Physics-Informed Neural Network with Transfer Learning for State Estimation in Lithium-Ion Batteries using the Single Particle Model with Electrolyte Gift Modekwe a and Qiugang Lu a,1
arXiv:2606.28220v1 [cs.LG] 26 Jun 2026
a Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409, USA
Abstract Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving nonlinear partial differential equations (PDEs), including battery electrochemical models. They typically en-force conservation laws within the loss function to ensure physically consistent solutions. Tradi-tional numerical methods such as finite difference, finite volume, and finite element techniques, re-ly on discretization and can be computationally expensive for nonlinear systems. To address this challenge, PINNs offer improved scalability, particularly for reduced-order models like the single particle model with electrolyte (SPMe). The SPMe describes lithium-ion battery dynamics through coupled diffusion, transport, reaction kinetics, and voltage equations. Despite these advantages, training SPMe-based PINNs from scratch for different battery chemistries or operating conditions is demanding and often leads to slow convergence. To overcome this limitation, this work introduces a transfer learning framework for SPMe-PINNs. The model is first pretrained to learn general elec-trochemical dynamics and then adapted to a target battery by transferring weights, freezing se-lected layers, and fine tuning the remaining parameters, including estimating key electrochemical variables. Validation using PyBaMM demonstrates accurate voltage prediction, indicating that the proposed approach preserves electrochemical consistency while reducing training time and ena-bling efficient generalization across batteries. Keywords Lithium-ion battery, Physics-informed neural networks, Transfer learning, Single particle model with electrolyte. 1. Introduction
variants, like the single particle model (SPM) (Santhanagopalan et al., 2006) and the SPM with electrolyte Lithium-ion batteries have emerged as a dominant (SPMe) (Moura et al., 2016), are grounded in fundamentechnology in modern energy storage, owing to their sutal electrochemical principles. These models characterperior energy density, extended cycle life, and high opize lithium-ion transport, charge conservation, and reacerational efficiency (Khan et al., 2023). Their versatility tion kinetics through systems of coupled PDEs. Thus, has led to widespread use in portable electronics, electhey offer strong physical interpretability and predictive tric vehicles, and grid-scale energy storage systems. As consistency across a wide range of operating conditions. the demand for reliable energy storage grows, accurate Despite these advantages, their practical deployment is battery modeling and monitoring have become essential often hindered by the need for detailed knowledge of for ensuring operational safety, optimizing performance, material-specific parameters, as well as significant comand extending battery lifespan. In particular, accurate putational demands that limit their suitability for realestimation of internal battery states and prediction of time applications. battery behavior are critical for effective battery manOn the other hand, data-driven models leverage maagement (Ekberg et al., 2025). Significant research efforts have focused on robust chine learning techniques to learn intricate relationships modeling of lithium-ion batteries. Notably, both directly from data without requiring explicit knowledge physics-based and data-driven modeling paradigms have of underlying physics. Specifically, deep learning framebeen extensively investigated in the literature. Physics- works have demonstrated strong capability in capturing based models, such as the Doyle–Fuller–Newman (DFN) nonlinear battery behaviors and degradation patterns model (Doyle et al., 1993) and its reduced-order (Alharbi et al., 2025). However, these models often suffer from limited interpretability, high data requirements, and reduced generalization to unseen operating 1 Corresponding author: Q. Lu (E-mail: [email protected]). conditions (Liu et al., 2022).
To address these limitations, PINNs have emerged as a promising hybrid framework that bridges the gap between physics-based and data-driven modeling. They incorporate governing physical equations directly into the neural network training process in order to enforce physical consistency, while retaining the flexibility of data-driven learning (Raissi et al., 2019). This paradigm has attracted considerable attention in battery modeling and state estimation. For instance, Singh et al. (2023) developed a PINN framework for lithium-ion battery state estimation, demonstrating improved accuracy over purely data-driven approaches by embedding electrochemical constraints into the loss function. Similarly, Nascimento et al. (2021) proposed a hybrid PINN framework for lithium-ion battery health prognostics, demonstrating reliable prognosis performance even with limited data. Building upon these advancements, Xue et al. (2023) introduced the PINN SPM, which utilizes a gated recurrent unit to solve electrolyte diffusion equations, effectively correcting voltage predictions of traditional SPM at high C-rates. Additionally, Méndez-Corbacho et al. (2024) demonstrated that a purely physics-driven PINN can solve the SPM across broad parameter ranges without any reliance on labeled experimental or simulation data. Despite these advances, a critical and largely unresolved challenge remains: existing PINN-based approaches are predominantly trained on individual battery basis, limiting their transferability across batteries with differing chemistries, aging histories, and operating conditions. In this study, we focus on improving the scalability and generalization of PINNs for lithium-ion battery modeling by integrating transfer learning within an SPMe-based PINN framework. The proposed approach leverages knowledge learned from a source battery to efficiently adapt to a target battery while preserving electrochemical consistency. The main contributions of this work are as follows:
The remainder of this paper is organized as follows. Section 2 introduces the fundamentals of the SPMe model. Section 3 presents the proposed framework. Section 4 provides the results and performance evaluation. Finally, conclusions are drawn in Section 5.
