本研究探討物理知識導向神經網路在求解四階偏微分方程、特別是四階卜瓦松－比克曼方程上的應用。該模型涵蓋體積排斥、電荷關聯與介電極化效應，適用於模擬高濃度電解質中離子電位分佈。為驗證方法的穩定性與準確性，本研究分別在二維正方形定義域與三維球體定義域中進行數值模擬，並利用常數解析解作為基準，結果顯示模型可穩定收斂。模型實作採用一種深度學習框架 PyTorch 與圖形處理單元 (GPU) 加速，並使用 Sigmoid 與 Tanh 激活函數強化非線性表現，L-BFGS 最佳化器則提升在高階偏微分方程問題中的收斂效率。透過區域劃分與訓練點配置策略，本研究成功模擬 Mg$^{2+}$ 在電解質中的電位分佈，結果與理論模型一致，證實了 PINNs 處理高階偏微分方程的有效性與潛力。

This study investigates the application of physics-informed neural networks (PINNs) to the solution of fourth-order partial differential equations, with a particular focus on the fourth-order Poisson–-Bikerman equation (4PBik). This model incorporates steric effects, charge correlation, and dielectric polarization, making it suitable for simulating ionic potential distributions in highly concentrated electrolyte solutions. To verify the stability and accuracy of the proposed method, numerical experiments were conducted on both a two-dimensional square domain and a three-dimensional spherical domain. Constant analytical solutions were used as references, and the results demonstrated stable convergence of the neural network. The model implementation employs the deep learning framework PyTorch, accelerated by graphics processing units (GPUs). Sigmoid and Tanh activation functions were used to enhance the model’s nonlinear representation capability, while the L-BFGS optimizer improved convergence efficiency in high-order PDE problems. By dividing the domain into distinct regions and strategically allocating training points, the model successfully reconstructed the electrostatic potential distribution of divalent cations Mg$^{2+}$ in an electrolyte environment. The results are consistent with theoretical predictions, confirming the effectiveness and potential of PINNs in solving high-order partial differential equations.