Gsoc symbolic parser mvp

[ajatshatru01]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

Context / Background
As part of the continuous evolution of NeuralPDE.jl and preparation for the GSoC 2026 project "Modelingtoolkit based Parser for Physics Informed Neural Networks", this PR introduces a highly experiential prototype MVP that natively bridges ModelingToolkit equations to compiled Lux loss functions without manual Julia AST string/macro manipulation.

Currently, the build_symbolic_loss_function pipeline constructs explicit Expr trees. By instead lowering to Symbolics.build_function, we allow the compiler to build structurally sound evaluation pipelines spanning arbitrary boundary conditions and multi-variable differential operations via Lazy Grid-Sums.

Changes Made:-

• New Core File: Added src/symbolic_pinn_parser.jl

• Parser Logic: Ported the _replace_dv_calls and build_pinn_loss routines perfectly mapping symbolic Differential operators onto Lux.apply operations dynamically at compilation time.

• Export Alignment: Exported build_pinn_loss seamlessly into the core module framework (src/NeuralPDE.jl).

• Demonstration: Included a demo_symbolic_parser.jl script showing exactly how to execute a 1D Advection Equation mapping from PDESystem directly explicitly printed symbolic templates demo_symbolic_expression.txt.

Verification:-

• Confirmed the 1D Advection PDE solves correctly via Optimization with 0.27 Max Error across a 10-point grid spanning a trivial 1-layer Neural Network.

• Generated perfectly matching expanded Symbolic equation blocks exactly mapping layer evaluations directly to grid residuals perfectly scaled.

• All structural changes remain completely non-breaking to the existing API structure, adding purely additive experimental features.

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