• A PINN based on the SPMe is developed to capture coupled solid-phase diffusion, electrolyte transport, and electrochemical reaction kinetics within a unified learning framework.
where cs,k (r,t) denotes the solid-phase lithium concentration, Ds,k is the solid diffusivity, and r is the radial coordinate inside the spherical particle. The corresponding boundary conditions are
2. Fundamentals of the SPMe The SPMe is an electrochemical model used to describe the behavior of lithium-ion batteries, extending the classic SPM model by incorporating electrolyte dynamics. In SPM, each electrode is represented as a single spherical particle, simplifying the complex porous electrode structure while capturing essential solid-phase lithium diffusion and intercalation kinetics. The SPMe enhances this by adding electrolyte concentration and potential variations across the cell to account for ionic transport through the separator and electrolyte resistance as in Fig. 1. These effects are typically more significant at higher C-rates (Mehta and Gupta, 2021). V(t)
0
r
I(t) 0 Negative Electrode x
Rn Electrolyte
+
0
r
Rp
Separator
Positive Electrode
Ls
Lp
Ln
Current Collector
Current Collector
-
L
I(t)
Figure 1: Schematics of the SPMe model. 2.1 Solid-Phase Diffusion
In the SPMe, lithium-ion diffusion within the active material particles of positive (p) and negative (n) electrodes is governed by Fick’s second law in spherical coordinates: • To the best of our knowledge, this is the first work to integrate transfer learning within a phys-ics- ∂cs,k Ds,k ∂ 2 ∂cs,k = 2 r , k ∈ {n, p}, (1) informed SPMe framework for lithium-ion battery ∂t ∂rk rk ∂rk modeling.
∂cs,k jk • A transfer learning strategy is introduced, where a ∂cs,k = 0, −Ds,k = , (2) ∂rk rk =0 ∂rk rk =Rk F pretrained SPMe-PINN is adapted to a target battery to capture and adapt to battery-specific characteristics. where Rk is the particle radius, jk is the interfacial reaction flux, and F is Faraday’s constant. • Key electrochemical variables are incorporated as learnable parameters during fine-tuning, ena-bling 2.2 Electrolyte Dynamics adaptation to different chemistries and op-erating The electrolyte concentration dynamics are described conditions with simultaneous parameter estimaacross the negative electrode, separator, and positive tion.
electrode domains by
εe,k
∂ce,k ∂ce,k ∂ = Deff e,k ∂t ∂x ∂x
activation overpotentials obtained from Butler–Volmer kinetics, ∆ϕe is the electrolyte potential drop, and ∆ϕs is the solid-phase ohmic drop. The electrolyte potential drop (Moura et al., 2016) is expressed as:
1 − t+ I(t), k ∈ {n, p}, FLk + 0, k = s, (3)
where εe,k is the electrolyte volume fraction, ce,k is the electrolyte concentration, Deff e,k is the effective electrolyte diffusivity, t+ is the lithium-ion transference number, Lk is the thickness of region k, and I(t) is the applied current density. The source term is zero in the separator region (k = s) because no electrochemical reactions occur within the separator. The effective electrolyte diffusivity is corrected using the Bruggeman relation: b Deff e = De ε e ,
(4)
where b is the Bruggeman coefficient. Zero-flux boundary conditions are imposed at the current collectors ∂ce = 0, ∂x x=0
∂ce = 0, ∂x x=L
∆ϕelec =
0 (Ln + 2Ls + L p ) 2RT c I(t) + k f (1 − t+ ) ln Le , 2κ F ce (10)
and the solid-phase ohmic drop is I(t) L p Ln ∆ϕsolid = − + , 3 σ p σn
(11)
with κ denoting electrolyte conductivity, σn and σ p as the electrode conductivities (Marquis et al., 2019) and k f as the electrolyte thermodynamic factor that accounts for non-ideal electrolyte behavior. 3. Proposed Transfer Learning-based SPMe-PINN
In this section, the proposed transfer learning framework for the SPMe-based physics-informed neural network (SPMe-PINN) is presented. The objective is to effiwhere L is the total cell thickness. Flux and concentraciently transfer electrochemical knowledge learned from tion continuity are also assumed at the boundaries. a source battery to a target battery while preserving the electrochemical consistency imposed by the governing 2.3 Electrochemical Kinetics equations. The framework consists of two major stages The electrochemical reaction kinetics are modeled us- (see Fig. 2): (i) pretraining on a source battery and (ii) ing the Butler–Volmer relation. The exchange current fine-tuning on a target battery. density is expressed as Pretrained PINN model α Back-propagation i0,k = kk cαe cαs,k cmax − c , (6) s,k 𝒩𝒩𝜃𝜃 (inputs) s,k (5)
