用精确边界条件的高斯过程模型，比传统有限元法更准两倍。
Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity
- 用特征流形导出的指数多项式基函数，精确满足方程和边界条件。
- 在相同自由度下，空间-时间L²误差和最大瞬时误差均低两个数量级。
- 适合追求高精度数值解的计算物理与工程仿真研究者。
我们提出一项新的基准测试，比较边界约束的埃伦普雷伊斯-帕拉莫多夫高斯过程（B-EPGP）代理模型与经典有限元方法结合克兰克-尼科尔森时间积分（CN-FEM）求解二维波动方程（齐次狄利克雷边界条件）的性能。B-EPGP利用从特征流形导出的指数多项式基函数，精确满足微分方程和边界条件，并通过惩罚最小二乘法估计系数。为确保跨范式公平比较，引入自由度（DoF）匹配协议。在匹配自由度条件下，B-EPGP始终优于CN-FEM，在空间-时间L²误差和最大瞬时空间L²误差上均提升约两个数量级。
原文摘要 · Abstract (English)
We present a new benchmarking study comparing a boundary-constrained Ehrenpreis--Palamodov Gaussian Process (B-EPGP) surrogate with a classical finite element method combined with Crank--Nicolson time stepping (CN-FEM) for solving the two-dimensional wave equation with homogeneous Dirichlet boundary conditions. The B-EPGP construction leverages exponential-polynomial bases derived from the characteristic variety to enforce the PDE and boundary conditions exactly and employs penalized least squares to estimate the coefficients. To ensure fairness across paradigms, we introduce a degrees-of-freedom (DoF) matching protocol. Under matched DoF, B-EPGP consistently attains lower space-time $L^2$-error and maximum-in-time $L^{2}$-error in space than CN-FEM, improving accuracy by roughly two orders of magnitude.
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