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

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.

• The paper demonstrates that structured Monte Carlo error can be corrected using a transformer-based neural network for high-accuracy PDE solutions.
• It introduces PDEZoo, a large benchmark with 2 million instances covering diverse elliptic PDE families to ensure reproducible evaluations.
• The method effectively balances variance reduction and bounded bias, achieving robust generalization even on out-of-distribution PDE tasks.

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

Motivation and Problem Formulation

Efficient and accurate solution of elliptic PDEs is central to numerous domains including geometry processing, scientific computing, and simulation pipelines. Classical deterministic solvers (FEM, FDM) are often impractical for irregular geometries due to meshing requirements, while Monte Carlo solvers like Walk-on-Spheres (WoS) offer mesh-free, unbiased estimates but require prohibitive sampling budgets to achieve low-variance results. Learning-based neural solvers (e.g., PINNs, neural operators) deliver fast inference but suffer from bias, lack robustness under distribution shift, and are not reliable for general PDE types.

This paper proposes a hybrid PDE solver, MC$^2$, that leverages low-budget WoS estimators as noisy but structured proxies for the true solution and applies a transformer-based neural network correction in a single forward pass. The central insight is that finite-sample Monte Carlo error exhibits spatial structure that is consistent and learnable across diverse PDEs and geometries, making it amenable to supervised error correction.

Methodology: Structured Post-processing of Monte Carlo Outputs

MC$^2$ operates in the finite-compute regime: a WoS estimator at low budget $B$ yields $\hat{u}_B = u + \mathcal{E}_B$, where $\mathcal{E}_B$ is spatially-correlated, zero-mean noise. Rather than amplifying sampling or modifying solver internals, MC$^2$ treats the WoS output as a structured observation and regresses directly from it (along with the PDE source term $f$) to the ground truth $u$. The correction model is instantiated as a Swin-UNet transformer, capable of capturing long-range dependencies in Monte Carlo noise and adapting to diverse geometries.

Discretization is performed on uniform 256×256 grids, ensuring compatibility with procedural domain representation. The neural corrector is trained on pairs of low-budget WoS fields and analytic ground truth fields spanning the Laplace, Poisson, Yukawa, Biharmonic, and Helmholtz PDE families. Ablative studies confirm the importance of source-term conditioning and multi-budget training for generalization and precision.

Benchmark Construction: PDEZoo

To ensure reproducibility and promote standardized evaluation of PDE solvers in the finite-compute regime, the authors introduce PDEZoo, a large-scale benchmark comprising 2 million elliptic PDE instances. Each instance specifies explicit analytical solutions, manufactured source terms, diverse domain geometries via signed distance functions, and Monte Carlo solver trajectories at multiple budgets. PDEZoo includes five elliptic families and supports in- and out-of-distribution testing (with held-out families).

Key features:

• Exact analytic ground truth for all problems
• Diversity in geometry (eight primitives and composition)
• Multi-budget WoS outputs for tracking convergence and variance
• Controlled difficulty calibration (four difficulty tiers)
• Procedural generation for unlimited training sets and fixed test splits

Experimental Evaluation

MC$^2$ was evaluated against classical denoising methods, neural architectures (FNO, PINO), and diffusion-based solvers (CoCoGen, DiffusionPDE) using the 5,000-instance PDEZoo test split. MC$^2$0 takes a WoS estimator at budget $^2$1 as input, achieving the following:

• PSNR 44.90dB from $^2$2 input, outperforming raw WoS at $^2$3 (15.26dB) and matching WoS at $^2$4—a 1000x wall-clock compute reduction.
• Lowest MSE and LPIPS among learned and post-processing methods, with perceptually negligible error.
• Superior generalization: MC$^2$5 achieves high PSNR (34.88dB, 35.87dB) on out-of-distribution Biharmonic and Helmholtz families—tasks unreachable by baseline neural and diffusion solvers.

Scaling studies demonstrate that MC$^2$6 performance improves with training set size and WoS input budget but saturates well below the high-budget Monte Carlo accuracy curve. Ablations confirm that multi-budget training and conditioning on $^2$7 are critical, with significant PSNR drops when these are omitted.

Theoretical Implications: Bias-Variance Tradeoff

A detailed analysis of estimator bias-variance decomposition shows:

• WoS estimators are nearly unbiased but their variance decays at $^2$8, making accuracy expensive.
• Pure neural operators eliminate variance but induce persistent approximation bias and suffer under distribution shift.
• MC$^2$9 applies a Lipschitz contraction to the Monte Carlo noise, resulting in decreased variance and only bounded bias. The learned corrector is strictly better (in mean-squared error) than either a standalone WoS estimator or a neural operator, provided the same information is available.

Empirical evidence confirms the bias-variance improvement persists across both in-distribution and held-out PDE families.

Practical and Theoretical Implications

The MC$^2$0 paradigm advances a general recipe: leveraging the structured error in unbiased stochastic estimators (such as Monte Carlo solvers) via a learned correction, applied once and trained across the problem distribution. This approach offers exceptionally fast inference with robustness, combines the unbiasedness of Monte Carlo with data-driven variance reduction, and eliminates the need for slow iterative sampling or per-problem retraining.

PDEZoo fills a critical infrastructure gap for reproducible benchmarking and evaluation. The MC$^2$1 framework is immediately transferable to other unbiased estimators in scientific computing (e.g., path-traced rendering, particle filters, MCMC sampling), contingent on the existence of structured finite-sample error and reference solutions for supervision.

A current limitation is applicability to elliptic PDEs on domains amenable to Green's function machinery; extension to hyperbolic, parabolic, or strongly nonlinear PDEs would require different unbiased solvers.

Conclusion

MC$^2$2 establishes Monte Carlo error correction as a robust, efficient paradigm for solving elliptic PDEs, achieving high-fidelity solutions at a fraction of the traditional compute cost, and demonstrating strong generalization across diverse PDE families and geometries. PDEZoo provides the scalable, standardized infrastructure necessary for head-to-head algorithmic comparison and accelerating progress in scientific machine learning. MC$^2$3 closes most of the gap to high-budget Monte Carlo for elliptic PDEs and is poised as a general template for the broader class of unbiased stochastic solvers.