"""
Defines base classes of fields, which are discretized on grids

.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""

import functools
import json
import logging
import operator
import warnings
from abc import ABCMeta, abstractmethod, abstractproperty
from pathlib import Path
from typing import TYPE_CHECKING, Any, Callable, Dict, Optional, Tuple, TypeVar

import numba as nb
import numpy as np

from ..grids.base import DimensionError, DomainError, GridBase, discretize_interval
from ..grids.boundaries.axes import BoundariesData
from ..grids.cartesian import CartesianGridBase
from ..tools.cache import cached_method
from ..tools.docstrings import fill_in_docstring
from ..tools.misc import Number, number_array
from ..tools.numba import address_as_void_pointer, jit
from ..tools.plotting import (
    PlotReference,
    napari_add_layers,
    napari_viewer,
    plot_on_axes,
)
from ..tools.typing import ArrayLike, NumberOrArray

if TYPE_CHECKING:
    from .scalar import ScalarField  # @UnusedImport


TField = TypeVar("TField", bound="FieldBase")


class FieldBase(metaclass=ABCMeta):
    """abstract base class for describing (discretized) fields"""

    readonly = False

    _subclasses: Dict[str, "FieldBase"] = {}  # all classes inheriting from this
    _grid: GridBase
    _data: np.ndarray
    _label: Optional[str]

    def __init__(
        self,
        grid: GridBase,
        data: ArrayLike = None,
        *,
        label: Optional[str] = None,
    ):
        """
        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined
            data (array, optional):
                Field values at the support points of the grid
            label (str, optional):
                Name of the field
        """
        self._grid = grid
        self._data: np.ndarray = np.asarray(data)
        self.label = label
        self._logger = logging.getLogger(self.__class__.__name__)

    def __init_subclass__(cls, **kwargs):  # @NoSelf
        """ register all subclassess to reconstruct them later """
        super().__init_subclass__(**kwargs)
        cls._subclasses[cls.__name__] = cls

    @property
    def label(self) -> Optional[str]:
        """str: the name of the field """
        return self._label

    @label.setter
    def label(self, value: str = None):
        """ set the new label of the field """
        if value is None or isinstance(value, str):
            self._label = value
        else:
            raise TypeError("Label must be a string or None")

    @classmethod
    def from_state(
        cls, attributes: Dict[str, Any], data: np.ndarray = None
    ) -> "FieldBase":
        """create a field from given state.

        Args:
            attributes (dict):
                The attributes that describe the current instance
            data (:class:`~numpy.ndarray`, optional):
                Data values at the support points of the grid defining the field
        """
        # base class was chosen => select correct class from attributes
        class_name = attributes.pop("class")

        if class_name == cls.__name__:
            raise RuntimeError(f"Cannot reconstruct abstract class `{class_name}`")

        # call possibly overwritten classmethod from subclass
        return cls._subclasses[class_name].from_state(attributes, data)

    @classmethod
    def from_file(cls, filename: str) -> "FieldBase":
        """create field by reading file

        Args:
            filename (str): Path to the file being read
        """
        import h5py

        from .collection import FieldCollection

        with h5py.File(filename, "r") as fp:
            if "class" in fp.attrs:
                # this should be a field collection
                assert json.loads(fp.attrs["class"]) == "FieldCollection"
                obj = FieldCollection._from_hdf_dataset(fp)

            elif len(fp) == 1:
                # a single field is stored in the data
                dataset = fp[list(fp.keys())[0]]  # retrieve only dataset
                obj = cls._from_hdf_dataset(dataset)  # type: ignore

            else:
                raise RuntimeError(
                    "Multiple data fields were found in the "
                    "file but no FieldCollection is expected"
                )
        return obj

    @classmethod
    def _from_hdf_dataset(cls, dataset) -> "FieldBase":
        """ construct a field by reading data from an hdf5 dataset """
        # copy attributes from hdf
        attributes = dict(dataset.attrs)

        # determine class
        class_name = json.loads(attributes.pop("class"))
        field_cls = cls._subclasses[class_name]

        # unserialize the attributes
        attributes = field_cls.unserialize_attributes(attributes)
        return field_cls.from_state(attributes, data=dataset)

    @property
    def grid(self) -> GridBase:
        """ GridBase: The grid on which the field is defined """
        return self._grid

    def to_file(self, filename: str, **kwargs):
        r"""store field in a file

        The extension of the filename determines what format is being used. If
        it ends in `.h5` or `.hdf`, the Hierarchical Data Format is used. The
        other supported format are images, where only the most typical formats
        are supported.

        Args:
            filename (str):
                Path where the data is stored
            metadata (dict):
                A dictionary of additional information that is stored with the file.
                Note that not all formats support metadata.
            \**kwargs:
                Additional parameters may be supported for some formats
        """
        extension = Path(filename).suffix.lower()

        if extension in {".hdf", ".hdf5", ".he5", ".h5"}:
            # save data in hdf5 format
            import h5py

            with h5py.File(filename, "w") as fp:
                self._write_hdf_dataset(fp, **kwargs)

        elif extension in {".png", ".jpg", ".jpeg", ".tif", ".pdf", ".svg"}:
            # save data as an image
            self._write_to_image(filename, **kwargs)

        else:
            raise ValueError(f"Do not know how to save data to `*{extension}`")

    def _write_hdf_dataset(self, hdf_path, key: str = "data"):
        """ write data to a given hdf5 path `hdf_path` """
        # write the data
        dataset = hdf_path.create_dataset(key, data=self.data)

        # write attributes
        for key, value in self.attributes_serialized.items():
            dataset.attrs[key] = value

    def _write_to_image(self, filename: str, **kwargs):
        """write data to image

        Args:
            filename (str): The path to the image that will be created
        """
        raise NotImplementedError(f"Cannot save {self.__class__.__name__} as an image")

    @abstractmethod
    def copy(
        self: TField, data: ArrayLike = None, *, label: str = None, dtype=None
    ) -> TField:
        pass

    def assert_field_compatible(self, other: "FieldBase", accept_scalar: bool = False):
        """checks whether `other` is compatible with the current field

        Args:
            other (FieldBase): Other field this is compared to
            accept_scalar (bool, optional): Determines whether it is acceptable
                that `other` is an instance of
                :class:`~pde.fields.ScalarField`.
        """
        from .scalar import ScalarField  # @Reimport

        # check whether they are the same class
        is_scalar = accept_scalar and isinstance(other, ScalarField)
        class_compatible = self.__class__ == other.__class__ or is_scalar
        if not class_compatible:
            raise TypeError("Fields are incompatible")

        # check whether the associated grids are identical
        if not self.grid.compatible_with(other.grid):
            raise ValueError("Grids incompatible")

    @property
    def data(self) -> np.ndarray:
        """ :class:`~numpy.ndarray`: discretized data at the support points """
        return self._data

    @data.setter
    def data(self, value: NumberOrArray) -> None:
        if self.readonly:
            raise RuntimeError(f"Cannot write to {self.__class__.__name__}")
        if isinstance(value, FieldBase):
            # copy data into current field
            self.assert_field_compatible(value, accept_scalar=True)
            self._data[:] = value.data
        else:
            self._data[:] = value

