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

2025-11-06

AI Summary

This work addresses the two-dimensional wave equation with homogeneous Dirichlet boundary conditions. We propose the Boundary-constrained Ehrenpreis–Palamodov Gaussian Process (B-EPGP) surrogate model, and conduct a fair comparison against the Crank–Nicolson finite element method (CN-FEM) under strict equality of degrees of freedom. B-EPGP employs an exponential-polynomial basis derived from characteristic varieties, ensuring exact satisfaction of both the PDE and boundary conditions; its coefficients are estimated via penalized least squares. Numerical experiments demonstrate that, at identical DOF counts, B-EPGP achieves significantly lower spatiotemporal L² error and temporal maximum-in-time L² spatial error than CN-FEM—improving accuracy by up to two orders of magnitude. To our knowledge, this is the first integration of the Ehrenpreis–Palamodov fundamental principle into a Gaussian process framework with explicit boundary constraints, establishing a new mesh-free, physics-informed paradigm for surrogate modeling.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

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• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Compares B-EPGP surrogates with CN-FEM for solving 2D wave equations
• Introduces DoF matching protocol to ensure fair benchmarking across methods
• Evaluates accuracy improvements in space-time and maximum-in-time errors

Innovation

• B-EPGP uses exponential-polynomial bases from characteristic variety
• Penalized least squares estimates coefficients for exact PDE enforcement
• Degrees-of-freedom matching protocol ensures fair comparison accuracy