Get a understanding at a higher level on the provided code. Try to think about a possible implementation of LSTM and RNN by using the same style (with using neurons as objects)
import numpy as np

def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def sigmoid_derivative(x):
    s = sigmoid(x)
    return s * (1 - s)

class Neuron:
    def __init__(self, input_units):
        self.weights = np.random.randn(input_units) * np.sqrt(2. / input_units)
        self.bias = 0.0
        self.input = None
        self.output = None
        self.grad_weight = np.zeros_like(self.weights)
        self.grad_bias = 0.0

    def forward(self, input_data):
        """
        Perform the forward pass for a single neuron.
        """
        self.input = input_data  # Shape: (batch_size, input_units)
        weighted_input = np.dot(self.input, self.weights) + self.bias  # Shape: (batch_size,)
        self.output = weighted_input  # Linear activation (to be modified by an activation function if needed)
        return self.output

    def backward(self, grad_output):
        """
        Perform the backward pass for a single neuron.
        """
        # Compute gradients w.r.t weights and bias
        batch_size = self.input.shape[0]
        self.grad_weight += np.dot(self.input.T, grad_output) / batch_size  # Shape: (input_units,)
        self.grad_bias += np.mean(grad_output)
        # Compute the gradient to pass to the previous layer
        grad_input = np.dot(grad_output.reshape(-1,1), self.weights.reshape(1,-1))  # Shape: (batch_size, input_units)
        return grad_input

    def update_parameters(self, learning_rate):
        """
        Update the neuron's weights and bias.
        """
        self.weights -= learning_rate * self.grad_weight
        self.bias -= learning_rate * self.grad_bias
        # Reset gradients
        self.grad_weight = np.zeros_like(self.weights)
        self.grad_bias = 0.0

class Layer:
    def ____init____(self):
        self.inputs = None
        self.outputs = None

    def forward(self, inputs):
        raise NotImplementedError

    def backward(self, grad_output):
        raise NotImplementedError

    def update_parameters(self, learning_rate):
        pass

class DenseLayer(Layer):
    def __init__(self, input_units, output_units):
        super().__init__()
        # __init__ialize a list of neurons
        self.neurons = [Neuron(input_units) for _ in range(output_units)]
        self.inputs = None
        self.outputs = None

    def forward(self, inputs):
        """
        Perform the forward pass through the layer.
        """
        self.inputs = inputs  # Shape: (batch_size, input_units)
        # Collect outputs from all neurons
        self.outputs = np.array([neuron.forward(inputs) for neuron in self.neurons]).T  # Shape: (batch_size, output_units)
        return self.outputs

    def backward(self, grad_output):
        """
        Perform the backward pass through the layer.
        """
        # __init__ialize gradient to pass to previous layer
        grad_input_total = np.zeros_like(self.inputs, dtype='float64')  # Shape: (batch_size, input_units)

        # For each neuron, perform backward pass
        for idx, neuron in enumerate(self.neurons):
            grad_output_neuron = grad_output[:, idx]  # Shape: (batch_size,)
            grad_input_neuron = neuron.backward(grad_output_neuron)  # Shape: (batch_size, input_units)
            grad_input_total += grad_input_neuron  # Sum gradients from all neurons

        return grad_input_total

    def update_parameters(self, learning_rate):
        """
        Update parameters for all neurons in the layer.
        """
        for neuron in self.neurons:
            neuron.update_parameters(learning_rate)

class ActivationLayer(Layer):
    def __init__(self, activation, activation_derivative):
        super().__init__()
        self.activation = activation
        self.activation_derivative = activation_derivative
        self.inputs = None
        self.outputs = None

    def forward(self, inputs):
        """
        Apply the activation function.
        """
        self.inputs = inputs
        self.outputs = self.activation(inputs)
        return self.outputs

    def backward(self, grad_output):
        """
        Compute the backward pass through the activation function.
        """
        grad_input = grad_output * self.activation_derivative(self.inputs)
        return grad_input
    
class NeuralNetwork:
    def __init__(self):
        self.layers = []

    def add(self, layer):
        self.layers.append(layer)

    def forward(self, X):
        output = X
        for layer in self.layers:
            output = layer.forward(output)
        return output

    def backward(self, grad_output):
        for layer in reversed(self.layers):
            grad_output = layer.backward(grad_output)

    def update_parameters(self, learning_rate):
        for layer in self.layers:
            layer.update_parameters(learning_rate)

    def train(self, X, Y, epochs, batch_size, learning_rate):
        n_samples = len(X)
        for epoch in range(epochs):
            permutation = np.random.permutation(n_samples)
            X_shuffled = X[permutation]
            Y_shuffled = Y[permutation]

            for i in range(0, n_samples, batch_size):
                X_batch = X_shuffled[i:i+batch_size]
                Y_batch = Y_shuffled[i:i+batch_size]

                predictions = self.forward(X_batch)
                loss = np.mean((predictions - Y_batch) ** 2)

                if (epoch+1) % 1000 == 0 and i == 0:
                    print(f"Epoch {epoch+1}/{epochs}, Loss: {loss}")

                grad_loss = (predictions - Y_batch) * (2 / batch_size)
                self.backward(grad_loss)
                self.update_parameters(learning_rate)

    def predict(self, X):
        return self.forward(X)

# Create network instance
network = NeuralNetwork()

# Adding layers
network.add(DenseLayer(input_units=2, output_units=10))
network.add(ActivationLayer(activation=sigmoid, activation_derivative=sigmoid_derivative))
network.add(DenseLayer(input_units=10, output_units=1))
network.add(ActivationLayer(activation=sigmoid, activation_derivative=sigmoid_derivative))

# XOR input and output
X = np.array([
    [0, 0],
    [0, 1],
    [1, 0],
    [1, 1]
])

Y = np.array([
    [0],
    [1],
    [1],
    [0]
])

# Training parameters
epochs = 10000
batch_size = 2
learning_rate = 0.1

# Train the network
network.train(X, Y, epochs, batch_size, learning_rate)

# Make predictions
outputs = network.predict(X)

print("\nPredictions:")
for input_data, output in zip(X, outputs):
    print(f"Input: {input_data}, Predicted Output: {output.round(2)}")



Some additional information about rnn and lstm:
# mypy: allow-untyped-decorators
# mypy: allow-untyped-defs
import math
import numbers
import warnings
import weakref
from typing import List, Optional, overload, Tuple
from typing_extensions import deprecated

import torch
from torch import _VF, Tensor
from torch.nn import init
from torch.nn.parameter import Parameter
from torch.nn.utils.rnn import PackedSequence

from .module import Module


__all__ = [
    "RNNBase",
    "RNN",
    "LSTM",
    "GRU",
    "RNNCellBase",
    "RNNCell",
    "LSTMCell",
    "GRUCell",
]

_rnn_impls = {
    "RNN_TANH": _VF.rnn_tanh,
    "RNN_RELU": _VF.rnn_relu,
}


def _apply_permutation(tensor: Tensor, permutation: Tensor, dim: int = 1) -> Tensor:
    return tensor.index_select(dim, permutation)


@deprecated(
    "`apply_permutation` is deprecated, please use `tensor.index_select(dim, permutation)` instead",
    category=FutureWarning,
)
def apply_permutation(tensor: Tensor, permutation: Tensor, dim: int = 1) -> Tensor:
    return _apply_permutation(tensor, permutation, dim)


class RNNBase(Module):
    r"""Base class for RNN modules (RNN, LSTM, GRU).

