that turns an abundant element into a dependable pillar of the energy transition.

Subject of Research: Comparative dynamic kinetics and battery management implications of commercial sodium-ion versus lithium-ion battery chemistries

Article Title: Benchmarking the dynamic kinetics of a commercial layered-oxide/hard-carbon sodium-ion cell versus lithium iron phosphate and nickel cobalt manganese lithium-ion batteries: a normalized parameter sensitivity and clustering analysis

Article References: Zhang, H., Liu, T., Zhao, Y., Tang, R., Hu, J., Zuo, Q., & Jiang, L. (2026). Benchmarking the dynamic kinetics of a commercial layered-oxide/hard-carbon sodium-ion cell versus lithium iron phosphate and nickel cobalt manganese lithium-ion batteries: a normalized parameter sensitivity and clustering analysis. Ionics. https://doi.org/10.1007/s11581-026-07543-7

Image Credits: AI Generated

DOI: 10.1007/s11581-026-07543-7

Keywords: sodium-ion batteries, lithium-ion batteries, equivalent circuit model, battery management system, kinetic asymmetry, low-temperature performance, Joule self-heating, K-Means clustering, principal component analysis, parameter identification, hard carbon anode, layered oxide cathode

The research, led by Yutong Cai and Zhaoyue Zhong, with Zefeng Zhang, Bo-Wei Qin and senior author Wei Lin at Fudan University in Shanghai, introduces a rigorous mathematical treatment of what the authors call modulatability. Rather than asking how to push an oscillator toward a target behavior through trial and error, the team asked a more fundamental question about the geometry of the problem itself. If you want to impose several properties on an oscillating system at once, say a specific frequency, a specific amplitude, and a specific phase relationship across parts of a network, how many adjustable knobs do you need? Their central theoretical result is strikingly elegant: the local uniqueness of a solution is generally guaranteed when the number of free parameters equals the number of properties being targeted. Too few knobs and the problem is overconstrained, with most target combinations simply unreachable. Too many and solutions proliferate, making it hard to know which one a solver will find.

This dimensional matching principle may sound abstract, but it has immediate practical consequences. In the biological sciences, researchers routinely try to reprogram oscillators whose dynamics involve dozens of interacting variables. Synthetic biologists, for example, have spent two decades engineering genetic circuits such as the repressilator, a ring of three genes that repress one another in turn and thereby generate oscillating protein concentrations. Tuning such circuits to display both a desired period and a desired amplitude simultaneously has historically required laborious computational redesign or experimental screening. The Fudan team’s framework reframes the whole enterprise: instead of simulating trajectories forward and hoping to stumble on good parameters, the modulation task is treated as a property-based inverse problem, in which the desired properties are fixed and the equations are solved backward for the parameters that produce them.

The technical machinery behind the framework combines several ideas from dynamical systems and numerical analysis. The oscillatory trajectories of the system are represented through a Fourier expansion, allowing periodic solutions to be described by a finite set of coefficients once a truncation order is chosen. The desired properties, such as minima of particular protein activities or the period of the collective rhythm, are expressed as functions of these coefficients and of the system’s kinetic parameters. Solving for the parameters then becomes a root-finding problem in a space whose dimension is chosen deliberately to match the number of targets. The authors deploy Newton-type iterations to converge on solutions, and they use continuation strategies, stepping gradually through parameter space, to follow modulation paths even when the targets are far from the system’s natural behavior. Sweeps over system dimension, Fourier truncation order, number of targets, iteration tolerance, and continuation steps, reported in the paper’s extended data, map out how computational cost scales with each factor.

Crucially, the framework is not merely a simulation exercise. The team validated it on electronic analogs of genetic circuits, building hardware implementations of the repressilator concept in which voltages play the role of protein concentrations. From recorded voltage time courses alone, the researchers inferred the underlying parameters of the oscillating circuit and then modulated them to hit orthogonal targets, meaning goals that do not interfere with one another, such as independently shifting the period while holding amplitude fixed. The experiments included orthogonal period modulation tasks with two and with three simultaneous targets, and the inferred parameter distributions were tight and unimodal, suggesting that the inverse problem is well-posed in precisely the sense the theory predicts. This hardware demonstration matters because electronic circuits are a standard proving ground for ideas intended eventually for living cells, capturing the nonlinear feedback structure of genetic networks while remaining far easier to measure and perturb.

