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name: deep_ritz_variational_energy_loss description: Compute Deep Ritz variational energy losses with gradient energy and boundary penalties.

Deep Ritz Variational Energy Loss

When To Use

Use this skill to turn sampled PDE points and a differentiable trial function into a Ritz objective. It is appropriate for Poisson-type energy minimization and can be extended to Rayleigh quotient eigenvalue objectives.

Inputs

• Trial model u(x; theta).
• Interior points with input gradients enabled.
• Boundary points and exact boundary values.
• Forcing function values f(x).
• Boundary penalty coefficient beta.

Outputs

• Scalar total loss.
• Interior energy, boundary penalty, and gradient-energy diagnostics.

Workflow

1. Evaluate the trial function at interior points.
2. Differentiate outputs with respect to coordinates.
3. Estimate mean(0.5 * |grad u|^2 - f * u).
4. Add beta * mean((u_boundary - g_boundary)^2).
5. Backpropagate the resulting scalar during training.

Validation

Run:

python tests/test_energy_loss.py

Limitations

Automatic differentiation requires PyTorch or another autograd runtime. Standard-library helpers provide exact-solution functions and numeric diagnostics but cannot replace full gradient-energy training.