Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity

TL;DR

The paper addresses efficient and accurate PDE solvers by benchmarking a boundary-constrained Ehrenpreis--Palamodov Gaussian Process (B-EPGP) surrogate against Crank--Nicolson finite elements (CN-FEM) for the 2D wave equation with Dirichlet boundaries. It introduces a DoF-matching framework, constructs an operator-informed basis via the characteristic variety, and uses ridge regression with Generalized Cross-Validation to estimate coefficients, enabling exact PDE/boundary satisfaction and uncertainty quantification. Across two initial data scenarios, B-EPGP achieves two to three orders of magnitude lower space-time $L^{2}$-error under matched DoF, with rapid post-training evaluation. This work demonstrates that operator-informed GP surrogates can substantially outperform classical solvers in accuracy while providing probabilistic uncertainty and meshless flexibility, suggesting a practical path for multi-query PDE tasks and high-dimensional problems.

Abstract

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.

Figures (2)

• Figure 1:
• Figure 2: $2$-D contour plots of the solution with mollifier initial condition at time stamps ($t = 0.0, 0.25, 0.5, 0.75, 1.0$). Each row shows the reference solution (top), B-EPGP (middle), and CN-FEM approximation (bottom).