求解非线性偏微分方程的高斯过程方法
Gaussian Process Methods for Solving Nonlinear Partial Differential Equations
摘要: 非线性偏微分方程(NPDEs)的求解在流体力学、金融数学等领域中具有广泛的应用。鉴于NPDEs通常难以求得解析解，本文详细探讨了基于高斯过程的偏微分方程求解方法：GPPDE方法和TD-GPsolver方法。考虑到NPDEs的非线性系统复杂性，我们将GPPDE方法和TD-GPsolver方法扩展至两种代表性的非线性偏微分方程——Allen-Cahn方程和Cahn-Hilliard方程，并与物理信息神经网络方法(PINN)进行对比，评估这些方法在不同离散化条件下的表现。结果表明，TD-GPsolver方法在处理大规模离散化数据集时不仅展现了较高的计算效率和稳定性，而且能够维持较低的误差率，GPPDE方法在处理小规模数据集时能提供较高精度，确保了计算效率和精度的平衡。PINN方法在高离散点设置下能显著提升数值精度，但其较长的计算时间限制了其实用性。
Abstract: The solving of nonlinear partial differential equations (NPDEs) finds wide applications in fields such as fluid mechanics and financial mathematics. Given the difficulty in obtaining analytical solutions for NPDEs, this study extensively explores Gaussian process-based methods for solving partial differential equations: GPPDE method and TD-GPsolver method. Considering the complexity of nonlinear systems in NPDEs, we extend the GPPDE method and TD-GPsolver method to two representative nonlinear partial differential equations—Allen-Cahn equation and Cahn-Hilliard equation, and compare them with the Physics-Informed Neural Network (PINN) method to evaluate their performance under different discretization conditions. The results indicate that the TD-GPsolver method demonstrates high computational efficiency and stability when dealing with large-scale discretized datasets while maintaining a low error rate. The GPPDE method offers higher accuracy when handling small-scale datasets, ensuring a balance between computational efficiency and accuracy. Although the PINN method significantly improves numerical accuracy under high discretization settings, its longer computational time limits its practicality.
文章引用：张维康. 求解非线性偏微分方程的高斯过程方法[J]. 应用数学进展, 2024, 13(6): 2753-2760. https://doi.org/10.12677/aam.2024.136264

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