## 1 Introduction
[PERSON] et al. [19] proposed a learning-based framework for online modeling of the distributed thermal process of soft-pack lithium-ion batteries under sparse sensing conditions, using KL decomposition for offline learning and achieving accurate temperature prediction through incremental updates. [PERSON] et al. [9] proposed a method combining nonlinear time domain transformation and spatiotemporal domain reconstruction, using local linear embedding for time transformation, extreme learning machine for modeling, and successful reconstruction of dynamic predictions, with the experimental results showing superior accuracy compared to traditional methods. [PERSON] et al. [20] proposed a sparse learning method based on physical information, effectively overcoming outlier influences and achieving robust DPS modeling, verified through the alternating direction method of multipliers optimization in complex experiments. [PERSON] et al. [21] introduced a new physics-informed machine learning method based on time/space separation, which integrates physical information and data for spatiotemporal modeling of DPS. The experimental results demonstrated its effectiveness. [PERSON] and [PERSON] [22] proposed a dual-scale incremental learning method for modeling complex time-varying DPS, which effectively handles dynamics at different scales by incrementally updating the spatial basis functions and temporal models. The experimental results verified the effectiveness of this method in the thermal process of a curing furnace. [PERSON] and [PERSON] [23] proposed a neural network method based on time-space separation for DPS modeling without process knowledge, achieved continuous space modeling through spatial basis functions and Gaussian process regression, and experimentally verified its effectiveness in catalytic reactions and battery thermal processes. Three-dimensional (3D) fuzzy systems have developed rapidly in the field of DPS modeling in recent years [24]. Compared with the traditional DPS modeling method based on time-space separation, the 3D fuzzy model naturally forms a framework of time-space separation and time-space integration, does not rely on model simplification, and possesses linguistic interpretability. [PERSON] et al. [25] proposed a time/space-separated 3D fuzzy modeling approach to model unknown nonlinear spatial distribution systems through input-output data. This approach uses the Mamdani fuzzy rule and KL decomposition to construct 3D fuzzy structures and a particle swarm optimization algorithm for parameter identification. Simulations and experiments have verified its effectiveness. [PERSON] et al. [26] proposed a 3D fuzzy modeling method based on clustering and support vector regression without model dimensionality reduction. The simulation results demonstrate the effectiveness of the proposed modeling method. However, most 3D fuzzy system-based modeling approaches rely on historical data for offline modeling. When the dynamic characteristics of the system change, such offline models based on historical data may not be able to adapt to the current system conditions, leading to a gradual decrease in model accuracy. Based on the literature review presented above, the main research gaps can be summarized as follows: Existing distributed parameter system modeling methods, such as those based on KL decomposition and time-space separation, although effective in linear or weakly nonlinear systems, have obvious limitations when facing strong nonlinear and time-varying systems. The KL decomposition method cannot effectively handle strong nonlinearities or time-varying dynamics due to its inherent linear characteristics, and the time-space separation method usually relies on experience or prior knowledge to construct spatial basis functions, which limits its application in complex industrial processes. In addition, most data-driven modeling methods rely on historical data for offline modeling and lack online learning capabilities.