Inputs
where kk is the reaction-rate constant, cmax s,k is the max-
(8)
where xn and x p denote the negative and positive electrode stoichiometries. 2.4 Terminal Voltage The measurable terminal voltage at the current collectors is obtained by combining the equilibrium potentials, reaction overpotentials, electrolyte potential drop, and solid-phase ohmic losses:
𝑐𝑐𝑛𝑛̂
𝒩𝒩𝜃𝜃 (inputs)
Inputs r x t
Fine-tune
LossPDE
Loss
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐 2 , , 𝜕𝜕𝜕𝜕 𝜕𝜕∅ 𝜕𝜕∅2
∅ = r, x
Back-propagation Lossdata
Outputs 𝑐𝑐1̂
𝑐𝑐2̂
Freeze
�) MSE(𝑽𝑽, 𝑽𝑽
Transfer learning
…
U p = U p (x p ),
𝑐𝑐2̂
𝑉𝑉�1 𝒄𝒄� 𝑉𝑉�2 � 𝑽𝑽 𝑉𝑉�𝑛𝑛
…
Un = Un (xn ),
r x t
𝑉𝑉�1 𝒄𝒄� 𝑉𝑉�2 � 𝑽𝑽 𝑉𝑉�𝑛𝑛
…
where R is the universal gas constant and T is temperature. The open-circuit potentials of the electrodes are represented as nonlinear functions of stoichiometry
𝑐𝑐1̂
…
imum solid concentration, and α is the charge-transfer coefficient. The reaction overpotential is given by jk 2RT sinh−1 , (7) ηk = F 2i0,k
Lossdata
Outputs
𝑐𝑐𝑛𝑛̂
�) MSE(𝑽𝑽, 𝑽𝑽
LossPDE
Loss
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐 2 , , 𝜕𝜕𝜕𝜕 𝜕𝜕∅ 𝜕𝜕∅2
Figure 2: Illustration of the proposed transfer learning framework for the SPMe-PINN model. 3.1 Pretraining on Source Battery
In the pretraining stage, a PINN is trained using electrochemical data generated from a source battery. The PINN takes normalized temporal and spatial coordinates as inputs and predicts the electrochemical states V (t) = (U p −Un ) + (η p − ηn ) + ∆ϕelec + ∆ϕsolid , (9) (concentrations) of the battery at the surface, including the solid-phase lithium concentrations in the negative where U p and Un denote the positive and negative elec- and positive particles and the electrolyte concentration trode open-circuit potentials, η p and ηn represent the at the boundaries.
The normalized variables are defined as t r x tˆ = , r̂ = , x̂ = , (12) tmax Rk L while the normalized concentrations are expressed as cs,k ce ĉs,k = max , ĉe = . (13) cs,k ce,0
3.2 Fine-Tuning on Target Battery
Three neural networks are employed to approximate the electrochemical states for the negative electrode, positive electrode and electrolyte. The PDE residuals are evaluated at a set of collocation points sampled within the normalized computational domains. The residual loss is written as: N 1 pde ( j) 2 ( j) ( j) (14) ∥Rs,n ∥ + ∥Rs,p ∥2 + ∥Re ∥2 , L pde = ∑ N pde j=1
Θt ← Θs ,
where Rs,n , Rs,p , and Re denote the residuals of the negative solid-phase, positive solid-phase, and electrolyte PDEs, respectively. In addition to enforcing the governing equations, the normalized PINN formulation also satisfies the corresponding boundary and initial conditions for the solid-phase and electrolyte concentration dynamics at selected points. After PINN predicts solid-phase and electrolyte concentrations, these states are used to compute the terminal voltage through the SPMe voltage formulation. The predicted surface concentrations are used to evaluate the open-circuit potentials and exchange current densities, while the electrolyte concentration accounts for electrolyte polarization. The PINN is trained by minimizing a composite loss function consisting of PDE residual, boundary condition, initial condition, and voltage losses. The optimization problem during pretraining is formulated as: Θ∗s = arg min w pde L pde + wbc Lbc + wic Lic + wV LV , (15)
puted on target battery measurements, LPDE enforces the SPMe governing equations on the target domain, and λ is a weighting factor. To preserve the shared electrochemical knowledge learned during pretraining, selected layers of the pretrained network are frozen during fine-tuning. During fine-tuning, parameters associated with target-specific electrochemical properties are updated, while the shared electrochemical dynamics encoded within the pretrained model are preserved. In this work, the fine-tuned parameters are the solid-phase diffusivities of the negative and positive electrodes.