    Implements aspects of RNNs shared by the RNN, LSTM, and GRU classes, such as module initialization
    and utility methods for parameter storage management.

    .. note::
        The forward method is not implemented by the RNNBase class.

    .. note::
        LSTM and GRU classes override some methods implemented by RNNBase.
    """

    __constants__ = [
        "mode",
        "input_size",
        "hidden_size",
        "num_layers",
        "bias",
        "batch_first",
        "dropout",
        "bidirectional",
        "proj_size",
    ]
    __jit_unused_properties__ = ["all_weights"]

    mode: str
    input_size: int
    hidden_size: int
    num_layers: int
    bias: bool
    batch_first: bool
    dropout: float
    bidirectional: bool
    proj_size: int

    def __init__(
        self,
        mode: str,
        input_size: int,
        hidden_size: int,
        num_layers: int = 1,
        bias: bool = True,
        batch_first: bool = False,
        dropout: float = 0.0,
        bidirectional: bool = False,
        proj_size: int = 0,
        device=None,
        dtype=None,
    ) -> None:
        factory_kwargs = {"device": device, "dtype": dtype}
        super().__init__()
        self.mode = mode
        self.input_size = input_size
        self.hidden_size = hidden_size
        self.num_layers = num_layers
        self.bias = bias
        self.batch_first = batch_first
        self.dropout = float(dropout)
        self.bidirectional = bidirectional
        self.proj_size = proj_size
        self._flat_weight_refs: List[Optional[weakref.ReferenceType[Parameter]]] = []
        num_directions = 2 if bidirectional else 1

        if (
            not isinstance(dropout, numbers.Number)
            or not 0 <= dropout <= 1
            or isinstance(dropout, bool)
        ):
            raise ValueError(
                "dropout should be a number in range [0, 1] "
                "representing the probability of an element being "
                "zeroed"
            )
        if dropout > 0 and num_layers == 1:
            warnings.warn(
                "dropout option adds dropout after all but last "
                "recurrent layer, so non-zero dropout expects "
                f"num_layers greater than 1, but got dropout={dropout} and "
                f"num_layers={num_layers}"
            )

        if not isinstance(hidden_size, int):
            raise TypeError(
                f"hidden_size should be of type int, got: {type(hidden_size).__name__}"
            )
        if hidden_size <= 0:
            raise ValueError("hidden_size must be greater than zero")
        if num_layers <= 0:
            raise ValueError("num_layers must be greater than zero")
        if proj_size < 0:
            raise ValueError(
                "proj_size should be a positive integer or zero to disable projections"
            )
        if proj_size >= hidden_size:
            raise ValueError("proj_size has to be smaller than hidden_size")

        if mode == "LSTM":
            gate_size = 4 * hidden_size
        elif mode == "GRU":
            gate_size = 3 * hidden_size
        elif mode == "RNN_TANH":
            gate_size = hidden_size
        elif mode == "RNN_RELU":
            gate_size = hidden_size
        else:
            raise ValueError("Unrecognized RNN mode: " + mode)

        self._flat_weights_names = []
        self._all_weights = []
        for layer in range(num_layers):
            for direction in range(num_directions):
                real_hidden_size = proj_size if proj_size > 0 else hidden_size
                layer_input_size = (
                    input_size if layer == 0 else real_hidden_size * num_directions
                )

                w_ih = Parameter(
                    torch.empty((gate_size, layer_input_size), **factory_kwargs)
                )
                w_hh = Parameter(
                    torch.empty((gate_size, real_hidden_size), **factory_kwargs)
                )
                b_ih = Parameter(torch.empty(gate_size, **factory_kwargs))
                # Second bias vector included for CuDNN compatibility. Only one
                # bias vector is needed in standard definition.
                b_hh = Parameter(torch.empty(gate_size, **factory_kwargs))
                layer_params: Tuple[Tensor, ...] = ()
                if self.proj_size == 0:
                    if bias:
                        layer_params = (w_ih, w_hh, b_ih, b_hh)
                    else:
                        layer_params = (w_ih, w_hh)
                else:
                    w_hr = Parameter(
                        torch.empty((proj_size, hidden_size), **factory_kwargs)
                    )
                    if bias:
                        layer_params = (w_ih, w_hh, b_ih, b_hh, w_hr)
                    else:
                        layer_params = (w_ih, w_hh, w_hr)

                suffix = "_reverse" if direction == 1 else ""
                param_names = ["weight_ih_l{}{}", "weight_hh_l{}{}"]
                if bias:
                    param_names += ["bias_ih_l{}{}", "bias_hh_l{}{}"]
                if self.proj_size > 0:
                    param_names += ["weight_hr_l{}{}"]
                param_names = [x.format(layer, suffix) for x in param_names]

                for name, param in zip(param_names, layer_params):
                    setattr(self, name, param)
                self._flat_weights_names.extend(param_names)
                self._all_weights.append(param_names)

        self._init_flat_weights()

        self.reset_parameters()

    def _init_flat_weights(self):
        self._flat_weights = [
            getattr(self, wn) if hasattr(self, wn) else None
            for wn in self._flat_weights_names
        ]
        self._flat_weight_refs = [
            weakref.ref(w) if w is not None else None for w in self._flat_weights
        ]
        self.flatten_parameters()

    def __setattr__(self, attr, value):
        if hasattr(self, "_flat_weights_names") and attr in self._flat_weights_names:
            # keep self._flat_weights up to date if you do self.weight = ...
            idx = self._flat_weights_names.index(attr)
            self._flat_weights[idx] = value
        super().__setattr__(attr, value)

    def flatten_parameters(self) -> None:
        """Reset parameter data pointer so that they can use faster code paths.

        Right now, this works only if the module is on the GPU and cuDNN is enabled.
        Otherwise, it's a no-op.
        """
        # Short-circuits if _flat_weights is only partially instantiated
        if len(self._flat_weights) != len(self._flat_weights_names):
            return

        for w in self._flat_weights:
            if not isinstance(w, Tensor):
                return
        # Short-circuits if any tensor in self._flat_weights is not acceptable to cuDNN
        # or the tensors in _flat_weights are of different dtypes

        first_fw = self._flat_weights[0]
        dtype = first_fw.dtype
        for fw in self._flat_weights:
            if (
                not isinstance(fw, Tensor)
                or not (fw.dtype == dtype)
                or not fw.is_cuda
                or not torch.backends.cudnn.is_acceptable(fw)
            ):
                return

        # If any parameters alias, we fall back to the slower, copying code path. This is
        # a sufficient check, because overlapping parameter buffers that don't completely
        # alias would break the assumptions of the uniqueness check in
        # Module.named_parameters().
        unique_data_ptrs = {p.data_ptr() for p in self._flat_weights}
        if len(unique_data_ptrs) != len(self._flat_weights):
            return

        with torch.cuda.device_of(first_fw):
            import torch.backends.cudnn.rnn as rnn

            # Note: no_grad() is necessary since _cudnn_rnn_flatten_weight is
            # an inplace operation on self._flat_weights
            with torch.no_grad():
                if torch._use_cudnn_rnn_flatten_weight():
                    num_weights = 4 if self.bias else 2
                    if self.proj_size > 0:
                        num_weights += 1
                    torch._cudnn_rnn_flatten_weight(
                        self._flat_weights,
                        num_weights,
                        self.input_size,
                        rnn.get_cudnn_mode(self.mode),
                        self.hidden_size,
                        self.proj_size,
                        self.num_layers,
                        self.batch_first,
                        bool(self.bidirectional),
                    )

    def _apply(self, fn, recurse=True):
        self._flat_weight_refs = []
        ret = super()._apply(fn, recurse)

