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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes MC^2, a single-pass Swin-UNet (SUNet) corrector that maps a low-budget Walk-on-Spheres PDE estimate to a high-fidelity solution, and releases PDEZoo, a 2M-instance elliptic PDE benchmark with analytic ground truth and multi-budget Monte Carlo trajectories.

Problem

Elliptic PDE solving under finite compute: Monte Carlo solvers like Walk-on-Spheres are unbiased but high-variance at low sample budgets, while learned solvers are fast but biased and brittle under distribution shift; there is also no standardized benchmark supporting finite-compute evaluation.

Approach

Treat a low-budget WoS estimate as a noisy observation of the true field and learn a single deterministic field-to-field neural correction (Swin-UNet) trained once across the PDE distribution, rather than per-instance training or iterative diffusion sampling.

Key results

• MC^2 from a B=8 WoS input reaches 44.10 dB average PSNR on the PDEZoo test set versus 15.26 dB for raw WoS at B=8 and 42.14 dB for WoS at B=16384.
• Best baseline FNO reaches 30.34 dB PSNR; CoCoGen and DiffusionPDE reach about 8.25 dB on the same averaged metric.
• MC^2 generalizes zero-shot to held-out Biharmonic and Helmholtz families at 34.88 and 35.87 dB PSNR.

Surprising findings

• PINO's physics regularization degrades performance below the raw B=8 WoS input on several families.
• Generic denoisers plateau below 27 dB on B=8 inputs while the learned corrector reaches about 44 dB.

Limitations

• Only applicable to PDEs that Walk-on-Spheres can solve (broadly elliptic problems on domains where Green's-function machinery applies).
• Evaluated only on 2D elliptic problems on a fixed 256x256 grid over [-1,1]^2.
• Some diffusion baselines are only evaluated on in-distribution families and cannot be assessed out-of-distribution.

Experimental setup

Datasets: PDEZoo

Benchmarks: PDEZoo

Metrics: PSNR SNR MSE LPIPS

Baselines: Walk-on-Spheres Gaussian filtering Total Variation denoising Non-Local Means FNO PINO CoCoGen DiffusionPDE

Builds on: Walk-on-Spheres Swin-UNet Fourier neural operator physics-informed neural operator EDM diffusion

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