用神经网络求解锂电池模型时，解决方程不稳和解错的问题。
Forward and Inverse Simulation of Pseudo-Two-Dimensional Model of Lithium-Ion Batteries Using Neural Networks
- 引入旁路项降低方程病态性，提升数值稳定性。
- 加入固相电势守恒律，避免参数估计失败。
- 适合做电池仿真与参数反演的研究者参考。
本文针对物理信息神经网络（PINN）在伪二维（P2D）锂电池模型正向与逆向模拟中因布特勒-沃尔默（BV）方程高非线性带来的挑战展开研究。BV方程中的双曲正弦项导致PINN损失函数的海森矩阵严重病态，难以求解。为此，我们提出一种旁路项，显著降低海森矩阵条件数，改善数值稳定性。此外，由于离子通量 $ j $ 幅值较小，常导致PINN收敛至错误解。我们证明，引入固相电势 $ ψ $ 的第二守恒律可有效防止此类收敛偏差，确保解的准确性。所提方法在正向模拟与逆向参数估计中均表现优异，实现高精度参数恢复与可靠解预测。
原文摘要 · Abstract (English)
In this work, we address the challenges posed by the high nonlinearity of the Butler-Volmer (BV) equation in forward and inverse simulations of the pseudo-two-dimensional (P2D) model using the physics-informed neural network (PINN) framework. The BV equation presents significant challenges for PINNs, primarily due to the hyperbolic sine term, which renders the Hessian of the PINN loss function highly ill-conditioned. To address this issue, we introduce a bypassing term that improves numerical stability by substantially reducing the condition number of the Hessian matrix. Furthermore, the small magnitude of the ionic flux \( j \) often leads to a common failure mode where PINNs converge to incorrect solutions. We demonstrate that incorporating a secondary conservation law for the solid-phase potential \( ψ\) effectively prevents such convergence issues and ensures solution accuracy. The proposed methods prove effective for solving both forward and inverse problems involving the BV equation. Specifically, we achieve precise parameter estimation in inverse scenarios and reliable solution predictions for forward simulations.
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