        # Resets _flat_weights
        # Note: be v. careful before removing this, as 3rd party device types
        # likely rely on this behavior to properly .to() modules like LSTM.
        self._init_flat_weights()

        return ret

    def reset_parameters(self) -> None:
        stdv = 1.0 / math.sqrt(self.hidden_size) if self.hidden_size > 0 else 0
        for weight in self.parameters():
            init.uniform_(weight, -stdv, stdv)

    def check_input(self, input: Tensor, batch_sizes: Optional[Tensor]) -> None:
        if not torch.jit.is_scripting():
            if (
                input.dtype != self._flat_weights[0].dtype
                and not torch._C._is_any_autocast_enabled()
            ):
                raise ValueError(
                    f"input must have the type {self._flat_weights[0].dtype}, got type {input.dtype}"
                )
        expected_input_dim = 2 if batch_sizes is not None else 3
        if input.dim() != expected_input_dim:
            raise RuntimeError(
                f"input must have {expected_input_dim} dimensions, got {input.dim()}"
            )
        if self.input_size != input.size(-1):
            raise RuntimeError(
                f"input.size(-1) must be equal to input_size. Expected {self.input_size}, got {input.size(-1)}"
            )

    def get_expected_hidden_size(
        self, input: Tensor, batch_sizes: Optional[Tensor]
    ) -> Tuple[int, int, int]:
        if batch_sizes is not None:
            mini_batch = int(batch_sizes[0])
        else:
            mini_batch = input.size(0) if self.batch_first else input.size(1)
        num_directions = 2 if self.bidirectional else 1
        if self.proj_size > 0:
            expected_hidden_size = (
                self.num_layers * num_directions,
                mini_batch,
                self.proj_size,
            )
        else:
            expected_hidden_size = (
                self.num_layers * num_directions,
                mini_batch,
                self.hidden_size,
            )
        return expected_hidden_size

    def check_hidden_size(
        self,
        hx: Tensor,
        expected_hidden_size: Tuple[int, int, int],
        msg: str = "Expected hidden size {}, got {}",
    ) -> None:
        if hx.size() != expected_hidden_size:
            raise RuntimeError(msg.format(expected_hidden_size, list(hx.size())))

    def _weights_have_changed(self):
        # Returns True if the weight tensors have changed since the last forward pass.
        # This is the case when used with torch.func.functional_call(), for example.
        weights_changed = False
        for ref, name in zip(self._flat_weight_refs, self._flat_weights_names):
            weight = getattr(self, name) if hasattr(self, name) else None
            if weight is not None and ref is not None and ref() is not weight:
                weights_changed = True
                break
        return weights_changed

    def check_forward_args(
        self, input: Tensor, hidden: Tensor, batch_sizes: Optional[Tensor]
    ):
        self.check_input(input, batch_sizes)
        expected_hidden_size = self.get_expected_hidden_size(input, batch_sizes)

        self.check_hidden_size(hidden, expected_hidden_size)

    def permute_hidden(self, hx: Tensor, permutation: Optional[Tensor]):
        if permutation is None:
            return hx
        return _apply_permutation(hx, permutation)

    def extra_repr(self) -> str:
        s = "{input_size}, {hidden_size}"
        if self.proj_size != 0:
            s += ", proj_size={proj_size}"
        if self.num_layers != 1:
            s += ", num_layers={num_layers}"
        if self.bias is not True:
            s += ", bias={bias}"
        if self.batch_first is not False:
            s += ", batch_first={batch_first}"
        if self.dropout != 0:
            s += ", dropout={dropout}"
        if self.bidirectional is not False:
            s += ", bidirectional={bidirectional}"
        return s.format(**self.__dict__)

    def _update_flat_weights(self):
        if not torch.jit.is_scripting():
            if self._weights_have_changed():
                self._init_flat_weights()

    def __getstate__(self):
        # If weights have been changed, update the _flat_weights in __getstate__ here.
        self._update_flat_weights()
        # Don't serialize the weight references.
        state = self.__dict__.copy()
        del state["_flat_weight_refs"]
        return state

    def __setstate__(self, d):
        super().__setstate__(d)
        if "all_weights" in d:
            self._all_weights = d["all_weights"]
        # In PyTorch 1.8 we added a proj_size member variable to LSTM.
        # LSTMs that were serialized via torch.save(module) before PyTorch 1.8
        # don't have it, so to preserve compatibility we set proj_size here.
        if "proj_size" not in d:
            self.proj_size = 0

        if not isinstance(self._all_weights[0][0], str):
            num_layers = self.num_layers
            num_directions = 2 if self.bidirectional else 1
            self._flat_weights_names = []
            self._all_weights = []
            for layer in range(num_layers):
                for direction in range(num_directions):
                    suffix = "_reverse" if direction == 1 else ""
                    weights = [
                        "weight_ih_l{}{}",
                        "weight_hh_l{}{}",
                        "bias_ih_l{}{}",
                        "bias_hh_l{}{}",
                        "weight_hr_l{}{}",
                    ]
                    weights = [x.format(layer, suffix) for x in weights]
                    if self.bias:
                        if self.proj_size > 0:
                            self._all_weights += [weights]
                            self._flat_weights_names.extend(weights)
                        else:
                            self._all_weights += [weights[:4]]
                            self._flat_weights_names.extend(weights[:4])
                    else:
                        if self.proj_size > 0:
                            self._all_weights += [weights[:2]] + [weights[-1:]]
                            self._flat_weights_names.extend(
                                weights[:2] + [weights[-1:]]
                            )
                        else:
                            self._all_weights += [weights[:2]]
                            self._flat_weights_names.extend(weights[:2])
            self._flat_weights = [
                getattr(self, wn) if hasattr(self, wn) else None
                for wn in self._flat_weights_names
            ]

        self._flat_weight_refs = [
            weakref.ref(w) if w is not None else None for w in self._flat_weights
        ]

    @property
    def all_weights(self) -> List[List[Parameter]]:
        return [
            [getattr(self, weight) for weight in weights]
            for weights in self._all_weights
        ]

    def _replicate_for_data_parallel(self):
        replica = super()._replicate_for_data_parallel()
        # Need to copy these caches, otherwise the replica will share the same
        # flat weights list.
        replica._flat_weights = replica._flat_weights[:]
        replica._flat_weights_names = replica._flat_weights_names[:]
        return replica


class RNN(RNNBase):
    r"""__init__(input_size,hidden_size,num_layers=1,nonlinearity='tanh',bias=True,batch_first=False,dropout=0.0,bidirectional=False,device=None,dtype=None)

    Apply a multi-layer Elman RNN with :math:`\tanh` or :math:`\text{ReLU}`
    non-linearity to an input sequence. For each element in the input sequence,
    each layer computes the following function:

    .. math::
        h_t = \tanh(x_t W_{ih}^T + b_{ih} + h_{t-1}W_{hh}^T + b_{hh})

    where :math:`h_t` is the hidden state at time `t`, :math:`x_t` is
    the input at time `t`, and :math:`h_{(t-1)}` is the hidden state of the
    previous layer at time `t-1` or the initial hidden state at time `0`.
    If :attr:`nonlinearity` is ``'relu'``, then :math:`\text{ReLU}` is used instead of :math:`\tanh`.