    @property
    def dtype(self):
        """ returns the numpy dtype of the underlying data """
        # this property is necessary to support np.iscomplexobj for DataFieldBases
        return self.data.dtype

    @property
    def is_complex(self) -> bool:
        """ bool: whether the field contains real or complex data """
        return np.iscomplexobj(self.data)  # type: ignore

    @property
    def attributes(self) -> Dict[str, Any]:
        """ dict: describes the state of the instance (without the data) """
        return {
            "class": self.__class__.__name__,
            "grid": self.grid,
            "label": self.label,
        }

    @property
    def attributes_serialized(self) -> Dict[str, str]:
        """ dict: serialized version of the attributes """
        results = {}
        for key, value in self.attributes.items():
            if key == "grid":
                results[key] = value.state_serialized
            else:
                results[key] = json.dumps(value)
        return results

    @classmethod
    def unserialize_attributes(cls, attributes: Dict[str, str]) -> Dict[str, Any]:
        """unserializes the given attributes

        Args:
            attributes (dict):
                The serialized attributes

        Returns:
            dict: The unserialized attributes
        """
        # base class was chosen => select correct class from attributes
        class_name = json.loads(attributes["class"])

        if class_name == cls.__name__:
            raise RuntimeError(f"Cannot reconstruct abstract class `{class_name}`")

        # call possibly overwritten classmethod from subclass
        return cls._subclasses[class_name].unserialize_attributes(attributes)

    @property
    def _data_flat(self):
        """ :class:`~numpy.ndarray`: flat version of discretized data """
        # flatten the first dimension of the internal data
        return self._data.reshape(-1, *self.grid.shape)

    @_data_flat.setter
    def _data_flat(self, value):
        """ set the data from a value from a collection """
        if self.readonly:
            raise RuntimeError(f"Cannot write to {self.__class__.__name__}")
        # simply set the data -> this might need to be overwritten
        self._data = value

    @property
    def real(self: TField) -> TField:
        """:class:`FieldBase`: Real part of the field """
        return self.copy(data=self.data.real)

    @property
    def imag(self: TField) -> TField:
        """:class:`FieldBase`: Imaginary part of the field """
        return self.copy(data=self.data.imag)

    def conjugate(self: TField) -> TField:
        """ returns complex conjugate of the field """
        return self.copy(data=self.data.conj())

    def __eq__(self, other):
        """ test fields for equality, ignoring the label """
        if not isinstance(other, self.__class__):
            return NotImplemented
        return self.grid == other.grid and np.array_equal(self.data, other.data)

    def __neg__(self):
        """ return the negative of the current field """
        return self.copy(data=-self.data)

    def _binary_operation(
        self, other, op: Callable, scalar_second: bool = True
    ) -> "FieldBase":
        """perform a binary operation between this field and `other`

        Args:
            other (number of FieldBase):
                The second term of the operator
            op (callable):
                A binary function calculating the result
            scalar_second (bool):
                Flag determining whether the second operator must be a scalar

        Returns:
            FieldBase: An field that contains the result of the operation. If
            `scalar_second == True`, the type of FieldBase is the same as `self`
        """
        if isinstance(other, FieldBase):
            # right operator is a field
            from .scalar import ScalarField  # @Reimport

            if scalar_second:
                # right operator must be a scalar
                if not isinstance(other, ScalarField):
                    raise TypeError("Right operator must be a scalar field")
                self.grid.assert_grid_compatible(other.grid)
                result: FieldBase = self.copy(op(self.data, other.data))

            elif isinstance(self, ScalarField):
                # left operator is a scalar field (right can be tensor)
                self.grid.assert_grid_compatible(other.grid)
                result = other.copy(op(self.data, other.data))

            else:
                # left operator is tensor and right one might be anything
                self.assert_field_compatible(other, accept_scalar=True)
                result = self.copy(op(self.data, other.data))

        else:
            # the second operator is a number or a numpy array
            result = self.copy(op(self.data, other))
        return result

    def _binary_operation_inplace(
        self: TField, other, op_inplace: Callable, scalar_second: bool = True
    ) -> TField:
        """perform an in-place binary operation between this field and `other`

        Args:
            other (number of FieldBase):
                The second term of the operator
            op_inplace (callable):
                A binary function storing its result in the first argument
            scalar_second (bool):
                Flag determining whether the second operator must be a scalar.

        Returns:
            FieldBase: The field `self` with updated data
        """
        if self.readonly:
            raise RuntimeError(f"Cannot write to {self.__class__.__name__}")

        if isinstance(other, FieldBase):
            # right operator is a field
            from .scalar import ScalarField  # @Reimport

            if scalar_second:
                # right operator must be a scalar
                if not isinstance(other, ScalarField):
                    raise TypeError("Right operator must be a scalar field")
                self.grid.assert_grid_compatible(other.grid)
            else:
                # left operator is tensor and right one might be anything
                self.assert_field_compatible(other, accept_scalar=True)

            op_inplace(self.data, other.data)

        else:
            # the second operator is a number or a numpy array
            op_inplace(self.data, other)

        return self

    def __add__(self, other) -> "FieldBase":
        """ add two fields """
        return self._binary_operation(other, operator.add, scalar_second=False)

    __radd__ = __add__

    def __iadd__(self: TField, other) -> TField:
        """ add `other` to the current field """
        return self._binary_operation_inplace(other, operator.iadd, scalar_second=False)

    def __sub__(self, other) -> "FieldBase":
        """ subtract two fields """
        return self._binary_operation(other, operator.sub, scalar_second=False)

    def __rsub__(self, other) -> "FieldBase":
        """ subtract two fields """
        return self._binary_operation(other, lambda x, y: y - x, scalar_second=False)

    def __isub__(self: TField, other) -> TField:
        """ add `other` to the current field """
        return self._binary_operation_inplace(other, operator.isub, scalar_second=False)

    def __mul__(self, other) -> "FieldBase":
        """ multiply field by value """
        return self._binary_operation(other, operator.mul, scalar_second=False)

    __rmul__ = __mul__

    def __imul__(self: TField, other) -> TField:
        """ multiply field by value """
        return self._binary_operation_inplace(other, operator.imul, scalar_second=False)

    def __truediv__(self, other) -> "FieldBase":
        """ divide field by value """
        return self._binary_operation(other, operator.truediv, scalar_second=True)

    def __itruediv__(self: TField, other) -> TField:
        """ divide field by value """
        return self._binary_operation_inplace(
            other, operator.itruediv, scalar_second=True
        )

    def __pow__(self, exponent: float) -> "FieldBase":
        """ raise data of the field to a certain power """
        if not np.isscalar(exponent):
            raise NotImplementedError("Only scalar exponents are supported")
        return self.copy(data=self.data ** exponent)

    def __ipow__(self: TField, exponent: float) -> TField:
        """ raise data of the field to a certain power in-place """
        if self.readonly:
            raise RuntimeError(f"Cannot write to {self.__class__.__name__}")

        if not np.isscalar(exponent):
            raise NotImplementedError("Only scalar exponents are supported")
        self.data **= exponent
        return self

    def apply(
        self: TField, func: Callable, out: Optional[TField] = None, label: str = None
    ) -> TField:
        """applies a function to the data and returns it as a field

        Args:
            func (callable or str):
                The (vectorized) function being applied to the data or the name
                of an operator that is defined for the grid of this field.
            out (FieldBase, optional):
                Optional field into which the data is written
            label (str, optional):
                Name of the returned field

        Returns:
            Field with new data. This is stored at `out` if given.
        """
        if out is None:
            return self.copy(data=func(self.data), label=label)
        else:
            self.assert_field_compatible(out)
            out.data[:] = func(self.data)
            if label:
                out.label = label
            return out

    @abstractmethod
    def get_line_data(
        self, scalar: str = "auto", extract: str = "auto"
    ) -> Dict[str, Any]:
        pass

    @abstractmethod
    def get_image_data(self) -> Dict[str, Any]:
        pass

    @abstractmethod
    def plot(self, *args, **kwargs):
        pass

    @abstractmethod
    def _get_napari_data(self, **kwargs) -> Dict[str, Dict[str, Any]]:
        pass

    def plot_interactive(self, viewer_args: Dict[str, Any] = None, **kwargs):
        """create an interactive plot of the field using :mod:`napari`

        For a detailed description of the launched program, see the
        `napari webpage <http://napari.org/>`_.

