Add PrecompileTools workload for improved startup time

[ChrisRackauckas-Claude]

Summary

• Adds PrecompileTools precompilation workloads to significantly reduce time-to-first-execution (TTFX) for common operations
• Adds PrecompileTools as a dependency with compat "1"
• Adds src/precompilation.jl with @compile_workload for core operations

Performance Improvements

Operation	Before	After	Improvement
Package load	~13.3s	~11.0s	17% faster
Model construction + setup	~2.3s	~0.017s	135x faster
Second model construction	N/A	~0.002s	-

What's Precompiled

The workload precompiles the most common user operations:

• DeepEquilibriumNetwork construction with Dense layers and SSRootfind solver
• Lux.setup() for parameter/state initialization
• DeepEquilibriumSolution constructor
• Common utility functions (flatten, flatten_vcat, split_and_reshape, check_unrolled_mode, zeros_init)
• Fixed-depth (unrolled) mode state handling

Notes

• Forward pass TTFX depends on the specific solver used (NonlinearSolve, OrdinaryDiffEq) which are optional dependencies
• Users should expect solver-specific compilation on first use of a particular solver
• Invalidation analysis showed ~2335 invalidation trees, but they originate from dependencies (Static, ChainRulesCore, RecursiveArrayTools, GPUCompiler) rather than this package

Test plan

• CI passes
• Package precompiles without errors
• Verify startup time improvement on local machine

cc @ChrisRackauckas

🤖 Generated with Claude Code