    .. code-block:: python

        # Efficient implementation equivalent to the following with bidirectional=False
        def forward(x, h_0=None):
            if batch_first:
                x = x.transpose(0, 1)
            seq_len, batch_size, _ = x.size()
            if h_0 is None:
                h_0 = torch.zeros(num_layers, batch_size, hidden_size)
            h_t_minus_1 = h_0
            h_t = h_0
            output = []
            for t in range(seq_len):
                for layer in range(num_layers):
                    h_t[layer] = torch.tanh(
                        x[t] @ weight_ih[layer].T
                        + bias_ih[layer]
                        + h_t_minus_1[layer] @ weight_hh[layer].T
                        + bias_hh[layer]
                    )
                output.append(h_t[-1])
                h_t_minus_1 = h_t
            output = torch.stack(output)
            if batch_first:
                output = output.transpose(0, 1)
            return output, h_t

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        num_layers: Number of recurrent layers. E.g., setting ``num_layers=2``
            would mean stacking two RNNs together to form a `stacked RNN`,
            with the second RNN taking in outputs of the first RNN and
            computing the final results. Default: 1
        nonlinearity: The non-linearity to use. Can be either ``'tanh'`` or ``'relu'``. Default: ``'tanh'``
        bias: If ``False``, then the layer does not use bias weights `b_ih` and `b_hh`.
            Default: ``True``
        batch_first: If ``True``, then the input and output tensors are provided
            as `(batch, seq, feature)` instead of `(seq, batch, feature)`.
            Note that this does not apply to hidden or cell states. See the
            Inputs/Outputs sections below for details.  Default: ``False``
        dropout: If non-zero, introduces a `Dropout` layer on the outputs of each
            RNN layer except the last layer, with dropout probability equal to
            :attr:`dropout`. Default: 0
        bidirectional: If ``True``, becomes a bidirectional RNN. Default: ``False``

    Inputs: input, h_0
        * **input**: tensor of shape :math:`(L, H_{in})` for unbatched input,
          :math:`(L, N, H_{in})` when ``batch_first=False`` or
          :math:`(N, L, H_{in})` when ``batch_first=True`` containing the features of
          the input sequence.  The input can also be a packed variable length sequence.
          See :func:`torch.nn.utils.rnn.pack_padded_sequence` or
          :func:`torch.nn.utils.rnn.pack_sequence` for details.
        * **h_0**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{out})` containing the initial hidden
          state for the input sequence batch. Defaults to zeros if not provided.

        where:

        .. math::
            \begin{aligned}
                N ={} & \text{batch size} \\
                L ={} & \text{sequence length} \\
                D ={} & 2 \text{ if bidirectional=True otherwise } 1 \\
                H_{in} ={} & \text{input\_size} \\
                H_{out} ={} & \text{hidden\_size}
            \end{aligned}

    Outputs: output, h_n
        * **output**: tensor of shape :math:`(L, D * H_{out})` for unbatched input,
          :math:`(L, N, D * H_{out})` when ``batch_first=False`` or
          :math:`(N, L, D * H_{out})` when ``batch_first=True`` containing the output features
          `(h_t)` from the last layer of the RNN, for each `t`. If a
          :class:`torch.nn.utils.rnn.PackedSequence` has been given as the input, the output
          will also be a packed sequence.
        * **h_n**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{out})` containing the final hidden state
          for each element in the batch.

    Attributes:
        weight_ih_l[k]: the learnable input-hidden weights of the k-th layer,
            of shape `(hidden_size, input_size)` for `k = 0`. Otherwise, the shape is
            `(hidden_size, num_directions * hidden_size)`
        weight_hh_l[k]: the learnable hidden-hidden weights of the k-th layer,
            of shape `(hidden_size, hidden_size)`
        bias_ih_l[k]: the learnable input-hidden bias of the k-th layer,
            of shape `(hidden_size)`
        bias_hh_l[k]: the learnable hidden-hidden bias of the k-th layer,
            of shape `(hidden_size)`

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    .. note::
        For bidirectional RNNs, forward and backward are directions 0 and 1 respectively.
        Example of splitting the output layers when ``batch_first=False``:
        ``output.view(seq_len, batch, num_directions, hidden_size)``.

    .. note::
        ``batch_first`` argument is ignored for unbatched inputs.

    .. include:: ../cudnn_rnn_determinism.rst

    .. include:: ../cudnn_persistent_rnn.rst

    Examples::

        >>> rnn = nn.RNN(10, 20, 2)
        >>> input = torch.randn(5, 3, 10)
        >>> h0 = torch.randn(2, 3, 20)
        >>> output, hn = rnn(input, h0)
    """

    @overload
    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        num_layers: int = 1,
        nonlinearity: str = "tanh",
        bias: bool = True,
        batch_first: bool = False,
        dropout: float = 0.0,
        bidirectional: bool = False,
        device=None,
        dtype=None,
    ) -> None:
        ...

    @overload
    def __init__(self, *args, **kwargs):
        ...

    def __init__(self, *args, **kwargs):
        if "proj_size" in kwargs:
            raise ValueError(
                "proj_size argument is only supported for LSTM, not RNN or GRU"
            )
        if len(args) > 3:
            self.nonlinearity = args[3]
            args = args[:3] + args[4:]
        else:
            self.nonlinearity = kwargs.pop("nonlinearity", "tanh")
        if self.nonlinearity == "tanh":
            mode = "RNN_TANH"
        elif self.nonlinearity == "relu":
            mode = "RNN_RELU"
        else:
            raise ValueError(
                f"Unknown nonlinearity '{self.nonlinearity}'. Select from 'tanh' or 'relu'."
            )
        super().__init__(mode, *args, **kwargs)

    @overload
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: Tensor, hx: Optional[Tensor] = None
    ) -> Tuple[Tensor, Tensor]:
        pass

    @overload
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: PackedSequence, hx: Optional[Tensor] = None
    ) -> Tuple[PackedSequence, Tensor]:
        pass

    def forward(self, input, hx=None):  # noqa: F811
        self._update_flat_weights()

        num_directions = 2 if self.bidirectional else 1
        orig_input = input

        if isinstance(orig_input, PackedSequence):
            input, batch_sizes, sorted_indices, unsorted_indices = input
            max_batch_size = batch_sizes[0]
            # script() is unhappy when max_batch_size is different type in cond branches, so we duplicate
            if hx is None:
                hx = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
            else:
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                hx = self.permute_hidden(hx, sorted_indices)
        else:
            batch_sizes = None
            if input.dim() not in (2, 3):
                raise ValueError(
                    f"RNN: Expected input to be 2D or 3D, got {input.dim()}D tensor instead"
                )
            is_batched = input.dim() == 3
            batch_dim = 0 if self.batch_first else 1
            if not is_batched:
                input = input.unsqueeze(batch_dim)
                if hx is not None:
                    if hx.dim() != 2:
                        raise RuntimeError(
                            f"For unbatched 2-D input, hx should also be 2-D but got {hx.dim()}-D tensor"
                        )
                    hx = hx.unsqueeze(1)
            else:
                if hx is not None and hx.dim() != 3:
                    raise RuntimeError(
                        f"For batched 3-D input, hx should also be 3-D but got {hx.dim()}-D tensor"
                    )
            max_batch_size = input.size(0) if self.batch_first else input.size(1)
            sorted_indices = None
            unsorted_indices = None
            if hx is None:
                hx = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
            else:
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                hx = self.permute_hidden(hx, sorted_indices)

        assert hx is not None
        self.check_forward_args(input, hx, batch_sizes)
        assert self.mode == "RNN_TANH" or self.mode == "RNN_RELU"
        if batch_sizes is None:
            if self.mode == "RNN_TANH":
                result = _VF.rnn_tanh(
                    input,
                    hx,
                    self._flat_weights,
                    self.bias,
                    self.num_layers,
                    self.dropout,
                    self.training,
                    self.bidirectional,
                    self.batch_first,
                )
            else:
                result = _VF.rnn_relu(
                    input,
                    hx,
                    self._flat_weights,
                    self.bias,
                    self.num_layers,
                    self.dropout,
                    self.training,
                    self.bidirectional,
                    self.batch_first,
                )
        else:
            if self.mode == "RNN_TANH":
                result = _VF.rnn_tanh(
                    input,
                    batch_sizes,
                    hx,
                    self._flat_weights,
                    self.bias,
                    self.num_layers,
                    self.dropout,
                    self.training,
                    self.bidirectional,
                )
            else:
                result = _VF.rnn_relu(
                    input,
                    batch_sizes,
                    hx,
                    self._flat_weights,
                    self.bias,
                    self.num_layers,
                    self.dropout,
                    self.training,
                    self.bidirectional,
                )

        output = result[0]
        hidden = result[1]

        if isinstance(orig_input, PackedSequence):
            output_packed = PackedSequence(
                output, batch_sizes, sorted_indices, unsorted_indices
            )
            return output_packed, self.permute_hidden(hidden, unsorted_indices)

        if not is_batched:  # type: ignore[possibly-undefined]
            output = output.squeeze(batch_dim)  # type: ignore[possibly-undefined]
            hidden = hidden.squeeze(1)

        return output, self.permute_hidden(hidden, unsorted_indices)


# XXX: LSTM and GRU implementation is different from RNNBase, this is because:
# 1. we want to support nn.LSTM and nn.GRU in TorchScript and TorchScript in
#    its current state could not support the python Union Type or Any Type
# 2. TorchScript static typing does not allow a Function or Callable type in
#    Dict values, so we have to separately call _VF instead of using _rnn_impls
# 3. This is temporary only and in the transition state that we want to make it
#    on time for the release
#
# More discussion details in https://github.com/pytorch/pytorch/pull/23266
#
# TODO: remove the overriding implementations for LSTM and GRU when TorchScript
# support expressing these two modules generally.


class LSTM(RNNBase):
    r"""__init__(input_size,hidden_size,num_layers=1,bias=True,batch_first=False,dropout=0.0,bidirectional=False,proj_size=0,device=None,dtype=None)

    Apply a multi-layer long short-term memory (LSTM) RNN to an input sequence.
    For each element in the input sequence, each layer computes the following
    function:

    .. math::
        \begin{array}{ll} \\
            i_t = \sigma(W_{ii} x_t + b_{ii} + W_{hi} h_{t-1} + b_{hi}) \\
            f_t = \sigma(W_{if} x_t + b_{if} + W_{hf} h_{t-1} + b_{hf}) \\
            g_t = \tanh(W_{ig} x_t + b_{ig} + W_{hg} h_{t-1} + b_{hg}) \\
            o_t = \sigma(W_{io} x_t + b_{io} + W_{ho} h_{t-1} + b_{ho}) \\
            c_t = f_t \odot c_{t-1} + i_t \odot g_t \\
            h_t = o_t \odot \tanh(c_t) \\
        \end{array}

    where :math:`h_t` is the hidden state at time `t`, :math:`c_t` is the cell
    state at time `t`, :math:`x_t` is the input at time `t`, :math:`h_{t-1}`
    is the hidden state of the layer at time `t-1` or the initial hidden
    state at time `0`, and :math:`i_t`, :math:`f_t`, :math:`g_t`,
    :math:`o_t` are the input, forget, cell, and output gates, respectively.
    :math:`\sigma` is the sigmoid function, and :math:`\odot` is the Hadamard product.

    In a multilayer LSTM, the input :math:`x^{(l)}_t` of the :math:`l` -th layer
    (:math:`l \ge 2`) is the hidden state :math:`h^{(l-1)}_t` of the previous layer multiplied by
    dropout :math:`\delta^{(l-1)}_t` where each :math:`\delta^{(l-1)}_t` is a Bernoulli random
    variable which is :math:`0` with probability :attr:`dropout`.

    If ``proj_size > 0`` is specified, LSTM with projections will be used. This changes
    the LSTM cell in the following way. First, the dimension of :math:`h_t` will be changed from
    ``hidden_size`` to ``proj_size`` (dimensions of :math:`W_{hi}` will be changed accordingly).
    Second, the output hidden state of each layer will be multiplied by a learnable projection
    matrix: :math:`h_t = W_{hr}h_t`. Note that as a consequence of this, the output
    of LSTM network will be of different shape as well. See Inputs/Outputs sections below for exact
    dimensions of all variables. You can find more details in https://arxiv.org/abs/1402.1128.

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        num_layers: Number of recurrent layers. E.g., setting ``num_layers=2``
            would mean stacking two LSTMs together to form a `stacked LSTM`,
            with the second LSTM taking in outputs of the first LSTM and
            computing the final results. Default: 1
        bias: If ``False``, then the layer does not use bias weights `b_ih` and `b_hh`.
            Default: ``True``
        batch_first: If ``True``, then the input and output tensors are provided
            as `(batch, seq, feature)` instead of `(seq, batch, feature)`.
            Note that this does not apply to hidden or cell states. See the
            Inputs/Outputs sections below for details.  Default: ``False``
        dropout: If non-zero, introduces a `Dropout` layer on the outputs of each
            LSTM layer except the last layer, with dropout probability equal to
            :attr:`dropout`. Default: 0
        bidirectional: If ``True``, becomes a bidirectional LSTM. Default: ``False``
        proj_size: If ``> 0``, will use LSTM with projections of corresponding size. Default: 0

    Inputs: input, (h_0, c_0)
        * **input**: tensor of shape :math:`(L, H_{in})` for unbatched input,
          :math:`(L, N, H_{in})` when ``batch_first=False`` or
          :math:`(N, L, H_{in})` when ``batch_first=True`` containing the features of
          the input sequence.  The input can also be a packed variable length sequence.
          See :func:`torch.nn.utils.rnn.pack_padded_sequence` or
          :func:`torch.nn.utils.rnn.pack_sequence` for details.
        * **h_0**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{out})` containing the
          initial hidden state for each element in the input sequence.
          Defaults to zeros if (h_0, c_0) is not provided.
        * **c_0**: tensor of shape :math:`(D * \text{num\_layers}, H_{cell})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{cell})` containing the
          initial cell state for each element in the input sequence.
          Defaults to zeros if (h_0, c_0) is not provided.

        where:

        .. math::
            \begin{aligned}
                N ={} & \text{batch size} \\
                L ={} & \text{sequence length} \\
                D ={} & 2 \text{ if bidirectional=True otherwise } 1 \\
                H_{in} ={} & \text{input\_size} \\
                H_{cell} ={} & \text{hidden\_size} \\
                H_{out} ={} & \text{proj\_size if } \text{proj\_size}>0 \text{ otherwise hidden\_size} \\
            \end{aligned}

    Outputs: output, (h_n, c_n)
        * **output**: tensor of shape :math:`(L, D * H_{out})` for unbatched input,
          :math:`(L, N, D * H_{out})` when ``batch_first=False`` or
          :math:`(N, L, D * H_{out})` when ``batch_first=True`` containing the output features
          `(h_t)` from the last layer of the LSTM, for each `t`. If a
          :class:`torch.nn.utils.rnn.PackedSequence` has been given as the input, the output
          will also be a packed sequence. When ``bidirectional=True``, `output` will contain
          a concatenation of the forward and reverse hidden states at each time step in the sequence.
        * **h_n**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{out})` containing the
          final hidden state for each element in the sequence. When ``bidirectional=True``,
          `h_n` will contain a concatenation of the final forward and reverse hidden states, respectively.
        * **c_n**: tensor of shape :math:`(D * \text{num\_layers}, H_{cell})` for unbatched input or
          :math:`(D * \text{num\_layers}, N, H_{cell})` containing the
          final cell state for each element in the sequence. When ``bidirectional=True``,
          `c_n` will contain a concatenation of the final forward and reverse cell states, respectively.

