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

arXiv 2511.04518 ↗First posted 6 Nov 2025cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

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 L2-error and maximum-in-time L2-error in space than CN-FEM, improving accuracy by roughly two orders of magnitude.

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References · 14

  1. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  3. 2017Linearly constrained Gaussian processescited by 41
  4. 2022Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solverscited by 40
  5. 2018Algorithmic Linearly Constrained Gaussian Processescited by 20
  6. 2020Gaussian Process Regression constrained by Boundary Value Problemscited by 15
  7. 2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
  8. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  9. 2022Efficient Hyperparameter Tuning for Large Scale Kernel Ridge Regressioncited by 11
  10. 2013Effective Degrees of Freedom: A Flawed Metaphorcited by 5
  11. 2024Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equationscited by 2
  12. 2025Gaussian Process Regression for Inverse Problems in Linear PDEscited by 2
2 more
  1. 2022Automatic Search Intervals for the Smoothing Parameter in Penalized Splinescited by 1
  2. 2025Linear ordinary differential equations constrained Gaussian Processes for solving optimal control problemscited by 1

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