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

arXiv: https://arxiv.org/abs/2511.04518 · HTML: https://mlsanity.com/paper/2511.04518/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 48 references that are in the map.
- First posted on arXiv 6 Nov 2025.

## Builds on (most cited first, 14 of 14)

- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Gaussian Process Regression constrained by Boundary Value Problems](https://mlsanity.com/paper/2012.11857/) (2020)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Efficient Hyperparameter Tuning for Large Scale Kernel Ridge Regression](https://mlsanity.com/paper/2201.06314/) (2022)
- [Effective Degrees of Freedom: A Flawed Metaphor](https://mlsanity.com/paper/1312.7851/) (2013)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Gaussian Process Regression for Inverse Problems in Linear PDEs](https://mlsanity.com/paper/2502.04276/) (2025)
- [Automatic Search Intervals for the Smoothing Parameter in Penalized Splines](https://mlsanity.com/paper/2205.15157/) (2022)
- [Linear ordinary differential equations constrained Gaussian Processes for solving optimal control problems](https://mlsanity.com/paper/2504.12775/) (2025)

## 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.