MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving

Ethan Hsu; Hong Meng Yam; Ivan Ge

cs.LGMay 10, 2026

MC $^2$ : Monte Carlo Correction for Fast Elliptic PDE Solving

Authors: Ethan Hsu, Hong Meng Yam, Ivan Ge

Organizations: Stanford University

Abstract

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC $^2$ }, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC $^2$ matches the accuracy of solutions using over $1000\times$ more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC $^2$ } and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs $\sim$ \textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.

Explore similar work

Error-Conditioned Neural Solvers
Jun 25, 2026Haina Jiang, Liam Wang, Peng-Chen Chen +4Neural Solvers Partial Differential Equations
Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing
Aug 10, 2026Konrad Kleinberg, Thomas KruseNeural-Network Approximation Stochastic Differential Equations
MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions
May 12, 2026Zichuan YangPartial Differential Equation Benchmarks Partial Differential Equations

Jun 25, 2026 · cs.LGJ/K move · Enter open · S save

Error-Conditioned Neural Solvers

Haina Jiang, Liam Wang, Peng-Chen Chen, Min Seop Kwak+3

University of Michigan · KAIST AI · Los Alamos National Laboratory

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

10\times

on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Neural Solvers Partial Differential Equations

Open paper Ask about it PDF

Abstract

Explore similar work

Error-Conditioned Neural Solvers

Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing

MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions