PDEZoo: 2D Elliptic PDE Benchmark

• PDEZoo is a benchmark for 2D elliptic PDE boundary-value problems on [-1,1]^2 with analytic ground truth and Monte Carlo Walk-on-Spheres approximations.
• It employs a method-of-manufactured-solutions pipeline across five elliptic families to derive exact forcing terms and boundary conditions, ensuring numerical accuracy.
• The benchmark supports finite-compute studies through multi-budget solver trajectories, enabling direct accuracy–compute comparisons and training of hybrid solvers like MC².

Searching arXiv for PDEZoo and closely related benchmark papers to ground the article. PDEZoo is a large-scale benchmark for elliptic PDE boundary-value problems in $2$D. Each instance is a complete problem $(\Omega,\mathcal{E},f,g,u)$ consisting of a domain $\Omega$, an elliptic operator family $\mathcal{E}$, a forcing term $f$, a Dirichlet boundary condition $g$, an analytic solution $u$, and multi-budget Monte Carlo Walk-on-Spheres approximations $\{\hat u_B\}$. It was introduced together with the hybrid solver MC$^2$, and is described as “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” (Hsu et al., 10 May 2026).

1. Scope and defining problem class

PDEZoo is organized around elliptic PDE boundary-value problems on bounded domains $\Omega\subset\mathbb{R}^2$, with all benchmark instances defined on subsets of $(\Omega,\mathcal{E},f,g,u)$0. The benchmark’s general tuple is

$(\Omega,\mathcal{E},f,g,u)$1

where $(\Omega,\mathcal{E},f,g,u)$2 is known analytically, $(\Omega,\mathcal{E},f,g,u)$3 is derived from $(\Omega,\mathcal{E},f,g,u)$4, and $(\Omega,\mathcal{E},f,g,u)$5. The paper presents, for context, the broader elliptic template

$(\Omega,\mathcal{E},f,g,u)$6

while emphasizing that PDEZoo itself uses five explicit elliptic families with Dirichlet boundary conditions (Hsu et al., 10 May 2026).

The benchmark is restricted to $(\Omega,\mathcal{E},f,g,u)$7D elliptic problems, and this restriction is central to its design. A stated motivation is that existing PDE benchmarks largely emphasize time-dependent parabolic or hyperbolic systems on fixed rectangular grids, whereas PDEZoo targets elliptic problems with irregular geometries, analytic ground truth, and explicit finite-compute solver trajectories. A plausible implication is that PDEZoo is intended not merely as a dataset of converged solutions, but as infrastructure for studying solver behavior under compute constraints (Hsu et al., 10 May 2026).

2. Elliptic families and manufactured solutions

PDEZoo covers five elliptic PDE families. For all of them, the analytic solution is first constructed, after which the forcing is derived exactly from the relevant operator. The boundary condition is then obtained by restriction to $(\Omega,\mathcal{E},f,g,u)$8, so $(\Omega,\mathcal{E},f,g,u)$9 (Hsu et al., 10 May 2026).

Family	PDE	Derived forcing or parameter
Laplace	$\Omega$0	$\Omega$1
Poisson	$\Omega$2	$\Omega$3
Yukawa	$\Omega$4	$\Omega$5, $\Omega$6
Biharmonic	$\Omega$7	$\Omega$8
Helmholtz	$\Omega$9	$\mathcal{E}$0, $\mathcal{E}$1

The benchmark uses a method-of-manufactured-solutions pipeline. For Poisson, Yukawa, Biharmonic, and Helmholtz, $\mathcal{E}$2 is assembled from a general smooth function family. For Laplace, $\mathcal{E}$3 is assembled from harmonic atoms, so $\mathcal{E}$4 analytically. The construction samples a random number of atoms

$\mathcal{E}$5

then forms

$\mathcal{E}$6

The general pool contains 20 atom types, and the harmonic pool contains 12 harmonic atoms. “Hard atoms” are over-sampled using hard_atom_extra = 4, which increases their probability from roughly $\mathcal{E}$7 to approximately $\mathcal{E}$8 (Hsu et al., 10 May 2026).

