AI 中文总结
本文针对具有各向异性电导率的随机感应加热问题，提出结合SPD流形与NODE框架的CMNN代理模型，通过蒙特卡洛模拟量化不确定性，评估不同NODE积分方案以实现高效不确定性传播。
AI 中文摘要
感应焊接是实现由纤维增强热塑性复合材料制成的轻质一体化结构的核心工艺。从建模角度来看，感应焊接过程可近似为单向耦合的电磁与传热方程。实际中，电导率等材料参数因纤维排列偏差而存在显著差异，细观结构中的纤维-纤维接触受材料固结质量的支配。在宏观尺度上明确表征这种变异性，对捕捉感应加热过程所需的闭合电流回路至关重要。为解决该问题，本文引入一种随机材料模型，其尊重电导率张量的对称正定（SPD）性质，并分离尺度与取向不确定性，构成代理框架的基础。所得随机电导率模型首先用于通过大量蒙特卡洛模拟量化感应加热过程的不确定性，为感应电流及生成的温度场提供详细洞察。随后，为实现高效的不确定性传播，利用包含标记材料状态与温度场的模拟数据子集，训练感知SPD的代理模型。这些代理模型被构建为本构流形神经网络（CMNNs），其明确尊重底层SPD流形结构，并与神经常微分方程（NODE）框架集成以捕捉时间动态。本文评估并比较了多种NODE积分方案。
英文摘要
Induction welding plays a central role in enabling lightweight, integrated structures made from fibre-reinforced thermoplastic composites. From a modelling perspective, the induction welding process can be approximated by one-way coupled electromagnetic and heat-transfer equations. In practice, material parameters such as electrical conductivity vary significantly resulting from the deviations in the placement of fibres and hence the fibre-fibre contacts in the mesostructure are governed by consolidation quality of the material. Explicit representation of this variability on the macroscopic scale is essential to capture the closed current loops required for the induction heating process. To address this, a stochastic material model is introduced that respects the symmetric positive-definite (SPD) nature of the conductivity tensor and separates scaling and orientation uncertainties, forming the basis of a surrogate framework. The resulting stochastic conductivity model is first used to quantify the uncertainty in the induction heating process through extensive Monte Carlo simulations, providing detailed insight into the induced currents and the resulting temperature field. Subsequently, to enable efficient uncertainty propagation, SPD-aware surrogate models are trained with a subset of the simulation data, consisting of labelled material states and temperature fields. The surrogates are formulated as Constitutive Manifold Neural Networks (CMNNs) that explicitly respect the underlying SPD manifold structure and are integrated with a Neural Ordinary Differential Equation (NODE) framework to capture temporal dynamics. Several NODE integration schemes are evaluated and compared.