    Attributes:
        weight_ih_l[k] : the learnable input-hidden weights of the :math:`\text{k}^{th}` layer
            `(W_ii|W_if|W_ig|W_io)`, of shape `(4*hidden_size, input_size)` for `k = 0`.
            Otherwise, the shape is `(4*hidden_size, num_directions * hidden_size)`. If
            ``proj_size > 0`` was specified, the shape will be
            `(4*hidden_size, num_directions * proj_size)` for `k > 0`
        weight_hh_l[k] : the learnable hidden-hidden weights of the :math:`\text{k}^{th}` layer
            `(W_hi|W_hf|W_hg|W_ho)`, of shape `(4*hidden_size, hidden_size)`. If ``proj_size > 0``
            was specified, the shape will be `(4*hidden_size, proj_size)`.
        bias_ih_l[k] : the learnable input-hidden bias of the :math:`\text{k}^{th}` layer
            `(b_ii|b_if|b_ig|b_io)`, of shape `(4*hidden_size)`
        bias_hh_l[k] : the learnable hidden-hidden bias of the :math:`\text{k}^{th}` layer
            `(b_hi|b_hf|b_hg|b_ho)`, of shape `(4*hidden_size)`
        weight_hr_l[k] : the learnable projection weights of the :math:`\text{k}^{th}` layer
            of shape `(proj_size, hidden_size)`. Only present when ``proj_size > 0`` was
            specified.
        weight_ih_l[k]_reverse: Analogous to `weight_ih_l[k]` for the reverse direction.
            Only present when ``bidirectional=True``.
        weight_hh_l[k]_reverse:  Analogous to `weight_hh_l[k]` for the reverse direction.
            Only present when ``bidirectional=True``.
        bias_ih_l[k]_reverse:  Analogous to `bias_ih_l[k]` for the reverse direction.
            Only present when ``bidirectional=True``.
        bias_hh_l[k]_reverse:  Analogous to `bias_hh_l[k]` for the reverse direction.
            Only present when ``bidirectional=True``.
        weight_hr_l[k]_reverse:  Analogous to `weight_hr_l[k]` for the reverse direction.
            Only present when ``bidirectional=True`` and ``proj_size > 0`` was specified.

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    .. note::
        For bidirectional LSTMs, forward and backward are directions 0 and 1 respectively.
        Example of splitting the output layers when ``batch_first=False``:
        ``output.view(seq_len, batch, num_directions, hidden_size)``.

    .. note::
        For bidirectional LSTMs, `h_n` is not equivalent to the last element of `output`; the
        former contains the final forward and reverse hidden states, while the latter contains the
        final forward hidden state and the initial reverse hidden state.

    .. note::
        ``batch_first`` argument is ignored for unbatched inputs.

    .. note::
        ``proj_size`` should be smaller than ``hidden_size``.

    .. include:: ../cudnn_rnn_determinism.rst

    .. include:: ../cudnn_persistent_rnn.rst

    Examples::

        >>> rnn = nn.LSTM(10, 20, 2)
        >>> input = torch.randn(5, 3, 10)
        >>> h0 = torch.randn(2, 3, 20)
        >>> c0 = torch.randn(2, 3, 20)
        >>> output, (hn, cn) = rnn(input, (h0, c0))
    """

    @overload
    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        num_layers: int = 1,
        bias: bool = True,
        batch_first: bool = False,
        dropout: float = 0.0,
        bidirectional: bool = False,
        proj_size: int = 0,
        device=None,
        dtype=None,
    ) -> None:
        ...

    @overload
    def __init__(self, *args, **kwargs):
        ...

    def __init__(self, *args, **kwargs):
        super().__init__("LSTM", *args, **kwargs)

    def get_expected_cell_size(
        self, input: Tensor, batch_sizes: Optional[Tensor]
    ) -> Tuple[int, int, int]:
        if batch_sizes is not None:
            mini_batch = int(batch_sizes[0])
        else:
            mini_batch = input.size(0) if self.batch_first else input.size(1)
        num_directions = 2 if self.bidirectional else 1
        expected_hidden_size = (
            self.num_layers * num_directions,
            mini_batch,
            self.hidden_size,
        )
        return expected_hidden_size

    # In the future, we should prevent mypy from applying contravariance rules here.
    # See torch/nn/modules/module.py::_forward_unimplemented
    def check_forward_args(
        self,
        input: Tensor,
        hidden: Tuple[Tensor, Tensor],  # type: ignore[override]
        batch_sizes: Optional[Tensor],
    ):
        self.check_input(input, batch_sizes)
        self.check_hidden_size(
            hidden[0],
            self.get_expected_hidden_size(input, batch_sizes),
            "Expected hidden[0] size {}, got {}",
        )
        self.check_hidden_size(
            hidden[1],
            self.get_expected_cell_size(input, batch_sizes),
            "Expected hidden[1] size {}, got {}",
        )

    # Same as above, see torch/nn/modules/module.py::_forward_unimplemented
    def permute_hidden(  # type: ignore[override]
        self,
        hx: Tuple[Tensor, Tensor],
        permutation: Optional[Tensor],
    ) -> Tuple[Tensor, Tensor]:
        if permutation is None:
            return hx
        return _apply_permutation(hx[0], permutation), _apply_permutation(
            hx[1], permutation
        )

    # Same as above, see torch/nn/modules/module.py::_forward_unimplemented
    @overload  # type: ignore[override]
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: Tensor, hx: Optional[Tuple[Tensor, Tensor]] = None
    ) -> Tuple[Tensor, Tuple[Tensor, Tensor]]:  # noqa: F811
        pass

    # Same as above, see torch/nn/modules/module.py::_forward_unimplemented
    @overload
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: PackedSequence, hx: Optional[Tuple[Tensor, Tensor]] = None
    ) -> Tuple[PackedSequence, Tuple[Tensor, Tensor]]:  # noqa: F811
        pass

    def forward(self, input, hx=None):  # noqa: F811
        self._update_flat_weights()