This design makes the forcing $\mathcal{E}$9 exact with respect to the benchmarked PDE family. It also separates benchmark error from reference-solver discretization error, because the “ground truth” field is obtained by direct evaluation of an analytic expression rather than by numerically solving the PDE (Hsu et al., 10 May 2026).

3. Geometry, signed distance functions, and domain composition

A distinctive feature of PDEZoo is its geometry model. Domains are represented by signed distance functions (SDFs) satisfying

• $f$0 inside the domain,
• $f$1 on the boundary,
• $f$2 outside the domain (Hsu et al., 10 May 2026).

PDEZoo defines 8 primitive domain families: disk, square, rectangle, ellipse, annulus, triangle, hexagon, and stadium. On top of these primitives it builds composed domains using Boolean operations. The SDFs for the compositions are

$f$3

$f$4

$f$5

The paper states that these composed SDFs are conservative underestimates of the true boundary distance, which is crucial for the correctness of Walk-on-Spheres because every WoS ball must remain entirely inside $f$6 (Hsu et al., 10 May 2026).

The procedural generator samples either a primitive or a composed domain. For the training distribution, each primitive is selected with probability $f$7, while composed domains are selected with probability $f$8. Compositions use two primitives drawn from disk, rectangle, ellipse, and triangle, with random parameters, rotations, offsets, and one of union, intersection, or difference (Hsu et al., 10 May 2026).

Instance generation is filtered by 8 quality filters, with up to 40 retries. These include finite-value checks for $f$9 and $g$0, domain non-degeneracy, a median-SDF threshold for composed domains, amplitude constraints on $g$1 and $g$2, bounded boundary values, and a requirement that all ground-truth values be finite on the $g$3 evaluation grid. This suggests that PDEZoo is engineered not only for diversity, but also for numerical regularity across a very large procedural corpus (Hsu et al., 10 May 2026).

4. Multi-budget Walk-on-Spheres trajectories and dataset organization

PDEZoo discretizes all fields on a fixed $g$4 grid over $g$5. Its canonical stochastic solver is Walk-on-Spheres (WoS), described as mesh-free, geometry-agnostic, and unbiased up to a small $g$6-shell termination bias. The WoS configuration uses

• grid $g$7,
• shell threshold $g$8,
• max steps per walk $g$9 for training and $u$0 for evaluation,
• walk batch size $u$1 (Hsu et al., 10 May 2026).

The benchmark’s scale is split into a large training corpus and a smaller but broader test corpus:

• Training split: 2,000,000 instances, containing only Laplace, Poisson, and Yukawa, with WoS budgets

$u$2

• Test split: 5,000 instances, spanning all five families, with 20 budgets from

$u$3

Helmholtz and Biharmonic are test-only families, so they function as OOD evaluation targets (Hsu et al., 10 May 2026).

For each budget $u$4, the field package stores three $u$5 float32 arrays:

• noisy: the WoS estimate $u$6,
• clean: the analytic field $u$7,
• mask: the interior-domain indicator (Hsu et al., 10 May 2026).

Metadata are stored separately in JSONL format. Each entry includes the case identifier, PDE family, family parameters such as $u$8 or $u$9, a PyTorch expression string for $\{\hat u_B\}$0, domain parameters, atom types and parameters, and hardness-related statistics. This makes the benchmark both grid-based and regenerable from symbolic descriptions (Hsu et al., 10 May 2026).

The paper also defines difficulty tiers using WoS MSE at $\{\hat u_B\}$1:

• Easy: MSE $\{\hat u_B\}$2, about $\{\hat u_B\}$3,
• Medium: $\{\hat u_B\}$4, about $\{\hat u_B\}$5,
• Hard: $\{\hat u_B\}$6, about $\{\hat u_B\}$7,
• Very Hard: $\{\hat u_B\}$8, about $\{\hat u_B\}$9 (Hsu et al., 10 May 2026).