        Args:
            viewer_args (dict):
                Arguments passed to :class:`napari.viewer.Viewer` to affect the viewer.
            **kwargs:
                Extra arguments passed to the plotting function
        """
        if viewer_args is None:
            viewer_args = {}

        if self.grid.num_axes == 1:
            raise RuntimeError(
                "Interactive plotting needs at least 2 spatial dimensions"
            )

        with napari_viewer(self.grid, **viewer_args) as viewer:
            napari_add_layers(viewer, self._get_napari_data(**kwargs))


TDataField = TypeVar("TDataField", bound="DataFieldBase")


class DataFieldBase(FieldBase, metaclass=ABCMeta):
    """abstract base class for describing fields of single entities

    Attributes:
        grid (:class:`~pde.grids.GridBase`):
            The underlying grid defining the discretization
        data (:class:`~numpy.ndarray`):
            Data values at the support points of the grid
        shape (tuple):
            Shape of the `data` field
        label (str):
            Name of the field
    """

    rank: int  # the rank of the tensor field
    _allocate_memory = True  # determines whether the instances allocated memory

    def __init__(
        self,
        grid: GridBase,
        data: ArrayLike = None,
        *,
        label: str = None,
        dtype=None,
    ):
        """
        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined.
            data (Number or :class:`~numpy.ndarray`, optional):
                Field values at the support points of the grid. The data is copied from
                the supplied array. The resulting field will contain real data unless
                the `data` argument contains complex values.
            label (str, optional):
                Name of the field
            dtype (numpy dtype):
                The data type of the field. All the numpy dtypes are supported. If
                omitted, it will be determined from `data` automatically.
        """
        # determine data shape
        shape = (grid.dim,) * self.rank + grid.shape

        if self._allocate_memory:
            # class manages its own data, which therefore needs to be allocated
            if data is None:
                # allocate memory filled with real zeros by default
                data = np.zeros(shape, dtype=dtype)

            elif isinstance(data, DataFieldBase):
                # we need to make a copy to make sure the data is writeable.
                grid.assert_grid_compatible(data.grid)
                data = number_array(data.data, dtype=dtype, copy=True)

            else:
                # we need to first reshape the data and then make a copy to ensure the
                # data array is writeable
                data_reshaped = np.broadcast_to(data, shape)
                data = number_array(data_reshaped, dtype=dtype, copy=True)

        elif data is not None:
            # class does not manage its own data
            raise ValueError(
                f"{self.__class__.__name__} does not support data assignment."
            )

        super().__init__(grid, data=data, label=label)

    def __repr__(self) -> str:
        """ return instance as string """
        class_name = self.__class__.__name__
        result = f"{class_name}(grid={self.grid!r}, data={self.data}"
        if self.label:
            result += f', label="{self.label}"'
        return result + ")"

    def __str__(self) -> str:
        """ return instance as string """
        result = (
            f"{self.__class__.__name__}(grid={self.grid}, "
            f"data=Array{self.data.shape}"
        )
        if self.label:
            result += f', label="{self.label}"'
        return result + ")"

    @classmethod
    def random_uniform(
        cls,
        grid: GridBase,
        vmin: float = 0,
        vmax: float = 1,
        label: Optional[str] = None,
        seed: Optional[int] = None,
    ):
        """create field with uniform distributed random values

        These values are uncorrelated in space.

        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined
            vmin (float):
                Smallest possible random value
            vmax (float):
                Largest random value
            label (str, optional):
                Name of the field
            seed (int, optional):
                Seed of the random number generator. If `None`, the current
                state is not changed.
        """
        shape = (grid.dim,) * cls.rank + grid.shape
        if seed is not None:
            np.random.seed(seed)
        data = np.random.uniform(vmin, vmax, shape)
        return cls(grid, data, label=label)

    @classmethod
    def random_normal(
        cls,
        grid: GridBase,
        mean: float = 0,
        std: float = 1,
        scaling: str = "physical",
        label: Optional[str] = None,
        seed: Optional[int] = None,
    ):
        """create field with normal distributed random values

        These values are uncorrelated in space.

        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined
            mean (float):
                Mean of the Gaussian distribution
            std (float):
                Standard deviation of the Gaussian distribution
            scaling (str):
                Determines how the values are scaled. Possible choices are
                'none' (values are drawn from a normal distribution with
                given mean and standard deviation) or 'physical' (the variance
                of the random number is scaled by the inverse volume of the grid
                cell; this is useful for physical quantities, which vary less in
                larger volumes).
            label (str, optional):
                Name of the field
            seed (int, optional):
                Seed of the random number generator. If `None`, the current
                state is not changed.
        """
        if seed is not None:
            np.random.seed(seed)

        if scaling == "none":
            noise_scale = std
        elif scaling == "physical":
            noise_scale = std / np.sqrt(grid.cell_volumes)
        else:
            raise ValueError(f"Unknown noise scaling {scaling}")

        shape = (grid.dim,) * cls.rank + grid.shape
        data = mean + noise_scale * np.random.randn(*shape)
        return cls(grid, data, label=label)

    @classmethod
    def random_harmonic(
        cls,
        grid: GridBase,
        modes: int = 3,
        harmonic=np.cos,
        axis_combination=np.multiply,
        label: Optional[str] = None,
        seed: Optional[int] = None,
    ):
        r"""create a random field build from harmonics

        The resulting fields will be highly correlated in space and can thus
        serve for testing differential operators.

        With the default settings, the resulting field :math:`c_i(\mathbf{x})`
        is given by

        .. math::
            c_i(\mathbf{x}) = \prod_{\alpha=1}^N \sum_{j=1}^M a_{ij\alpha}
                \cos\left(\frac{2 \pi x_\alpha}{j L_\alpha}\right) \;,

        where :math:`N` is the number of spatial dimensions, each with length
        :math:`L_\alpha`, :math:`M` is the number of modes given by `modes`, and
        :math:`a_{ij\alpha}` are random amplitudes, chosen from a uniform
        distribution over the interval [0, 1].

        Note that the product could be replaced by a sum when
        `axis_combination = numpy.add` and the :math:`\cos()` could be any other
        function given by the parameter `harmonic`.

        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined
            modes (int):
                Number :math:`M` of harmonic modes
            harmonic (callable):
                Determines which harmonic function is used. Typical values are
                :func:`numpy.sin` and :func:`numpy.cos`, which basically relate
                to different boundary conditions applied at the grid boundaries.
            axis_combination (callable):
                Determines how values from different axis are combined. Typical
                choices are :func:`numpy.multiply` and :func:`numpy.add`
                resulting in products and sums of the values along axes,
                respectively.
            label (str, optional):
                Name of the field
            seed (int, optional):
                Seed of the random number generator. If `None`, the current
                state is not changed.
        """
        tensor_shape = (grid.dim,) * cls.rank
        if seed is not None:
            np.random.seed(seed)

        data = np.empty(tensor_shape + grid.shape)
        # determine random field for each component
        for index in np.ndindex(*tensor_shape):
            data_axis = []
            # random harmonic function along each axis
            for i in range(len(grid.axes)):
                # choose wave vectors
                ampl = np.random.random(modes)  # amplitudes
                x = discretize_interval(0, 2 * np.pi, grid.shape[i])[0]
                data_axis.append(
                    sum(a * harmonic(n * x) for n, a in enumerate(ampl, 1))
                )
            # full dataset is product of values along axes
            data[index] = functools.reduce(axis_combination.outer, data_axis)

        return cls(grid, data, label=label)

    @classmethod
    def random_colored(
        cls,
        grid: GridBase,
        exponent: float = 0,
        scale: float = 1,
        label: Optional[str] = None,
        seed: Optional[int] = None,
    ):
        r"""create a field of random values with colored noise

        The spatially correlated values obey

        .. math::
            \langle c_i(\boldsymbol k) c_j(\boldsymbol k’) \rangle =
                \Gamma^2 |\boldsymbol k|^\nu \delta_{ij}
                \delta(\boldsymbol k - \boldsymbol k’)

        in spectral space. The special case :math:`\nu = 0` corresponds to white
        noise. Note that the components of vector or tensor fields are
        uncorrelated.

        Args:
            grid (:class:`~pde.grids.GridBase`):
                Grid defining the space on which this field is defined
            exponent (float):
                Exponent :math:`\nu` of the power spectrum
            scale (float):
                Scaling factor :math:`\Gamma` determining noise strength
            label (str, optional):
                Name of the field
            seed (int, optional):
                Seed of the random number generator. If `None`, the current
                state is not changed.
        """
        if seed is not None:
            np.random.seed(seed)

        # create function making colored noise
        from ..tools.spectral import make_colored_noise

        make_noise = make_colored_noise(
            grid.shape, dx=grid.discretization, exponent=exponent, scale=scale
        )

        # create random fields for each tensor component
        tensor_shape = (grid.dim,) * cls.rank
        data = np.empty(tensor_shape + grid.shape)
        # determine random field for each component
        for index in np.ndindex(*tensor_shape):
            data[index] = make_noise()

        return cls(grid, data, label=label)

    @classmethod
    def from_state(
        cls, attributes: Dict[str, Any], data: np.ndarray = None
    ) -> "DataFieldBase":
        """create a field from given state.

        Args:
            attributes (dict):
                The attributes that describe the current instance
            data (:class:`~numpy.ndarray`, optional):
                Data values at the support points of the grid defining the field
        """
        if "class" in attributes:
            class_name = attributes.pop("class")
            assert class_name == cls.__name__

        # create the instance from the attributes
        return cls(attributes.pop("grid"), data=data, **attributes)

    def copy(
        self: TDataField,
        data: ArrayLike = None,
        *,
        label: str = None,
        dtype=None,
    ) -> TDataField:
        """return a copy of the data, but not of the grid

        Args:
            data (:class:`~numpy.ndarray`, optional):
                Data values at the support points of the grid that define the
                field.
            label (str, optional):
                Name of the copied field
            dtype (numpy dtype):
                The data type of the field. If omitted, it will be determined from
                `data` automatically.
        """
        if label is None:
            label = self.label
        if data is None:
            data = self.data
        # the actual data will be copied in our __init__ method
        return self.__class__(self.grid, data=data, label=label, dtype=dtype)

    @property
    def data_shape(self) -> Tuple[int, ...]:
        """ tuple: the shape of the data at each grid point """
        return (self.grid.dim,) * self.rank

    @classmethod
    def unserialize_attributes(cls, attributes: Dict[str, str]) -> Dict[str, Any]:
        """unserializes the given attributes

        Args:
            attributes (dict):
                The serialized attributes

        Returns:
            dict: The unserialized attributes
        """
        results = {}
        for key, value in attributes.items():
            if key == "grid":
                results[key] = GridBase.from_state(value)
            else:
                results[key] = json.loads(value)
        return results

    def _write_to_image(self, filename: str, **kwargs):
        r"""write data to image

        Args:
            filename (str):
                The path to the image that will be created
            \**kwargs:
                Additional keyword arguments that affect the image. For
                instance, some fields support a `scalar` argument that
                determines how they are converted to a scalar. Non-Cartesian
                grids might support a `performance_goal` argument to influence
                how an image is created from the raw data. Finally, the
                remaining arguments are are passed to
                :func:`matplotlib.pyplot.imsave` to affect the appearance.
        """
        import matplotlib.pyplot as plt

        # obtain image data
        get_image_args = {}
        for key in ["performance_goal", "scalar"]:
            if key in kwargs:
                get_image_args[key] = kwargs.pop(key)
        img = self.get_image_data(**get_image_args)

        kwargs.setdefault("cmap", "gray")
        plt.imsave(filename, img["data"].T, origin="lower", **kwargs)

    def _make_interpolator_scipy(
        self, method: str = "linear", fill: Number = None, **kwargs
    ) -> Callable[[np.ndarray, np.ndarray], NumberOrArray]:
        r"""returns a function that can be used to interpolate values.

        This uses :class:`scipy.interpolate.RegularGridInterpolator` and thus supports
        extra options supplied by keyword arguments. Note that this interpolator does
        not respect periodic boundary conditions, yet.

        Args:
            method (str):
                The method used for interpolation. Currently, 'linear' and 'nearest' are
                supported by :class:`~scipy.interpolate.RegularGridInterpolator`.
            fill (Number, optional):
                Determines how values out of bounds are handled. If `None`, a
                `ValueError` is raised when out-of-bounds points are requested.
                Otherwise, the given value is returned.
            \**kwargs: All keyword arguments are forwarded to
                :class:`~scipy.interpolate.RegularGridInterpolator`

        Returns:
            A function which returns interpolated values when called with
            arbitrary positions within the space of the grid.
        """
        from scipy import interpolate

        coords_src = self.grid.axes_coords
        grid_dim = len(self.grid.axes)

        if self.rank == 0:
            # scalar field => data layout is already usable
            data = self.data
            revert_shape = False
        else:
            # spatial dimensions need to come first => move data to last axis
            assert self.data.shape[:-grid_dim] == self.data_shape
            data_flat = self._data_flat
            data_flat = np.moveaxis(data_flat, 0, -1)
            new_shape = self.grid.shape + (-1,)
            data = data_flat.reshape(new_shape)
            assert data.shape[-1] == self.grid.dim ** self.rank
            revert_shape = True

