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name: deep_ritz_stochastic_quadrature_sampling description: Generate stochastic interior and boundary quadrature samples for Deep Ritz variational PDE training.

Deep Ritz Stochastic Quadrature Sampling

When To Use

Use this skill when implementing a Deep Ritz or variational neural PDE experiment that needs Monte Carlo estimates of domain and boundary integrals. Do not use it for fixed-grid finite differences or finite elements.

Inputs

• dimension: positive integer spatial dimension.
• lower, upper: scalar hypercube bounds.
• interior_count: number of interior samples.
• boundary_count: number of boundary samples.
• seed: optional deterministic seed.

Outputs

• Interior points inside [lower, upper]^dimension.
• Boundary points with at least one coordinate fixed to a lower or upper face.
• Face metadata for each boundary point.

Workflow

1. Use fresh random interior points for each training iteration.
2. Sample boundary faces uniformly across dimensions and low/high sides.
3. Use simple batch means as equal-weight Monte Carlo quadrature estimates.
4. Use fixed seeds only for validation, tests, and reproducible reduced recovery.

Validation

Run:

python tests/test_sampler.py

Limitations

The helper targets hypercube domains, matching the paper's high-dimensional Poisson recovery. L-shaped domains require a custom rejection sampler.