Solver mixin

[GiovanniCanali]

Description

This PR fixes #628 #782

TODO list:

• Fix docstring for all solver-related files
• Add one-line doc header
• Add rst files
• Add old PINN variants + tests

Checklist

• Code follows the project’s Code Style Guidelines
• Tests have been added or updated
• Documentation has been updated if necessary
• Pull request is linked to an open issue

[GiovanniCanali]

BaseSolver, SingleModelSolver, MultiModelSolver, and EnsembleSolver are base classes that aggregate one or more mixins into a single behavioral unit. They are intended to be specialized by concrete solver implementations that inherit from them.

[dario-coscia]

@GiovanniCanali I write here as I see stuff but I did not finish full review:

1. I would divide mixing and solvers in the doc

2. Why are the mixing classes with _? They are not private; people will use them.
3. Why do we remove the reduction in the loss here?
4. The _EnsembleMixin forward shuould call the forward super of _MultiModelMixin and mean the super forward result.
5. How do you envision the integration of weighting, both per residual sample and per conditional components?
6. I would remove typechecker to a safer approach, like pydantic models.

[GiovanniCanali]

@GiovanniCanali I write here as I see stuff but I did not finish full review:

1. I would divide mixing and solvers in the doc

2. Why are the mixing classes with `_`? They are not private; people will use them. 3. Why do we remove the reduction in the loss [here](https://github.com/GiovanniCanali/PINA/blob/ea9d2443bd288da0bc4aa1f6dbc66c8db3584b3c/pina/_src/solver/base_solver.py#L126-L134)? 4. The [_EnsembleMixin forward](https://github.com/GiovanniCanali/PINA/blob/ea9d2443bd288da0bc4aa1f6dbc66c8db3584b3c/pina/_src/solver/mixin/ensemble_mixin.py#L5) shuould call the forward super of [_MultiModelMixin](https://github.com/GiovanniCanali/PINA/blob/solver_mixin/pina/_src/solver/mixin/multi_model_mixin.py) and mean the super forward result. 5. How do you envision the integration of weighting, both per residual sample and per conditional components? 6. I would remove typechecker to a safer approach, like [pydantic](https://pydantic.dev/docs/) models.

3. The loss reduction is stored in self._reduction and removed from self._loss_fn. This separates loss computation into two steps: first, tensor_loss is computed from the residuals; then, the reduction is applied to obtain scalar_loss. See _loss_from_residual and _apply_reduction in BaseSolver.

4. While your reasoning is correct, I would keep the forward logics separate because merging it looks more error-prone. Also, since this is only a two-line function, I do not think it introduces a meaningful code-duplication issue.

5. Per-condition weighting is handled by ConditionAggregatorMixin and works as it did in the previous version. Point-wise weights, such as those used in residual-based attention, are handled by dedicated mixins.

6. I did not get your point, can you explain this better?

I agree on the remaining two points, I'll fix them soon

[dario-coscia]

Stellar update!!!!