Θ
where:
Lbc =
1 Nbc ( j) ( j) 2 BCpred − BCtrue , ∑ Nbc j=1
In the second stage, the pretrained SPMe-PINN is adapted to a target battery with potentially different chemistry, operating conditions, or material properties. The pretrained parameters are first transferred to initialize the target model according to (19)
after which the network is further optimized using target battery data by minimizing the fine-tuning loss (t) (t) Lfine-tune = LPDE + λ Ldata ,
(20)
(t)
where Ldata = MSE(V, V̂) is the data fidelity term com(t)
3.4 Computational Efficiency and Model Benefits Compared with training from scratch, the proposed transfer learning can significantly reduce training time and enhance convergence stability. Additionally, by leveraging pretrained electrochemical knowledge, the model achieves improved prediction accuracy under different data scenarios.
(16) 4. Results and Discussion
In this section, the performance of the proposed framework is evaluated through a series of case studies (17) designed to assess its accuracy, robustness, and generalization capability. The analysis begins with validation 1 NV ( j) ( j) 2 LV = VPINN −Vre f , (18) on a source battery using the Chen et al. (2020) param∑ NV j=1 eter set (hereafter referred to as B1) to establish the Vre f denotes the reference terminal voltage obtained baseline performance of the model. The PINN archifrom the source battery data and Θs denotes the train- tecture consists of four hidden layers with 64 neurons able parameters of the pretrained model, including the per layer. This is followed by transfer learning studies neural-network weights, biases, and learnable electro- on different battery datasets, incorporating variations chemical kinetic parameters. Since the voltage computa- in cell configuration, operating conditions, and chemtion remains fully differentiable, the voltage loss is back- istry. Specifically, the Ecker et al. (2015) dataset (herepropagated through the SPMe voltage equations and after referred to as B2) is used to examine cross-domain subsequently through the predicted concentration fields, transfer within similar chemistry, while the Prada et al. enabling the network parameters and learnable electro- (2013) dataset (hereafter referred to as B3) is employed chemical parameters to be updated simultaneously dur- to evaluate cross-chemistry generalization. During fineing training. Through this process, the PINN learns tuning, the first two layers of the pretrained network generalized electrochemical dynamics that can be trans- are frozen to preserve the learned electrochemical representations while adapting the remaining parameters to ferred across different battery systems. 1 Nic ( j) ( j) 2 Lic = ∑ ICpred − ICtrue , Nic j=1
4.2
PyBaMM PINN surf Cs,n [mol/m3 ]
4
3
3.8 3.6 3.4
×104 PyBaMM PINN
2
1
3.2 0
1000
2000
0 0
3000
1000
Time [s]
(a) 5
PyBaMM PINN
4 3 2 1000
2000
3000
4.2
PyBaMM PINN
4.1 4 3.9 3.8 3.7 0
1000
Terminal Voltage [V]
PyBaMM PINN
3.5 3 2.5 0
1000
2000
3000
Time [s] Figure 3: Comparison of terminal voltage predictions obtained from the PyBaMM SPMe model and the SPMe-PINN framework for B1. 4.2 Cross-Domain Transfer To evaluate the transferability of the proposed framework across different cell configurations, we consider the B2 dataset, which is based on a Kokam cell with graphite/NMC chemistry and a pouch cell format. In contrast to the source battery, which is cylindrical, this introduces variations in cell geometry and electrochemical response, making it a suitable test case for transfer learning. In this stage, the pretrained SPMe-PINN model is adapted to the target battery by freezing some layers and fine-tuning the remaining network parameters. The model is evaluated under a 1C discharge condition. The terminal voltage prediction, shown in Fig. 4a, demonstrates strong agreement with the PyBaMM reference solution, although an increase in error is observed relative to the source case due to domain shift.
