Model-based approaches rely on mathematical representations of battery electrochemical processes. The Kalman Filter (KF) is widely used due to its interpretability [15,16], but is limited by linear assumptions. To address this, nonlinear variants such as the Extended Kalman Filter (EKF) [17], Unscented Kalman Filter (UKF) [18], and Particle Filter (PF) [19] were introduced. While the PF excels in modeling highly nonlinear systems, this accuracy comes at a significant computational cost, with a complexity of approximately O(N) per time step for the number of particles N, hindering scalability [20].

Data-driven methods emerged to model complex patterns without requiring detailed physical knowledge [21]. Traditional machine learning algorithms such as Gaussian Process Regression (GPR) [8], Support Vector Machines (SVM) [10], Fuzzy Logic [22], and Random Forests [9] have demonstrated robust performance but often depend on manual feature engineering. Deep learning models, particularly Recurrent Neural Networks (RNNs) [23] and Long Short-Term Memory (LSTM) networks [24], automated this process and improved accuracy. However, their sequential nature, with a time complexity of O(L) for a sequence of length L [25], makes them inefficient for modeling long-range dependencies. The Transformer architecture [14] marked a significant advancement with its parallel self-attention mechanism. However, this introduced a critical bottleneck: a time and memory complexity of O(L2), making the standard Transformer computationally prohibitive for the very long time-series data required in battery lifecycle prognostics, which is a core task for a DT. Efficient architectures like the Informer [26], which reduces complexity to O(Llog⁡L), became necessary to make true Long-Sequence Time-series Forecasting (LSTF) feasible.

A recent innovation that enhances robustness, particularly in multivariate tasks, is Channel Independence. This approach processes each variable independently and has shown strong performance in various backbones, including CNNs [27], linear models [28], and Transformers [29,30].

Hybrid approaches combine the strengths of both paradigms [12]. For instance, Shi [31] used a KF for pre-processing before an Transformer, achieving a 25% improvement in Mean Squared Error. However, a critical analysis reveals two persistent limitations. First, many existing hybrid models create a significant computational bottleneck by naively combining two computationally expensive models, such as a standard Transformer (O(L2)) and a Particle Filter (O(N)). Second, the vast majority, including the aforementioned study, remain focused on univariate targets such as SOH or State of Charge (SOC) [13,32].

The Digital Twin (DT) paradigm, originally introduced by NASA [33], aims to create a real-time, virtual replica of a physical battery system for predictive diagnostics and optimized control [34,35]. However, a high-fidelity DT, often integrated into EVs [1], ESSs [4], and smart grids [36] to enhance reliability [3], imposes stricter requirements that reveal limitations in the prognostic models described above.

First, a true DT requires a comprehensive, multivariate understanding of the battery’s state. As noted, most studies focus on univariate predictions, a critical shortcoming for a holistic system view. Second, the real-time nature of a DT makes computational efficiency paramount, as high-frequency data updates create significant computational demands [37]. The quadratic complexity of standard Transformers, for example, is often too slow. Finally, the reliability of a DT depends on its physical plausibility. The battery’s operational complexity makes accurate modeling difficult [38], and purely data-driven models, even efficient ones, can produce physically unrealistic predictions, reducing the reliability of AI-driven control strategies [39].

The preceding review highlights a clear research gap for a framework that can satisfy the demanding requirements of a practical battery DT. This gap can be summarized by two key needs: 1.

Computational efficiency for LSTF: While standard Transformers are too slow (O(L2)), efficient variants like the Informer exist but have been seldom utilized in hybrid frameworks for battery prognostics.

To address these specific gaps, this study makes the following contributions:

We propose a multivariate time-series prediction model that extends beyond SOH estimation, enhancing the scope and interpretability of battery digital twins.

This study addresses the Long Sequence Time-series Forecasting (LSTF) problem. Using a fixed-size sliding window approach, the objective is to predict a sequence of Ly future values given an input sequence of length Lx at time t: (1)Xt={xt(1),…,xt(Lx)|xt(i)∈Rdx},Yt={yt(1),…,yt(Ly)|yt(i)∈Rdy}where Lx is the length of the input sequence, and dx and dy are the dimensions of the input and output sequences, respectively. LSTF problems typically involve long prediction horizons (large Ly), and in this study, a multivariate case is considered where dy≥1.