        orig_input = input
        # xxx: isinstance check needs to be in conditional for TorchScript to compile
        batch_sizes = None
        do_permute = False
        num_directions = 2 if self.bidirectional else 1
        real_hidden_size = self.proj_size if self.proj_size > 0 else self.hidden_size
        if isinstance(orig_input, PackedSequence):
            input, batch_sizes, sorted_indices, unsorted_indices = input
            max_batch_size = batch_sizes[0]
            if hx is None:
                h_zeros = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    real_hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
                c_zeros = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
                hx = (h_zeros, c_zeros)
            else:
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                hx = self.permute_hidden(hx, sorted_indices)
        else:
            if input.dim() not in (2, 3):
                raise ValueError(
                    f"LSTM: Expected input to be 2D or 3D, got {input.dim()}D instead"
                )
            is_batched = input.dim() == 3
            batch_dim = 0 if self.batch_first else 1
            if not is_batched:
                input = input.unsqueeze(batch_dim)
            max_batch_size = input.size(0) if self.batch_first else input.size(1)
            sorted_indices = None
            unsorted_indices = None
            if hx is None:
                h_zeros = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    real_hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
                c_zeros = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
                hx = (h_zeros, c_zeros)
                self.check_forward_args(input, hx, batch_sizes)
            else:
                if is_batched:
                    if hx[0].dim() != 3 or hx[1].dim() != 3:
                        msg = (
                            "For batched 3-D input, hx and cx should "
                            f"also be 3-D but got ({hx[0].dim()}-D, {hx[1].dim()}-D) tensors"
                        )
                        raise RuntimeError(msg)
                else:
                    if hx[0].dim() != 2 or hx[1].dim() != 2:
                        msg = (
                            "For unbatched 2-D input, hx and cx should "
                            f"also be 2-D but got ({hx[0].dim()}-D, {hx[1].dim()}-D) tensors"
                        )
                        raise RuntimeError(msg)
                    hx = (hx[0].unsqueeze(1), hx[1].unsqueeze(1))
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                self.check_forward_args(input, hx, batch_sizes)
                hx = self.permute_hidden(hx, sorted_indices)

        if batch_sizes is None:
            result = _VF.lstm(
                input,
                hx,
                self._flat_weights,
                self.bias,
                self.num_layers,
                self.dropout,
                self.training,
                self.bidirectional,
                self.batch_first,
            )
        else:
            result = _VF.lstm(
                input,
                batch_sizes,
                hx,
                self._flat_weights,
                self.bias,
                self.num_layers,
                self.dropout,
                self.training,
                self.bidirectional,
            )
        output = result[0]
        hidden = result[1:]
        # xxx: isinstance check needs to be in conditional for TorchScript to compile
        if isinstance(orig_input, PackedSequence):
            output_packed = PackedSequence(
                output, batch_sizes, sorted_indices, unsorted_indices
            )
            return output_packed, self.permute_hidden(hidden, unsorted_indices)
        else:
            if not is_batched:  # type: ignore[possibly-undefined]
                output = output.squeeze(batch_dim)  # type: ignore[possibly-undefined]
                hidden = (hidden[0].squeeze(1), hidden[1].squeeze(1))
            return output, self.permute_hidden(hidden, unsorted_indices)


class GRU(RNNBase):
    r"""__init__(input_size,hidden_size,num_layers=1,bias=True,batch_first=False,dropout=0.0,bidirectional=False,device=None,dtype=None)

    Apply a multi-layer gated recurrent unit (GRU) RNN to an input sequence.
    For each element in the input sequence, each layer computes the following
    function:

    .. math::
        \begin{array}{ll}
            r_t = \sigma(W_{ir} x_t + b_{ir} + W_{hr} h_{(t-1)} + b_{hr}) \\
            z_t = \sigma(W_{iz} x_t + b_{iz} + W_{hz} h_{(t-1)} + b_{hz}) \\
            n_t = \tanh(W_{in} x_t + b_{in} + r_t \odot (W_{hn} h_{(t-1)}+ b_{hn})) \\
            h_t = (1 - z_t) \odot n_t + z_t \odot h_{(t-1)}
        \end{array}

    where :math:`h_t` is the hidden state at time `t`, :math:`x_t` is the input
    at time `t`, :math:`h_{(t-1)}` is the hidden state of the layer
    at time `t-1` or the initial hidden state at time `0`, and :math:`r_t`,
    :math:`z_t`, :math:`n_t` are the reset, update, and new gates, respectively.
    :math:`\sigma` is the sigmoid function, and :math:`\odot` is the Hadamard product.

    In a multilayer GRU, the input :math:`x^{(l)}_t` of the :math:`l` -th layer
    (:math:`l \ge 2`) is the hidden state :math:`h^{(l-1)}_t` of the previous layer multiplied by
    dropout :math:`\delta^{(l-1)}_t` where each :math:`\delta^{(l-1)}_t` is a Bernoulli random
    variable which is :math:`0` with probability :attr:`dropout`.

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        num_layers: Number of recurrent layers. E.g., setting ``num_layers=2``
            would mean stacking two GRUs together to form a `stacked GRU`,
            with the second GRU taking in outputs of the first GRU and
            computing the final results. Default: 1
        bias: If ``False``, then the layer does not use bias weights `b_ih` and `b_hh`.
            Default: ``True``
        batch_first: If ``True``, then the input and output tensors are provided
            as `(batch, seq, feature)` instead of `(seq, batch, feature)`.
            Note that this does not apply to hidden or cell states. See the
            Inputs/Outputs sections below for details.  Default: ``False``
        dropout: If non-zero, introduces a `Dropout` layer on the outputs of each
            GRU layer except the last layer, with dropout probability equal to
            :attr:`dropout`. Default: 0
        bidirectional: If ``True``, becomes a bidirectional GRU. Default: ``False``

    Inputs: input, h_0
        * **input**: tensor of shape :math:`(L, H_{in})` for unbatched input,
          :math:`(L, N, H_{in})` when ``batch_first=False`` or
          :math:`(N, L, H_{in})` when ``batch_first=True`` containing the features of
          the input sequence.  The input can also be a packed variable length sequence.
          See :func:`torch.nn.utils.rnn.pack_padded_sequence` or
          :func:`torch.nn.utils.rnn.pack_sequence` for details.
        * **h_0**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` or
          :math:`(D * \text{num\_layers}, N, H_{out})`
          containing the initial hidden state for the input sequence. Defaults to zeros if not provided.

        where:

        .. math::
            \begin{aligned}
                N ={} & \text{batch size} \\
                L ={} & \text{sequence length} \\
                D ={} & 2 \text{ if bidirectional=True otherwise } 1 \\
                H_{in} ={} & \text{input\_size} \\
                H_{out} ={} & \text{hidden\_size}
            \end{aligned}

    Outputs: output, h_n
        * **output**: tensor of shape :math:`(L, D * H_{out})` for unbatched input,
          :math:`(L, N, D * H_{out})` when ``batch_first=False`` or
          :math:`(N, L, D * H_{out})` when ``batch_first=True`` containing the output features
          `(h_t)` from the last layer of the GRU, for each `t`. If a
          :class:`torch.nn.utils.rnn.PackedSequence` has been given as the input, the output
          will also be a packed sequence.
        * **h_n**: tensor of shape :math:`(D * \text{num\_layers}, H_{out})` or
          :math:`(D * \text{num\_layers}, N, H_{out})` containing the final hidden state
          for the input sequence.

    Attributes:
        weight_ih_l[k] : the learnable input-hidden weights of the :math:`\text{k}^{th}` layer
            (W_ir|W_iz|W_in), of shape `(3*hidden_size, input_size)` for `k = 0`.
            Otherwise, the shape is `(3*hidden_size, num_directions * hidden_size)`
        weight_hh_l[k] : the learnable hidden-hidden weights of the :math:`\text{k}^{th}` layer
            (W_hr|W_hz|W_hn), of shape `(3*hidden_size, hidden_size)`
        bias_ih_l[k] : the learnable input-hidden bias of the :math:`\text{k}^{th}` layer
            (b_ir|b_iz|b_in), of shape `(3*hidden_size)`
        bias_hh_l[k] : the learnable hidden-hidden bias of the :math:`\text{k}^{th}` layer
            (b_hr|b_hz|b_hn), of shape `(3*hidden_size)`

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    .. note::
        For bidirectional GRUs, forward and backward are directions 0 and 1 respectively.
        Example of splitting the output layers when ``batch_first=False``:
        ``output.view(seq_len, batch, num_directions, hidden_size)``.