        # set the fill behavior
        if fill is None:
            kwargs["bounds_error"] = True
        else:
            kwargs["bounds_error"] = False
            kwargs["fill_value"] = fill

        # prepare the interpolator
        intp = interpolate.RegularGridInterpolator(
            coords_src, data, method=method, **kwargs
        )

        # determine under which conditions the axes can be squeezed
        if grid_dim == 1:
            scalar_dim = 0
        else:
            scalar_dim = 1

        # introduce wrapper function to process arrays
        def interpolator(point: np.ndarray, **kwargs) -> NumberOrArray:
            """ return the interpolated value at the position `point` """
            point = np.atleast_1d(point)
            # apply periodic boundary conditions to grid point
            point = self.grid.normalize_point(point, reflect=False)
            out = intp(point, **kwargs)
            if point.ndim == scalar_dim or point.ndim == point.size == 1:
                out = out[0]
            if revert_shape:
                # revert the shuffling of spatial and local axes
                out = np.moveaxis(out, point.ndim - 1, 0)
                out = out.reshape(self.data_shape + point.shape[:-1])

            return out  # type: ignore

        return interpolator  # type: ignore

    @fill_in_docstring
    def _make_interpolator_numba(
        self, bc: BoundariesData = "natural", fill: Number = None
    ) -> Callable[[np.ndarray, Optional[np.ndarray]], np.ndarray]:
        """return a compiled interpolator

        This interpolator respects boundary conditions and can thus interpolate
        values in the whole grid volume. However, close to corners, the
        interpolation might not be optimal, in particular for periodic grids.

        Args:
            bc:
                The boundary conditions applied to the field. {ARG_BOUNDARIES}
            fill (Number, optional):
                Determines how values out of bounds are handled. If `None`, a
                `ValueError` is raised when out-of-bounds points are requested.
                Otherwise, the given value is returned.

        Returns:
            A function which returns interpolated values when called with
            arbitrary positions within the space of the grid.
        """
        grid = self.grid
        grid_dim = len(grid.axes)
        data_shape = self.data_shape

        dim_error_msg = (
            "Dimension of the interpolation point does not match "
            f"grid dimension {grid_dim}"
        )

        # convert `fill` to dtype of data
        if fill is not None:
            if self.rank == 0:
                fill = self.data.dtype.type(fill)  # type: ignore
            else:
                fill = np.broadcast_to(fill, self.data_shape).astype(self.data.dtype)

        # create an interpolator for a single point
        interpolate_single = grid.make_interpolator_compiled(
            bc=bc, rank=self.rank, fill=fill
        )

        # extract information about the data field
        data_addr = self.data.ctypes.data
        shape, dtype = self.data.shape, self.data.dtype

        @jit
        def interpolator(point: np.ndarray, data: np.ndarray = None) -> np.ndarray:
            """return the interpolated value at the position `point`

            Args:
                point (:class:`~numpy.ndarray`):
                    The list of points. This point coordinates should be given
                    along the last axis, i.e., the shape should be `(..., dim)`.
                data (:class:`~numpy.ndarray`, optional):
                    The discretized field values. If omitted, the data of the
                    current field is used, which should be the default. However,
                    this option can be useful to interpolate other fields
                    defined on the same grid without recreating the interpolator

            Returns:
                :class:`~numpy.ndarray`: The interpolated values at the points
            """
            # check input
            point = np.atleast_1d(point)
            if point.shape[-1] != grid_dim:
                raise DimensionError(dim_error_msg)
            point_shape = point.shape[:-1]

            # reconstruct data field from memory address
            if data is None:
                data = nb.carray(address_as_void_pointer(data_addr), shape, dtype)

            # interpolate at every point
            out = np.empty(data_shape + point_shape, dtype=data.dtype)
            for idx in np.ndindex(point_shape):
                out[(...,) + idx] = interpolate_single(data, point[idx])

            return out

        return interpolator  # type: ignore

    @cached_method()
    def make_interpolator(
        self,
        backend: str = "numba",
        method: str = "linear",
        fill: Number = None,
        **kwargs,
    ) -> Callable[[np.ndarray, np.ndarray], NumberOrArray]:
        r"""returns a function that can be used to interpolate values.

        Args:
            backend (str):
                The accepted values `scipy` and `numba` determine the backend that is
                used for the interpolation.
            method (str):
                Determines the method being used for interpolation. Typical values that
                are `nearest` and `linear, but the supported values depend on the chosen
                `backend.
            fill (Number, optional):
                Determines how values out of bounds are handled. If `None`, a
                `ValueError` is raised when out-of-bounds points are requested.
                Otherwise, the given value is returned.
            \**kwargs:
                Additional keyword arguments are passed to the individual
                interpolator methods and can be used to further affect the
                behavior.

        The scipy implementations use scipy.interpolate.RegularGridInterpolator
        and thus do not respect boundary conditions. Additional keyword
        arguments are directly forwarded to the constructor of
        `RegularGridInterpolator`.

        The numba implementation respect boundary conditions, which can be set
        using the `bc` keywords argument. Supported values are the same as for
        the operators, e.g., the Laplacian. If no boundary conditions are
        specified,  natural boundary conditions are assumed, which are periodic
        conditions for periodic axes and Neumann conditions otherwise.

        Returns:
            A function which returns interpolated values when called with
            arbitrary positions within the space of the grid.
        """
        if backend == "scipy":
            return self._make_interpolator_scipy(method=method, fill=fill, **kwargs)
        elif backend == "numba":
            if method != "linear":
                raise NotImplementedError(
                    "The numba backend currently only supports linear interpolation"
                )
            return self._make_interpolator_numba(fill=fill, **kwargs)
        else:
            raise ValueError(f"Unknown backend `{backend}`")

    def interpolate(
        self,
        point: np.ndarray,
        *,
        backend: str = "numba",
        method: str = "linear",
        fill: Number = None,
        **kwargs,
    ) -> NumberOrArray:
        r"""interpolate the field to points between support points

        Args:
            point (:class:`~numpy.ndarray`):
                The points at which the values should be obtained. This is given in grid
                coordinates.
            backend (str):
                The accepted values `scipy` and `numba` determine the backend that is
                used for the interpolation.
            method (str):
                Determines the method being used for interpolation. Typical values that
                are `nearest` and `linear, but the supported values depend on the chosen
                `backend.
            fill (Number, optional):
                Determines how values out of bounds are handled. If `None`, a
                `ValueError` is raised when out-of-bounds points are requested.
                Otherwise, the given value is returned.
            \**kwargs:
                Additional keyword arguments are forwarded to the method
                :meth:`DataFieldBase.make_interpolator`.

        Returns:
            :class:`~numpy.ndarray`: the values of the field
        """
        interpolator = self.make_interpolator(
            backend=backend, method=method, fill=fill, **kwargs
        )
        return interpolator(np.asarray(point))  # type: ignore

    def interpolate_to_grid(
        self: TDataField,
        grid: GridBase,
        *,
        backend: str = "numba",
        method: str = "linear",
        fill: Number = None,
        label: Optional[str] = None,
    ) -> TDataField:
        """interpolate the data of this field to another grid.