Multivariate time-series data in batteries consist of multiple independent time-series channels. Given a lookback window: (2)X={x1,x2,…,xC|xc∈RL}where C represents the total number of channels and L is the length of the observation window. Each univariate time-series of length L for channel i is represented as: (3)x1:L(i)=(x1(i),…,xL(i)),i=1,…,C

Each input sequence (x1,…,xL) is split into C univariate time-series, where each individual sequence x(i) is independently fed into the same backbone model. The backbone model then generates the corresponding predictions: (4)x^(i)=(x^L+1(i),…,x^L+T(i))where T>0 denotes the length of the prediction window.

In this study, the input variables are set as X={Vmean,Tmax,Tavg,Tmin,QCharge,QDischarge,IR,SOH}.

Each univariate time series of length L starting from time index 1 is denoted as V1:L=(V1,…,VL),…,SOH1:L=(SOH1,…,SOHL). Each univariate time series is independently input into the same backbone. Finally, the backbone provides prediction results V^=(V^L,…,V^L+T),…,SOH^=(SOH^L,…,SOH^L+T).

The Particle Filter (PF) is an algorithm used for state estimation in nonlinear and non-Gaussian systems. In this study, the PF is applied as a preprocessing step to incorporate the physical characteristics of the input data and filter out noise before it is fed into the deep learning model. The main steps of the Particle Filter algorithm, as applied in our framework, are detailed below.

First, a set of N weighted particles {xt(i),wt(i)}i=1N is used to approximate the posterior probability density. Each particle xt(i) is randomly initialized, and the initial weights are uniformly assigned as w0(i)=1/N.

The state of the system is governed by a state transition model, generally defined as: (5)xt=f(xt−1,ut−1)where f(⋅) is the state transition function and ut is the process noise. To specifically model the complex, nonlinear degradation path of a Li-ion battery, we define f(⋅) as a Gaussian Mixture state transition model. This function can approximate the multi-modal behaviors seen in different phases of a battery’s lifecycle: (6)f(x)=a⋅exp⁡(−((x−b)⋅c)2)+d⋅exp⁡(−((x−e)⋅f)2)where the parameters {a,b,c,d,e,f} are hyperparameters tuned to model characteristic degradation patterns. The measurement model maps the system state to the observed sensor measurements yt: (7)yt=h(xt,vt)where h(⋅) is the measurement function and vt represents measurement noise.

The PF algorithm then proceeds iteratively. The prediction step propagates each particle forward using the system dynamics. The posterior probability density function p(xt|y1:t) is estimated via the particle approximation: (8)p(xt|y1:t)≈∑i=1Nwt(i)δ(xt−xt(i))

The update step adjusts the particle weights based on the new observation yt. Using importance sampling, the new weight is computed as: (9)wt(i)∝wt−1(i)⋅p(yt|xt(i))⋅p(xt(i)|xt−1(i))q(xt(i)|x0:t−1(i),yt)where q(⋅) is the importance density function. The weights are then normalized to sum to one: w^t(i)=wt(i)/∑j=1Nwt(j).

To prevent particle degeneracy where a few particles have all the weight, a resampling step is performed. This step redistributes the particles, eliminating those with low weights and replicating those with high weights, typically resetting all weights to wt(i)=1/N.

Finally, the estimated state at time t is obtained by the weighted average of the resampled particles: (10)x^t=∑i=1Nwt(i)xt(i)

In this study, the Particle Filter is applied to the input time-series data X={V1:L,…,SOH1:L}. After the final estimation step, a refined data sequence, denoted as X′={V^,…,SOH^}, is obtained. This processed sequence X^′ is then utilized as the input for the encoder of the Informer model, providing it with a noise-reduced and physically constrained representation of the battery’s state.

The processed data, with filtered noise and enhanced physical characteristics, is then fed into the Informer-based data-driven model. The core mechanisms of Informer used in the proposed method are outlined below.

