# Summary

## Summary

Differentiable systems architecture extends automatic differentiation beyond isolated functions and neural network layers. The central idea is to treat larger systems as compositions of differentiable, partially differentiable, or optimization-aware components.

A system may include:

| Component | Differentiable Role |
|---|---|
| Pipeline | Propagates task loss through many stages |
| Database | Makes retrieval, ranking, and aggregation trainable |
| Renderer | Sends image loss back into geometry, pose, lighting, and material parameters |
| Physics engine | Sends trajectory loss back into forces, controls, and physical constants |
| Search system | Makes query encoding, ranking, and memory selection trainable |
| Compiler | Learns optimization, scheduling, and layout decisions |
| Runtime or OS | Optimizes resource management policies |
| Symbolic-numeric layer | Combines exact structure with continuous learning |

The architectural question is not whether every operation can be made smooth. Many important system operations are discrete, symbolic, stateful, or safety-critical. The question is where gradients provide useful credit assignment.

A practical differentiable system usually has three layers:

```text
symbolic structure
  -> differentiable computation
  -> optimization objective
```

The symbolic layer preserves invariants. It handles types, queries, programs, transactions, constraints, proofs, or legal actions.

The differentiable layer provides continuous parameters and gradient paths. It handles embeddings, scores, controllers, solvers, rankings, simulations, neural modules, and smooth approximations.

The objective layer defines what the system should improve: accuracy, latency, energy, memory, reconstruction quality, trajectory cost, ranking quality, or task success.

The hard part is the boundary between these layers. If the boundary is too hard, gradients stop. If it is too soft, the system may lose correctness, interpretability, or operational safety.

Good differentiable systems therefore use hybrid design. They keep exact mechanisms where exactness matters, and introduce differentiability where learning can improve behavior.

The recurring pattern is:

$$
\text{structured system} + \text{gradient path} + \text{measurable loss}
$$

This pattern turns a fixed pipeline into an adaptive one. Automatic differentiation supplies the local derivative machinery. Architecture determines whether those derivatives are meaningful, stable, and useful.