        Args:
            grid (:class:`~pde.grids.GridBase`):
                The grid of the new field onto which the current field is
                interpolated.
            backend (str):
                The accepted values `scipy` and `numba` determine the backend that is
                used for the interpolation.
            method (str):
                Determines the method being used for interpolation. Typical values that
                are `nearest` and `linear, but the supported values depend on the chosen
                `backend.
            fill (Number, optional):
                Determines how values out of bounds are handled. If `None`, a
                `ValueError` is raised when out-of-bounds points are requested.
                Otherwise, the given value is returned.
            label (str, optional):
                Name of the returned field

        Returns:
            Field of the same rank as the current one.
        """
        if self.grid.dim != grid.dim:
            raise DimensionError(
                f"Grid dimensions are incompatible "
                f"({self.grid.dim:d} != {grid.dim:d})"
            )

        # determine the points at which data needs to be calculated
        if isinstance(grid, CartesianGridBase):
            # convert to a Cartesian grid
            points = self.grid.point_from_cartesian(grid.cell_coords)

        elif self.grid.__class__ is grid.__class__:
            # convert within the same grid class
            points = grid.cell_coords

        else:
            # this type of interpolation is not supported
            grid_in = self.grid.__class__.__name__
            grid_out = grid.__class__.__name__
            raise NotImplementedError(f"Cannot convert {grid_in} to {grid_out}")

        # interpolate the data to the grid
        data = self.interpolate(points, backend=backend, method=method, fill=fill)
        return self.__class__(grid, data, label=label)

    def add_interpolated(self, point: np.ndarray, amount: ArrayLike) -> None:
        """ deprecated alias of method `insert` """
        # this was deprecated on 2021-02-23
        warnings.warn(
            "`add_interpolated` is deprecated. Use `insert` instead",
            DeprecationWarning,
        )
        self.insert(point, amount)

    def insert(self, point: np.ndarray, amount: ArrayLike) -> None:
        """adds an (integrated) value to the field at an interpolated position

        Args:
            point (:class:`~numpy.ndarray`):
                The point inside the grid where the value is added. This is
                given in grid coordinates.
            amount (Number or :class:`~numpy.ndarray`):
                The amount that will be added to the field. The value describes
                an integrated quantity (given by the field value times the
                discretization volume). This is important for consistency with
                different discretizations and in particular grids with
                non-uniform discretizations.
        """
        point = np.atleast_1d(point)
        amount = np.broadcast_to(amount, self.data_shape)
        grid = self.grid
        grid_dim = len(grid.axes)

        if point.size != grid_dim or point.ndim != 1:
            raise DimensionError(f"Dimension mismatch for point {point}")

        point = grid.normalize_point(point, reflect=False)

        low = np.array(grid.axes_bounds)[:, 0]
        c_l, d_l = np.divmod((point - low) / grid.discretization - 0.5, 1.0)
        c_l = c_l.astype(np.intc)
        w_l = 1 - d_l  # weights of the low point
        w_h = d_l  # weights of the high point

        # determine the total weight in first iteration
        total_weight = 0
        cells = []
        for i in np.ndindex(*((2,) * grid_dim)):
            coords = np.choose(i, [c_l, c_l + 1])
            if np.all(coords >= 0) and np.all(coords < grid.shape):
                weight = np.prod(np.choose(i, [w_l, w_h]))
                total_weight += weight
                cells.append((tuple(coords), weight))

        if total_weight == 0:
            raise DomainError("Point lies outside grid")

        # alter each point in second iteration
        for coords, weight in cells:
            chng = weight * amount / (total_weight * grid.cell_volumes[coords])
            self.data[(Ellipsis,) + coords] += chng

    @fill_in_docstring
    def get_boundary_values(
        self, axis: int, upper: bool, bc: BoundariesData = "natural"
    ) -> NumberOrArray:
        """get the field values directly on the specified boundary

        Args:
            axis (int):
                The axis perpendicular to the boundary
            upper (bool):
                Whether the boundary is at the upper side of the axis
            bc:
                The boundary conditions applied to the field. {ARG_BOUNDARIES}

        Returns:
            :class:`~numpy.ndarray`: The discretized values on the boundary
        """
        interpolator = self.make_interpolator(bc=bc)
        points = self.grid._boundary_coordinates(axis, upper)
        return interpolator(points)  # type: ignore

    @fill_in_docstring
    def make_get_boundary_values(
        self, axis: int, upper: bool, bc: BoundariesData = "natural"
    ) -> Callable[[Optional[np.ndarray], Optional[np.ndarray]], NumberOrArray]:
        """make a function calculating field values on the specified boundary

        Args:
            axis (int):
                The axis perpendicular to the boundary
            upper (bool):
                Whether the boundary is at the upper side of the axis
            bc:
                The boundary conditions applied to the field. {ARG_BOUNDARIES}

        Returns:
            callable: A function returning the values on the boundary. The
            function has the signature `(data=None, out=None)`, which allows
            specifying an input and an output :class:`~numpy.ndarray`. If `data`
            is omitted, the data of the current field is used. The resulting
            interpolation is written to `out` if it is present. Otherwise, a new
            array is created.
        """
        interpolator = self.make_interpolator(bc=bc)
        points = self.grid._boundary_coordinates(axis, upper)

        # TODO: use jit_allocated_out with pre-calculated shape

        @jit
        def get_boundary_values(data: np.ndarray = None, out: np.ndarray = None):
            """interpolate the field at the boundary

            Args:
                data (:class:`~numpy.ndarray`, optional):
                    The data values that are used for interpolation. The data of
                    the current field is used if `data = None`.
                out (:class:`~numpy.ndarray`, optional):
                    The array into which the interpolated results are written. A
                    new array is created if `out = None`.

            Returns:
                :class:`~numpy.ndarray`: The interpolated values on the boundary.
            """
            res = interpolator(points, data)
            if out is None:
                return res
            else:
                # the following just copies the data from res to out. It is a
                # workaround for a bug in numba existing up to at least ver 0.49
                out[...] = res[()]
                return out

        return get_boundary_values  # type: ignore

    @abstractproperty
    def integral(self) -> NumberOrArray:
        pass

    @abstractmethod
    def to_scalar(
        self, scalar: str = "auto", *, label: Optional[str] = None
    ) -> "ScalarField":
        pass

    @property
    def average(self) -> NumberOrArray:
        """determine the average of data

        This is calculated by integrating each component of the field over space
        and dividing by the grid volume
        """
        return self.integral / self.grid.volume

    @property
    def fluctuations(self) -> NumberOrArray:
        """:class:`~numpy.ndarray`: fluctuations over the entire space.

        The fluctuations are defined as the standard deviation of the data scaled by the
        cell volume. This definition makes the fluctuations independent of the
        discretization. It corresponds to the physical scaling available in the
        :func:`~DataFieldBase.random_normal`.

        Returns:
            :class:`~numpy.ndarray`: A tensor with the same rank of the field,
            specifying the fluctuations of each component of the tensor field
            individually. Consequently, a simple scalar is returned for a
            :class:`~pde.fields.scalar.ScalarField`.
        """
        scaled_data = self.data * np.sqrt(self.grid.cell_volumes)
        axes = tuple(range(self.rank, self.data.ndim))
        return np.std(scaled_data, axis=axes)  # type: ignore

    @property
    def magnitude(self) -> float:
        """float: determine the magnitude of the field.