Model Input The output Xen=X′={V′,…,SOH′} obtained from the Particle Filter is combined with the positional encoding matrix PE(pos,j) to create the encoder input sequence Xen′: (11)Xen′=Xen+PE(pos,j)

Subsequently, Xen′ is passed to encoder layers and distilling processes.

ProbSparse Self-attention The Self-attention mechanism in Informer exploits the long-tail distribution property of self-attention scores in Transformers. Instead of computing all attention scores, ProbSparse Self-attention selects only the most dominant queries, reducing computational complexity: (12)Attention(Q,K,V)=softmax(Q¯KTd)Vwhere Q, K and V represent the query, key, and value matrices, respectively, and d is scaling factor. Q¯ denotes a sparsified set of dominant queries. Kullback-Leibler divergence is used to distinguish important queries, and only the top u queries are selected for Q based on sparsity measurement criteria. This enables the creation of different sparse query-key pairs for each head in the multi-head attention mechanism, preventing significant information loss.

Encoder input Xen′ is computed as: (13)Q=WQXen′,K=WKXen′,V=WVXen′where WQ,WK,WV are learnable weight matrices. The output Xen′′ is obtained after ProbSparse Self-Attention and a feed-forward network. (14)Multi−Attention(Q,K,V)=Concat(h1,…,hH)where hi denotes the ProbSparse self-attention operation of the ith head.

Encoder’s Distilling Operation The distilling operation in the encoder filters out important information from the input sequence. It assigns higher weights to dominant features, reducing the length of the input sequence by half and decreasing spatial complexity. The distilling process propagates from the jth to the j+1th layer and is computed as follows: (15)Xj+1t=MaxPool(ELU(Conv1d([Xjt]AB)))where [⋅]AB includes key operations within the attention block and Multihead-ProbSparse self-attention, Conv1d represents a 1D convolution operation, ELU is the activation function, and MaxPool is a max-pooling layer with stride = 2, which downsamples Xt by half. This ensures that dominant features are prioritized, allowing the next layer to form a focused self-attention feature map.

Decoder’s Generating Structure The output is obtained using the decoder’s generating structure, which reduces decoding time and prevents cumulative error propagation during the prediction period. The start token is a sequence of length Ltoken selected from the input sequence. The decoder input is represented as follows: (16)Xdet=Concat(Xtokent,X0t)∈R(Ltoken+Ly)×dmodelwhere Xtokent∈RLtoken×dmodel is the start token, and X0t∈RLy×dmodel is a placeholder for the target sequence, initialized with scalar values of zero. The masked multi-head attention and ProbSparse self-attention are applied, with the masked dot product set to −1, ensuring that each position does not attend to future positions. Finally, a fully connected layer generates the final output.

In this study, the decoder input is defined as follows: (17)Xdet=Concat(Xtokent,X0t), (18)Xtoken=Vtoken,…,SOHtoken,X0={V0,…,SOH0}where {Vtoken,…,SOHtoken} are the label sequences for each variable, and {V0,…,SOH0} are placeholders for each variable. Adding positional encoding PE(pos,j), the decoder input sequence Xde′ is generated as: (19)Xde′=Xde+PE(pos,j)

Subsequently, Xde′ is passed to the decoder layer. Unlike the encoder layer, the decoder layer applies masked multi-head ProbSparse self-attention to prevent referencing future time steps. This is defined as: (20)Multi−Attention(Q~,K~,V~)=Concat(h1,…,hH)where Q~ represents the masked queries, K~,V~ are obtained from the encoder layer, and hi denotes the masked ProbSparse self-attention operation of the i-th head. The output of masked multi-head ProbSparse self-attention passes through a feed-forward network to yield the final decoder output X^=Xde′′, X^={V^,…,SOH^}.

The performance of a battery gradually deteriorates over time, potentially preventing it from fulfilling its expected operational lifespan. Therefore, monitoring the State of Health (SOH) of a battery is essential [40]. SOH is defined as the ratio of the currently available capacity to the initial capacity, serving as an indicator of battery degradation and overall health status over time: (21)SOH=CpresentCinitial×100where Cpresent represents the currently available capacity, and Cinitial denotes the initial capacity of the battery. Typically, when SOH decreases to 80%, the battery is considered unsuitable for further use.

