code: def value_and_grad(
      fn: Callable[..., Any],
      mdl: Module,
      *primals,
      has_aux: bool = False,
      reduce_axes=(),
      variables: lift.CollectionFilter = True,
      rngs: lift.PRNGSequenceFilter = True,
  ):
      """A limited, lifted equivalent of ``jax.value_and_grad``.
  
      Note that for this convenience function, gradients are only calculated for
      the function inputs, and not with respect to any module variables. The
      target function must return a scalar-valued output.  For a more general
      lifted vjp, see ``nn.vjp`` for the lifted vector-Jacobian product.
  
      Example::
  
        class LearnScale(nn.Module):
          @nn.compact
          def __call__(self, x, y):
            p = self.param('scale', nn.initializers.zeros_init(), ())
            return p * x * y
  
        class Foo(nn.Module):
          @nn.compact
          def __call__(self, x, y):
            z, (x_grad, y_grad) = nn.value_and_grad(
                lambda mdl, x, y: mdl(x, y), LearnScale(), x, y)
            return z, x_grad, y_grad
  
      Args:
        fn: Function to be differentiated. Its arguments should be arrays, scalars,
          or standard Python containers of arrays or scalars. It should return an
          array, scalar, or standard Python container of arrays or scalars. It will
          receive the scope and primals as arguments.
        mdl: The module of which the variables will be differentiated.
        *primals: A sequence of primal values at which the Jacobian of ``fn``
          should be evaluated. The length of ``primals`` should be equal to the
          number of positional parameters to ``fn``. Each primal value should be a
          tuple of arrays, scalar, or standard Python containers thereof.
        has_aux: Optional, bool. Indicates whether ``fn`` returns a pair where the
         first element is considered the output of the mathematical function to be
         differentiated and the second element is auxiliary data. Default ``False``.
        reduce_axes: Optional, tuple of axis names. If an axis is listed here, and
          ``fn`` implicitly broadcasts a value over that axis, the backward pass
          will perform a ``psum`` of the corresponding gradient. Otherwise, the
          grad will be per-example over named axes. For example, if ``'batch'``
          is a named batch axis, ``vjp(f, *args, reduce_axes=('batch',))`` will
          create a grad function that sums over the batch while ``grad(f, *args)``
          will create a per-example grad.
        variables: variables collections that are available inside ``fn`` but
          do not receive a cotangent.
        rngs: the prngs that are available inside ``fn``.
      Returns:
        If ``has_aux`` is ``False``, returns a ``primals_out, grads`` pair, where
        ``primals_out`` is ``fn(*primals)``.  ``grads`` are the gradients for the
        corresponding primals and do not include the gradients for module variables.
        If ``has_aux`` is ``True``, returns a
        ``(primals_out, aux), grads`` tuple where ``aux`` is the auxiliary data
        returned by ``fn``.
      """
  
      grad_partial = functools.partial(
          lift_direct_transform,
          lift.value_and_grad,
          (fn,),
          mdl,
          *primals,
          has_aux=has_aux,
          reduce_axes=reduce_axes,
          variables=variables,
          rngs=rngs,
      )
  
      if has_aux:
          out, aux, argument_grads = grad_partial()
          if out.shape != ():
              raise ValueError(
                  "grad can only work on functions with "
                  f"scalar-valued outputs. out shape={out.shape}"
              )
          return (out, aux), argument_grads
      else:
          out, argument_grads = grad_partial()
          if out.shape != ():
              raise ValueError(
                  "grad can only work on functions with "
                  f"scalar-valued outputs. out shape={out.shape}"
              )
          return out, argument_grads
  
imports: import dataclasses
  import functools
  import inspect
  from typing import Any, Callable, Dict, Iterable, Mapping, Optional, Sequence, Tuple, Type, TypeVar, Union
  from flax import errors, struct, traceback_util
  from flax.core import Scope, lift, meta
  from flax.core.frozen_dict import FrozenDict
  from flax.linen import module
  from flax.linen.module import Module, Variable, _derive_profiling_name, _get_unbound_fn, wrap_method_once
  import jax
label_desc: Use image node_id 18 for calling a global function with example usage: value_and_grad(fn, mdl) and returns: out,  aux,  argument_grads, out,  argument_grads