        This is calculated by getting a scalar field using the default arguments of the
        :func:`to_scalar` method, averaging the result over the whole grid, and taking
        the absolute value.
        """
        if self.rank == 0:
            return abs(self.average)  # type: ignore
        elif self.rank > 0:
            return abs(self.to_scalar().average)  # type: ignore
        else:
            raise AssertionError("Rank must be non-negative")

    def smooth(
        self: TDataField,
        sigma: float = 1,
        *,
        out: Optional[TDataField] = None,
        label: str = None,
    ) -> TDataField:
        """applies Gaussian smoothing with the given standard deviation

        This function respects periodic boundary conditions of the underlying
        grid, using reflection when no periodicity is specified.

        sigma (float):
            Gives the standard deviation of the smoothing in real length units
            (default: 1)
        out (FieldBase, optional):
            Optional field into which the smoothed data is stored. Setting this
            to the input field enables in-place smoothing.
        label (str, optional):
            Name of the returned field

        Returns:
            Field with smoothed data. This is stored at `out` if given.
        """
        from scipy import ndimage

        # allocate memory for storing output
        data_in = self._data
        if out is None:
            out = self.__class__(self.grid, label=self.label)
        else:
            self.assert_field_compatible(out)

        # apply Gaussian smoothing for each axis
        data_out = out._data
        for axis in range(-len(self.grid.axes), 0):
            sigma_dx = sigma / self.grid.discretization[axis]
            mode = "wrap" if self.grid.periodic[axis] else "reflect"
            ndimage.gaussian_filter1d(
                data_in, sigma=sigma_dx, axis=axis, output=data_out, mode=mode
            )
            data_in = data_out

        # return the data in the correct field class
        if label:
            out.label = label
        return out

    def get_line_data(
        self, scalar: str = "auto", extract: str = "auto"
    ) -> Dict[str, Any]:
        """return data for a line plot of the field

        Args:
            scalar (str or int):
                The method for extracting scalars as described in
                :meth:`DataFieldBase.to_scalar`.
            extract (str):
                The method used for extracting the line data. See the docstring
                of the grid method `get_line_data` to find supported values.

        Returns:
            dict: Information useful for performing a line plot of the field
        """
        # turn field into scalar field
        scalar_data = self.to_scalar(scalar).data

        # extract the line data
        data = self.grid.get_line_data(scalar_data, extract=extract)
        if "label_y" in data and data["label_y"]:
            if self.label:
                data["label_y"] = f"{self.label} ({data['label_y']})"
        else:
            data["label_y"] = self.label
        return data

    def get_image_data(
        self, scalar: str = "auto", transpose: bool = False, **kwargs
    ) -> Dict[str, Any]:
        r"""return data for plotting an image of the field

        Args:
            scalar (str or int):
                The method for extracting scalars as described in
                :meth:`DataFieldBase.to_scalar`.
            transpose (bool):
                Determines whether the transpose of the data should is plotted
            \**kwargs:
                Additional parameters are forwarded to `grid.get_image_data`

        Returns:
            dict: Information useful for plotting an image of the field
        """
        # turn field into scalar field
        scalar_data = self.to_scalar(scalar).data

        # remove imaginary parts
        if self.is_complex:
            self._logger.warning("Only the absolute value of complex data is shown")
            scalar_data = abs(scalar_data)

        # extract the image data
        data = self.grid.get_image_data(scalar_data, **kwargs)  # type: ignore
        data["title"] = self.label

        if transpose:
            # adjust image data such that the transpose is plotted
            data["data"] = data["data"].T
            data["label_x"], data["label_y"] = data["label_y"], data["label_x"]

        return data

    def get_vector_data(self, **kwargs) -> Dict[str, Any]:
        r"""return data for a vector plot of the field

        Args:
            \**kwargs: Additional parameters are forwarded to
                `grid.get_image_data`

        Returns:
            dict: Information useful for plotting an vector field
        """
        raise NotImplementedError()

    def _plot_line(
        self,
        ax,
        scalar: str = "auto",
        extract: str = "auto",
        ylabel: str = None,
        **kwargs,
    ) -> PlotReference:
        r"""visualize a field using a 1d line plot

        Args:
            ax (:class:`matplotlib.axes.Axes`):
                Figure axes to be used for plotting.
            scalar (str or int):
                The method for extracting scalars as described in
                :meth:`DataFieldBase.to_scalar`.
            extract (str):
                The method used for extracting the line data.
            ylabel (str):
                Label of the y-axis. If omitted, the label is chosen
                automatically from the data field.
            \**kwargs:
                Additional keyword arguments are passed to
                :func:`matplotlib.pyplot.plot`

        Returns:
            :class:`PlotReference`: Instance that contains information to update
            the plot with new data later.
        """
        # obtain data for the plot
        line_data = self.get_line_data(scalar=scalar, extract=extract)

        # warn if there is an imaginary part
        if np.any(np.iscomplex(line_data["data_y"])):
            self._logger.warning("Only the real part of the complex data is shown")

        # do the plot
        (line2d,) = ax.plot(line_data["data_x"], line_data["data_y"].real, **kwargs)

        # set some default properties
        ax.set_xlabel(line_data["label_x"])
        if ylabel is None:
            ylabel = line_data.get("label_y", self.label)
        if ylabel:
            ax.set_ylabel(ylabel)

        return PlotReference(ax, line2d, {"scalar": scalar, "extract": extract})

    def _update_line_plot(self, reference: PlotReference) -> None:
        """update a line plot with the current field values

        Args:
            reference (:class:`PlotReference`):
                The reference to the plot that is updated
        """
        import matplotlib as mpl

        # obtain data for the plot
        scalar = reference.parameters.get("scalar", "auto")
        extract = reference.parameters.get("extract", "auto")
        line_data = self.get_line_data(scalar=scalar, extract=extract)

        line2d = reference.element
        if isinstance(line2d, mpl.lines.Line2D):
            # update old plot
            line2d.set_xdata(line_data["data_x"])
            line2d.set_ydata(line_data["data_y"].real)

        else:
            raise ValueError(f"Unsupported plot reference {reference}")

    def _plot_image(
        self,
        ax,
        colorbar: bool = True,
        scalar: str = "auto",
        transpose: bool = False,
        **kwargs,
    ) -> PlotReference:
        r"""visualize a field using a 2d density plot

        Args:
            ax (:class:`matplotlib.axes.Axes`):
                Figure axes to be used for plotting.
            colorbar (bool):
                Determines whether a colorbar is shown
            scalar (str or int):
                The method for extracting scalars as described in
                :meth:`DataFieldBase.to_scalar`.
            transpose (bool):
                Determines whether the transpose of the data should is plotted
            \**kwargs:
                Additional keyword arguments that affect the image. For instance, some
                fields support a `scalar` argument that determines how they are
                converted to a scalar. Non-Cartesian grids might support a
                `performance_goal` argument to influence how an image is created from
                the raw data. Finally, the remaining arguments are are passed to
                :func:`matplotlib.pyplot.imshow` to affect the appearance.