The dataset used in this study is obtained from a publicly available dataset provided by MIT [41]. It consists of 124 lithium iron phosphate (LFP) battery cells, which have been cycled under 72 different operating conditions. Among them, 10 battery cells were selected for evaluation of the proposed method. The nominal capacity of these cells is 1.1 Ah, and the batteries were charged using a two-step fast-charging protocol. This charging protocol follows the format C1(Q1)-C2, where C1 and C2 are the first and second constant current (CC) charging phases, respectively, and Q1 represents the state of charge (SOC; %) at which the current transition occurs. The second current phase is terminated at 80% SOC, after which the cells are charged using a 1-C CC-CV mode. A 4-C constant current discharge is applied until the battery reaches its lower cutoff voltage. All battery tests were conducted in a temperature-controlled chamber at 30∘C. The selected battery cells and their respective charging/discharging policies are shown in Table 1.

The dataset provides detailed records of battery degradation trends over different cycles. Fig. 2 illustrates the aging trends of battery variables over increasing cycles. As the cycle count increases, distinct patterns can be observed across different variables. For instance, temperature-related variables tend to shift toward higher values as the cycle count increases. This indicates that, as the battery degrades, it reaches a higher surface temperature more rapidly during operation. Such trends provide valuable insights into battery dynamics and degradation mechanisms, which can be utilized for battery state estimation and predictive maintenance.

Fig. 3 demonstrates how each battery variable evolves over cycles until the SOH reaches 80%. These observed trends highlight the progressive degradation of lithium-ion batteries, which is crucial for developing accurate SOH estimation models.

This study analyzed data using the battery dataset provided by MIT, which includes cycle-level information on Qcharge (charge capacity), Qdischarge (discharge capacity), IR (internal resistance), Tavg (average temperature), Tmin (minimum temperature), Tmax (maximum temperature), V (mean voltage per cycle), and SOH (State of Health).

This study was conducted using data from 10 distinct battery cells. To ensure a rigorous evaluation of the model’s generalization capabilities and to prevent any data leakage, we employed a 10-fold cross-validation (CV) scheme based on the battery cell IDs.

Specifically, in each of the 10 folds, one unique battery cell was designated as the test set, while the remaining 9 cells were used for training and validation (maintaining a 6-cell training/3-cell validation split ratio within these 9 cells). This process was repeated 10 times, ensuring that each cell served as the test set exactly once. This cell-level, leave-one-out cross-validation approach guarantees that the model is always evaluated on entirely unseen cells, fundamentally preventing data leakage and providing a robust assessment of generalization performance by capturing variance across different test cells.

The final performance metrics reported in Section 5.1 (Table 2) represent the mean and standard deviation calculated across these 10 folds.

To validate the effectiveness of the proposed method, we selected Transformer, DeepVAR, DLinear, and NLinear as benchmark models. All models were trained and evaluated under the same experimental settings, including the 10-fold CV procedure, to ensure a fair comparison.

To evaluate the performance of the proposed CIPF-Informer model, we compared it with state-of-the-art models (Transformer, DeepVAR, DLinear, NLinear) across various prediction lengths (16, 32, 64). As detailed in Section 4.3, we employed a 10-fold cross-validation scheme to ensure a robust evaluation. The results, summarized in Table 2, represent the mean and standard deviation (μ±σ) across the 10 folds.

As shown in Table 2, the proposed model, CIPF-Informer, consistently achieved the lowest mean MSE and MAE values and the highest mean R2 scores across all prediction lengths. For a prediction length of 16, the proposed model achieved a mean MSE of 0.116, which represents a significant improvement of 22.1%, 38.9%, 74.2%, and 44.2% compared to the mean MSE of Transformer (0.149), DeepVAR (0.190), DLinear (0.449), and NLinear (0.208), respectively. Similarly, for the longest prediction horizon of 64, its mean MSE of 0.221 represents improvements of 22.7%, 25.3%, 62.7%, and 40.6% over Transformer (0.286), DeepVAR (0.296), DLinear (0.592), and NLinear (0.372), respectively.