(c)
2000
3000
Time [s]
Time [s]
4
3000
(b)
×104
1 0
2000
Time [s]
Terminal Voltage [V]
The B1 dataset is based on the LG M50 graphite/NMC lithium-ion cylindrical cell. We consider a constant 1C discharge condition. The objective of this experiment is to verify that the proposed PINN architecture can accurately reproduce the electrochemical dynamics of the SPMe model prior to transfer learning. The terminal voltage predicted by the SPMe-PINN is compared against the reference SPMe solution obtained from PyBaMM. As shown in Fig. 3, the proposed model closely tracks the reference voltage profile across the entire discharge window. The prediction error remains consistently low (a root mean square error of 8.1e−4 ), indicating that the model successfully captures both transient and steady-state electrochemical behavior. This agreement demonstrates that the PINN is able to learn the dynamics governing the system, thereby providing a reliable pretrained model for subsequent transfer learning tasks.
Terminal Voltage [V]
4.1 Source Model Validation
Additionally, the model is used to estimate internal electrochemical states, specifically the surface lithium concentration at both electrodes. As illustrated in Fig. 4b and Fig. 4c, the predicted concentration profiles closely match the ground truth, confirming that the model retains its ability to capture internal dynamics after transfer.
surf Cs,p [mol/m3 ]
the target battery domain. The results include comparisons of terminal voltage and internal state predictions against reference solutions obtained from PyBaMM.
(d)
Figure 4: Comparison between the PyBaMM SPMe model and the transfer learning in the SPMe-PINN framework for B2 showing (a) terminal voltage prediction at a 1C discharge rate, (b) negative electrode solidphase lithium concentration, (c) positive electrode solidphase lithium concentration, and (d) terminal voltage prediction at a 0.5C discharge rate. To further assess the robustness, we evaluate the model under a different operating condition using the same B2 dataset. Specifically, a 0.5C discharge rate is applied for 1 hour. The results, shown in Fig. 4d, indicate that the proposed model maintains a high prediction accuracy for the terminal voltage. 4.3 Cross-Chemistry Transfer To evaluate generalization across different battery chemistries, we consider the B3 dataset, which corresponds to an LFP cell. This represents a more challenging transfer scenario due to the distinct electrochemical characteristics of LFP compared to NMC systems. The same transfer learning procedure is applied, with partial freezing of the pretrained network and fine-tuning on the target dataset. The model is evaluated under both 1C and 1.2C discharge conditions. The predicted terminal voltage are shown in Fig. 5a and 5b. Despite the significant change in chemistry and operating conditions, the model demonstrates strong predictive capability and successfully captures the key features of the LFP voltage response, indicating good generalization performance across different discharge rates.
3 2.5 2 1.5 1 0
1000
2000
Terminal Voltage [V]
Terminal Voltage [V]
PyBaMM PINN
3.5
3000
Time [s]
(a)
PyBaMM PINN
3.5 3 2.5
ment of experimental techniques for parameterization of multi-scale lithium-ion battery models. Journal of The Electrochemical Society 167(8), 080534.
2 1.5 1 0
500
Doyle, M., T. F. Fuller, and J. Newman (1993). Modeling of galvanostatic charge and discharge of the 1000 1500 2000 2500 lithium/polymer/insertion cell. Journal of the ElecTime [s] trochemical society 140(6), 1526–1533. (b)
Figure 5: Comparison between the PyBaMM SPMe Ecker, M., T. K. D. Tran, P. Dechent, S. Käbitz, A. Warnecke, and D. U. Sauer (2015). Parameterimodel and the proposed transfer learning SPMe-PINN zation of a physico-chemical model of a lithium-ion framework for the B3 showing (a) terminal voltage prebattery: I. determination of parameters. Journal of diction at 1C discharge rate and (b) terminal voltage The Electrochemical Society 162(9), A1836–A1848. prediction at 1.2C discharge rate. 4.4 Parameter Estimation To evaluate the parameter identification capability of the proposed framework, the solid-phase diffusivities are modeled as learnable parameters during training. As shown in Table 1, the predicted diffusivity values exhibit strong agreement with the corresponding ground-truth values, demonstrating the effectiveness of the model in predicting key electrochemical parameters. Table 1: Comparison between true and predicted electrode diffusivities. Battery B2 B3
Negative Electrode True Predicted 8.332e−15 8.341e−15 3.000e−15 3.013e−15
5. Conclusion
Positive Electrode True Predicted 2.981e−13 2.983e−13 5.900e−13 5.896e−13
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