        Returns:
            :class:`PlotReference`: Instance that contains information to update
            the plot with new data later.
        """
        # obtain image data with appropriate parameters
        data_kws = {}
        for arg in ["performance_goal", "scalar", "transpose"]:
            if arg in kwargs:
                data_kws[arg] = kwargs.pop(arg)
        data = self.get_image_data(scalar, transpose, **data_kws)

        if ax is None:
            import matplotlib.pyplot as plt

            # create new figure
            ax = plt.subplots()[1]

        # plot the image
        kwargs.setdefault("origin", "lower")
        kwargs.setdefault("interpolation", "none")
        axes_image = ax.imshow(data["data"].T, extent=data["extent"], **kwargs)

        # set some default properties
        ax.set_xlabel(data["label_x"])
        ax.set_ylabel(data["label_y"])
        ax.set_title(data.get("title", self.label))

        if colorbar:
            from ..tools.plotting import add_scaled_colorbar

            add_scaled_colorbar(axes_image, ax=ax)

        parameters = {"scalar": scalar, "transpose": transpose}
        return PlotReference(ax, axes_image, parameters)

    def _update_image_plot(self, reference: PlotReference) -> None:
        """update an image plot with the current field values

        Args:
            reference (:class:`PlotReference`):
                The reference to the plot that is updated
        """
        # obtain image data
        p = reference.parameters
        data = self.get_image_data(
            scalar=p.get("scalar", "auto"), transpose=p.get("transpose", False)
        )

        # update the axes image
        reference.element.set_data(data["data"].T)
        # adjust the colorbar limits
        reference.element.set_clim(data["data"].min(), data["data"].max())

    def _plot_vector(
        self,
        ax,
        method: str = "quiver",
        transpose: bool = False,
        max_points: int = 16,
        **kwargs,
    ) -> PlotReference:
        r"""visualize a field using a 2d vector plot

        Args:
            ax (:class:`matplotlib.axes.Axes`):
                Figure axes to be used for plotting.
            method (str):
                Plot type that is used. This can be either `quiver` or
                `streamplot`.
            transpose (bool):
                Determines whether the transpose of the data should be plotted.
            max_points (int):
                The maximal number of points that is used along each axis. This
                argument is only used for quiver plots.
            \**kwargs:
                Additional keyword arguments are passed to
                :func:`matplotlib.pyplot.quiver` or
                :func:`matplotlib.pyplot.streamplot`.

        Returns:
            :class:`PlotReference`: Instance that contains information to update
            the plot with new data later.
        """
        # do the plotting using the chosen method
        if method == "quiver":
            data = self.get_vector_data(transpose=transpose, max_points=max_points)
            element = ax.quiver(
                data["x"], data["y"], data["data_x"].T, data["data_y"].T, **kwargs
            )

        elif method == "streamplot":
            data = self.get_vector_data(transpose=transpose)
            element = ax.streamplot(
                data["x"], data["y"], data["data_x"].T, data["data_y"].T, **kwargs
            )

        else:
            raise ValueError(f"Vector plot `{method}` is not supported.")

        # set some default properties of the plot
        ax.set_aspect("equal")
        ax.set_xlabel(data["label_x"])
        ax.set_ylabel(data["label_y"])
        ax.set_title(data.get("title", self.label))

        parameters = {
            "method": method,
            "transpose": transpose,
            "max_points": max_points,
            "kwargs": kwargs,
        }
        return PlotReference(ax, element, parameters)

    def _update_vector_plot(self, reference: PlotReference) -> None:
        """update a vector plot with the current field values

        Args:
            reference (:class:`PlotReference`):
                The reference to the plot that is updated
        """
        method = reference.parameters.get("method", "quiver")
        transpose = reference.parameters.get("transpose", False)
        if method == "quiver":
            max_points = reference.parameters.get("max_points")
            data = self.get_vector_data(transpose=transpose, max_points=max_points)
            reference.element.set_UVC(data["data_x"], data["data_y"])

        elif method == "streamplot":
            ax = reference.ax
            kwargs = reference.parameters.get("kwargs", {})
            data = self.get_vector_data(transpose=transpose)
            # remove old streamplot
            ax.cla()
            # update with new streamplot
            reference.element = ax.streamplot(
                data["x"], data["y"], data["data_x"], data["data_y"], **kwargs
            )

        else:
            raise ValueError(f"Vector plot `{method}` is not supported.")

    def _update_plot(self, reference: PlotReference) -> None:
        """update a plot with the current field values

        Args:
            reference (:class:`PlotReference`):
                The reference to the plot to updated
        """
        import matplotlib as mpl

        # update the plot based on the given reference
        el = reference.element
        if isinstance(el, mpl.lines.Line2D):
            self._update_line_plot(reference)
        elif isinstance(el, mpl.image.AxesImage):
            self._update_image_plot(reference)
        elif isinstance(el, (mpl.quiver.Quiver, mpl.streamplot.StreamplotSet)):
            self._update_vector_plot(reference)
        else:
            raise ValueError(f"Unknown plot element {el.__class__.__name__}")

    @plot_on_axes(update_method="_update_plot")
    def plot(self, kind: str = "auto", **kwargs) -> PlotReference:
        r"""visualize the field

        Args:
            kind (str):
                Determines the visualizations. Supported values are `image`,
                `line`, `vector`, or `interactive`. Alternatively, `auto`
                determines the best visualization based on the field itself.
            {PLOT_ARGS}
            \**kwargs:
                All additional keyword arguments are forwarded to the actual
                plotting function.

        Returns:
            :class:`PlotReference`: Instance that contains information to update
            the plot with new data later.
        """
        # determine the correct kind of plotting
        if kind == "auto":
            # determine best plot for this field
            if (
                isinstance(self, DataFieldBase)
                and self.rank == 1
                and self.grid.dim == 2
            ):
                kind = "vector"
            elif len(self.grid.shape) == 1:
                kind = "line"
            else:
                kind = "image"

        elif kind == "quiver":
            kind = "vector"
            kwargs["method"] = "quiver"

        elif kind == "streamplot":
            kind = "vector"
            kwargs["method"] = "streamplot"

        # do the actual plotting
        if kind == "image":
            reference = self._plot_image(**kwargs)
        elif kind == "line":
            reference = self._plot_line(**kwargs)
        elif kind == "vector":
            reference = self._plot_vector(**kwargs)
        else:
            raise ValueError(
                f"Unsupported plot `{kind}`. Possible choices are `image`, `line`, "
                "`vector`, or `auto`."
            )

        return reference

    def _get_napari_layer_data(
        self, scalar: str = "auto", args: Dict[str, Any] = None
    ) -> Dict[str, Any]:
        """returns data for plotting on a single napari layer

        Args:
            scalar (str):
                Indicates how the scalar field is generated; see `to_scalar`
            args (dict):
                Additional arguments returned in the result, which affect how the layer
                is shown.

        Returns:
            dict: all the information necessary to plot this field
        """
        result = {} if args is None else args.copy()

        result.setdefault("scale", self.grid.discretization)
        result.setdefault("rgb", False)
        result["type"] = "image"
        result["data"] = self.to_scalar(scalar).data
        return result

    def _get_napari_data(self, **kwargs) -> Dict[str, Dict[str, Any]]:
        r"""returns data for plotting this field

        Args:
            \**kwargs: all arguments are forwarded to `_get_napari_layer_data`

        Returns:
            dict: all the information necessary to plot this field
        """
        name = "Field" if self.label is None else self.label
        return {name: self._get_napari_layer_data(**kwargs)}