Fig. 4 presents a plot illustrating the performance of the proposed method for each variable.

Table 3 and Fig. 5 present the variable-specific prediction performance of the proposed method compared to state-of-the-art baselines (Transformer, DeepVAR, DLinear, and NLinear) under a prediction length of 16.

The evaluation metrics used are MSE, MAE, and R2, and the analysis is conducted across eight key variables: QDischarge, QCharge, IR, Tavg, Tmin, Tmax, V, and SOH. As shown in the table, the proposed method consistently outperforms the baseline models across all variables and evaluation metrics. For instance, in the QCharge variable, the proposed model achieves an MSE of 0.010, which represents improvements of 86.7%, 92.8%, 97.1%, and 44.4% compared to Transformer (0.075), DeepVAR (0.139), DLinear (0.339), and NLinear (0.018), respectively. Similarly, for the Tmax variable, CIPF-Informer achieves an MSE of 0.163, outperforming Transformer (0.266) and DLinear (0.474) by 38.7% and 65.6%, respectively, highlighting the model’s superior capability in temperature-related predictions. In the SOH and QDischarge variables, the proposed model achieves performance comparable to NLinear in terms of MSE, but still shows more than 50% improvement compared to Transformer, DeepVAR, and DLinear. Overall, the proposed model demonstrates MSE-based performance gains ranging from 3.7% to 97.1% across all variables, clearly validating the robustness and efficiency of CIPF-Informer in multivariate battery state prediction tasks.

To formally validate the statistical significance of these improvements, we conducted a series of paired t-tests on the 10 averaged fold results (one for each fold) for the MSE metric. The Bonferroni correction was applied to account for multiple comparisons, setting the significance level at α=0.0125 (0.05/4). For the prediction length of 16, the results confirmed that the CIPF-Informer is statistically superior to all four baseline models, with all p-values falling well below the adjusted significance level (all p < 0.0125). The specific p-values for each comparison are summarized in Table 4.

These results, supported by statistical significance testing, demonstrate that CIPF-Informer robustly outperforms all baseline models across all prediction horizons. In particular, the model maintains stable and superior performance in long-term forecasting, demonstrating its effectiveness with significantly less performance degradation compared to existing methods.

Our proposed framework operates within a DT ecosystem composed of two main domains, as shown in Fig. 7: a Physical System (the real-world asset, including the battery pack, sensors, and the Onboard BMS) and a Virtual Digital Twin (a cloud-based computational environment containing Cloud Storage, the CIPF-Informer predictive engine, and a Simulation and Decision Logic module). The true value of the DT is realized through a continuous, bidirectional feedback loop between these physical and virtual twins.

This process begins with the Physical-to-Virtual (‘Real-time data’) flow, where the Onboard BMS collects real-time sensor data and transmits it to the Cloud Storage. This allows for early detection of degradation patterns and anomalies [42] and provides the CIPF-Informer model with the necessary information for both real-time predictions and periodic retraining to adapt to the battery’s aging characteristics. The loop is completed by the Virtual-to-Physical (‘Control’) flow, which is the action-oriented part where intelligence from the virtual twin optimizes the operation of the physical asset. For instance, in an intelligent charging scenario, the CIPF-Informer might predict that a standard fast-charging profile will cause excessive future temperatures. The Simulation and Decision Logic module would then determine an optimal, safer charging current to prevent thermal risks [43] and even assess risks for extreme conditions like thermal runaway [44]. This is translated into a Control command, which is sent from the cloud back to the Onboard BMS to adjust the charging process in real-time.

A critical aspect for practical deployment is the model’s computational cost. All experiments were conducted on a single NVIDIA RTX 3090 GPU. The average training time for the proposed CIPF-Informer was approximately 42.7 min per epoch. While this introduces a significant overhead compared to a non-PF baseline, we argue it represents a highly favorable trade-off between cost and performance, as the PF’s contribution to accuracy and stability is substantial (see Section 5.2). In terms of inference, a single prediction can be processed in approximately 76.2 milliseconds, which is sufficient for many near-real-time monitoring applications in a cloud-based DT.