5. Finite-compute benchmarking and relation to MC$^2$0

PDEZoo was introduced alongside MC$^2$1, a hybrid WoS–neural-network solver that uses a low-budget Monte Carlo field as a structured estimator and learns a single-pass correction. In the benchmark, MC$^2$2 is trained on PDEZoo instances using low-budget WoS fields $^2$3, the source term field $^2$4, and the domain mask, with the analytic field $^2$5 as target. The training regime uses a random subset of 100k training instances and budgets $^2$6; evaluation commonly uses a fixed budget such as $^2$7 (Hsu et al., 10 May 2026).

The principal metrics reported on PDEZoo are masked MSE, PSNR, SNR, and LPIPS. Because the test split includes multiple WoS budgets for the same instance, the benchmark supports direct accuracy–compute comparisons. The paper reports that WoS MSE follows the expected $^2$8 slope over budget on sampled test cases, making the benchmark suitable for studying how much learned correction can replace brute-force Monte Carlo compute (Hsu et al., 10 May 2026).

Within that evaluation protocol, MC$^2$9 using only $\Omega\subset\mathbb{R}^2$0 WoS is reported to achieve PSNR $\Omega\subset\mathbb{R}^2$1–$\Omega\subset\mathbb{R}^2$2 dB overall, comparable to WoS at $\Omega\subset\mathbb{R}^2$3–$\Omega\subset\mathbb{R}^2$4. The paper characterizes this as solving PDEs approximately $\Omega\subset\mathbb{R}^2$5 faster than pure WoS at matched accuracy. It also reports zero-shot generalization from training families to held-out Biharmonic and Helmholtz test problems (Hsu et al., 10 May 2026).

PDEZoo is nevertheless method-agnostic in design. The benchmark is used to evaluate raw WoS at many budgets, classical denoisers, FNO, PINO, DiffusionPDE, and CoCoGen. The stated purpose is broader than benchmarking a single method: to support reproducible study of finite-compute elliptic PDE solving (Hsu et al., 10 May 2026).

6. Position within the broader PDE benchmark ecosystem

PDEZoo occupies a different niche from several other recent PDE resources. APEBench is a benchmark for autoregressive neural emulators of time-dependent PDEs, built around a differentiable pseudo-spectral solver and covering 46 distinct PDEs across 1D, 2D, and 3D; it emphasizes rollout metrics and temporal generalization rather than elliptic boundary-value solving under finite Monte Carlo budgets (Koehler et al., 2024). VisualPDE is an online interactive solver for a broad class of 1D and 2D PDE systems, designed for rapid exploration and sharing rather than standardized large-scale benchmarking with analytic elliptic ground truth (Walker et al., 2023). The diagnostic suite in “A Diagnostic Software Suite for Auditing Learned PDE Simulators” provides architecture-independent, post hoc diagnostics such as semigroup consistency, generator discrepancy, energy behavior, integral balance, admissibility constraints, perturbation response, and scaling-law consistency for learned evolution operators, which is complementary to PDEZoo’s focus on elliptic field accuracy and finite-compute Monte Carlo trajectories (Shikhman, 16 Jun 2026).

Within that landscape, PDEZoo’s defining characteristics are its restriction to 2D elliptic PDEs, its use of analytic ground truth rather than numerically converged reference solves, its SDF-based geometric diversity, and its release of multi-budget stochastic solver outputs. The paper also notes several limitations: the benchmark is confined to $\Omega\subset\mathbb{R}^2$6D elliptic problems, it is tied to Walk-on-Spheres and related Green’s-function machinery, and it relies on manufactured solutions rather than directly modeling application-specific physical datasets. This suggests that PDEZoo is best understood as a specialized benchmark for elliptic solver evaluation under controlled geometry and compute variation, rather than a universal PDE benchmark (Hsu et al., 10 May 2026).