The data-driven side of the work is where the framework arguably delivers its most compelling numbers. In benchmark comparisons against baseline inference methods, the property-based approach achieved shorter runtimes and higher accuracy, a combination that is rare in inverse problems, where speed is usually bought at the cost of precision. The authors also demonstrated the framework on a model of an engineered Sir2-HAP negative feedback loop, a genetic oscillator linked to cellular longevity in yeast. In that system, low activity of the SIR2 and HAP proteins corresponds to a detrimental aging zone that the oscillating state variables pass through. By identifying a parameter set that up-regulates the minimal activities of both proteins, the framework showed in stochastic simulations, which added Ornstein-Uhlenbeck noise to eight kinetic terms, that the modulated oscillator avoids the aging zone more effectively, with distributions of minimum protein levels shifted upward across one hundred independent runs. It is a concrete illustration of how abstract parameter identification could translate into a design principle for extending cellular lifespan.

Networks add another layer of complexity, and the paper addresses it head-on. In one extended demonstration, the team applied their method to a FitzHugh-Nagumo neuronal network of one hundred nodes arranged on a randomly generated Barabasi-Albert topology, a structure mimicking the heterogeneous, hub-rich connectivity seen in many real systems. With fifty target nodes whose amplitudes needed adjustment, the researchers modulated random subsets of edges and measured how often the modulation succeeded while the network remained oscillatory. The results, summarized across many repeated trials, revealed how success rates depend on the balance between the number of targeted nodes and the number of edges available for modulation, with the equal-dimension condition again emerging as the natural dividing line. For anyone trying to control synchronization patterns in power grids, neural tissue, or coupled laser arrays, this kind of systematic map of when control is feasible is exactly the missing piece.

What makes the study resonate beyond applied mathematics is the sheer breadth of oscillatory phenomena it touches. The references span circadian medicine, the bioelectrical phase transition that patterns the first vertebrate heartbeats, collective oscillations emerging in massive human crowds, amplitude and frequency modulation of subthalamic beta oscillations in Parkinson’s disease, and synthetic oscillators designed to slow cellular aging. In each of these domains, the practical question is the same: which properties of the rhythm can be moved, and by touching which parameters? The modulatability framework offers a principled answer, replacing intuition with a dimension-counting criterion and replacing forward simulation with inverse problem solving. It also connects to a broader theoretical conversation about controlling complex networks, complementing earlier work on controllability, functional control of oscillator networks, and inverse problems for dynamic patterns in coupled systems.

The authors have made their work unusually accessible. The source code is publicly available on GitHub, with a frozen release archived on Zenodo, and a figure-to-script mapping document allows readers to reproduce individual results directly. Source data, including network topologies, modulation parameters, oscillatory trajectories, stochastic realizations, and the experimental voltage recordings from the electronic circuit, are deposited without restrictions. For a field where reproducibility of nonlinear dynamics can be notoriously difficult, this level of transparency lowers the barrier for other groups to test the framework on their own oscillators, whether biological, neural, mechanical, or photonic.

The larger significance of the work lies in its reframing. For years, the modulation of oscillators has been pursued through two parallel traditions, one statistical and one rooted in dynamical systems theory, each producing partial answers. By proving a general condition for when modulation parameters exist uniquely and packaging the insight into an efficient computational pipeline, the Fudan team has given both theorists and experimentalists a common language for a problem that cuts across biology, neuroscience, and engineering. If rhythms govern health, and disrupted oscillations are implicated in conditions from diabetes to psychiatric illness, then knowing exactly when and how a rhythm can be retuned is more than a mathematical curiosity. It is a step toward making oscillators, whether in a cell, a circuit board, or a brain, into things we can genuinely engineer.

Subject of Research: Mathematical theory and computational framework for the modulatability of complex oscillators

Article Title: Modulatability of complex oscillators

Article References: Cai, Y., Zhong, Z., Zhang, Z., Qin, B.-W., & Lin, W. (2026). Modulatability of complex oscillators. Nature Computational Science. https://doi.org/10.1038/s43588-026-01050-5

Image Credits: AI Generated

DOI: 10.1038/s43588-026-01050-5

Keywords: oscillators, dynamical systems, inverse problems, complex networks, synchronization, genetic circuits, repressilator, circadian rhythms, parameter identification, nonlinear dynamics, Nature Computational Science, synthetic biology

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