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name: autodiff_pde_residual description: Compute Burgers-style PINN residual values and derivative diagnostics from differentiable or analytic surrogate predictions.

Automatic Differentiation PDE Residual

Use this skill when a recovery harness needs the physics residual that turns a neural surrogate into a PINN. It is designed for Burgers-equation reduced recovery and can also serve as a template for other PDE residual builders. Do not use it to create data items or perform optimizer steps.

Inputs

• A surrogate with analytic derivative methods or a differentiable backend wrapper.
• Collocation coordinates with t and x values.
• PDE coefficients, especially Burgers viscosity nu.

Outputs

• Residual values for each collocation point.
• Diagnostic derivatives u, u_t, u_x, and u_xx.
• Mean-squared residual loss helper for training objectives.

Workflow

1. Evaluate the surrogate at each collocation point.
2. Compute time and spatial derivatives using the available differentiable interface.
3. Assemble u_t + u * u_x - nu * u_xx.
4. Return residuals and diagnostics separately so recovery can log mechanism evidence.

Validation

Run python tests/test_residual.py or validate the tree with validate_skill_tree.py --run-tests.

Limitations

The included script uses an analytic tiny surrogate for deterministic reduced recovery. Full automatic differentiation backends can wrap the same residual contract when available.