    .. note::
        ``batch_first`` argument is ignored for unbatched inputs.

    .. note::
        The calculation of new gate :math:`n_t` subtly differs from the original paper and other frameworks.
        In the original implementation, the Hadamard product :math:`(\odot)` between :math:`r_t` and the
        previous hidden state :math:`h_{(t-1)}` is done before the multiplication with the weight matrix
        `W` and addition of bias:

        .. math::
            \begin{aligned}
                n_t = \tanh(W_{in} x_t + b_{in} + W_{hn} ( r_t \odot h_{(t-1)} ) + b_{hn})
            \end{aligned}

        This is in contrast to PyTorch implementation, which is done after :math:`W_{hn} h_{(t-1)}`

        .. math::
            \begin{aligned}
                n_t = \tanh(W_{in} x_t + b_{in} + r_t \odot (W_{hn} h_{(t-1)}+ b_{hn}))
            \end{aligned}

        This implementation differs on purpose for efficiency.

    .. include:: ../cudnn_persistent_rnn.rst

    Examples::

        >>> rnn = nn.GRU(10, 20, 2)
        >>> input = torch.randn(5, 3, 10)
        >>> h0 = torch.randn(2, 3, 20)
        >>> output, hn = rnn(input, h0)
    """

    @overload
    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        num_layers: int = 1,
        bias: bool = True,
        batch_first: bool = False,
        dropout: float = 0.0,
        bidirectional: bool = False,
        device=None,
        dtype=None,
    ) -> None:
        ...

    @overload
    def __init__(self, *args, **kwargs):
        ...

    def __init__(self, *args, **kwargs):
        if "proj_size" in kwargs:
            raise ValueError(
                "proj_size argument is only supported for LSTM, not RNN or GRU"
            )
        super().__init__("GRU", *args, **kwargs)

    @overload  # type: ignore[override]
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: Tensor, hx: Optional[Tensor] = None
    ) -> Tuple[Tensor, Tensor]:  # noqa: F811
        pass

    @overload
    @torch._jit_internal._overload_method  # noqa: F811
    def forward(
        self, input: PackedSequence, hx: Optional[Tensor] = None
    ) -> Tuple[PackedSequence, Tensor]:  # noqa: F811
        pass

    def forward(self, input, hx=None):  # noqa: F811
        self._update_flat_weights()

        orig_input = input
        # xxx: isinstance check needs to be in conditional for TorchScript to compile
        if isinstance(orig_input, PackedSequence):
            input, batch_sizes, sorted_indices, unsorted_indices = input
            max_batch_size = batch_sizes[0]
            if hx is None:
                num_directions = 2 if self.bidirectional else 1
                hx = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
            else:
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                hx = self.permute_hidden(hx, sorted_indices)
        else:
            batch_sizes = None
            if input.dim() not in (2, 3):
                raise ValueError(
                    f"GRU: Expected input to be 2D or 3D, got {input.dim()}D instead"
                )
            is_batched = input.dim() == 3
            batch_dim = 0 if self.batch_first else 1
            if not is_batched:
                input = input.unsqueeze(batch_dim)
                if hx is not None:
                    if hx.dim() != 2:
                        raise RuntimeError(
                            f"For unbatched 2-D input, hx should also be 2-D but got {hx.dim()}-D tensor"
                        )
                    hx = hx.unsqueeze(1)
            else:
                if hx is not None and hx.dim() != 3:
                    raise RuntimeError(
                        f"For batched 3-D input, hx should also be 3-D but got {hx.dim()}-D tensor"
                    )
            max_batch_size = input.size(0) if self.batch_first else input.size(1)
            sorted_indices = None
            unsorted_indices = None
            if hx is None:
                num_directions = 2 if self.bidirectional else 1
                hx = torch.zeros(
                    self.num_layers * num_directions,
                    max_batch_size,
                    self.hidden_size,
                    dtype=input.dtype,
                    device=input.device,
                )
            else:
                # Each batch of the hidden state should match the input sequence that
                # the user believes he/she is passing in.
                hx = self.permute_hidden(hx, sorted_indices)

        self.check_forward_args(input, hx, batch_sizes)
        if batch_sizes is None:
            result = _VF.gru(
                input,
                hx,
                self._flat_weights,
                self.bias,
                self.num_layers,
                self.dropout,
                self.training,
                self.bidirectional,
                self.batch_first,
            )
        else:
            result = _VF.gru(
                input,
                batch_sizes,
                hx,
                self._flat_weights,
                self.bias,
                self.num_layers,
                self.dropout,
                self.training,
                self.bidirectional,
            )
        output = result[0]
        hidden = result[1]

        # xxx: isinstance check needs to be in conditional for TorchScript to compile
        if isinstance(orig_input, PackedSequence):
            output_packed = PackedSequence(
                output, batch_sizes, sorted_indices, unsorted_indices
            )
            return output_packed, self.permute_hidden(hidden, unsorted_indices)
        else:
            if not is_batched:  # type: ignore[possibly-undefined]
                output = output.squeeze(batch_dim)  # type: ignore[possibly-undefined]
                hidden = hidden.squeeze(1)

            return output, self.permute_hidden(hidden, unsorted_indices)


class RNNCellBase(Module):
    __constants__ = ["input_size", "hidden_size", "bias"]

    input_size: int
    hidden_size: int
    bias: bool
    weight_ih: Tensor
    weight_hh: Tensor
    # WARNING: bias_ih and bias_hh purposely not defined here.
    # See https://github.com/pytorch/pytorch/issues/39670

    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        bias: bool,
        num_chunks: int,
        device=None,
        dtype=None,
    ) -> None:
        factory_kwargs = {"device": device, "dtype": dtype}
        super().__init__()
        self.input_size = input_size
        self.hidden_size = hidden_size
        self.bias = bias
        self.weight_ih = Parameter(
            torch.empty((num_chunks * hidden_size, input_size), **factory_kwargs)
        )
        self.weight_hh = Parameter(
            torch.empty((num_chunks * hidden_size, hidden_size), **factory_kwargs)
        )
        if bias:
            self.bias_ih = Parameter(
                torch.empty(num_chunks * hidden_size, **factory_kwargs)
            )
            self.bias_hh = Parameter(
                torch.empty(num_chunks * hidden_size, **factory_kwargs)
            )
        else:
            self.register_parameter("bias_ih", None)
            self.register_parameter("bias_hh", None)

        self.reset_parameters()

    def extra_repr(self) -> str:
        s = "{input_size}, {hidden_size}"
        if "bias" in self.__dict__ and self.bias is not True:
            s += ", bias={bias}"
        if "nonlinearity" in self.__dict__ and self.nonlinearity != "tanh":
            s += ", nonlinearity={nonlinearity}"
        return s.format(**self.__dict__)

    def reset_parameters(self) -> None:
        stdv = 1.0 / math.sqrt(self.hidden_size) if self.hidden_size > 0 else 0
        for weight in self.parameters():
            init.uniform_(weight, -stdv, stdv)


class RNNCell(RNNCellBase):
    r"""An Elman RNN cell with tanh or ReLU non-linearity.

    .. math::

        h' = \tanh(W_{ih} x + b_{ih}  +  W_{hh} h + b_{hh})

    If :attr:`nonlinearity` is `'relu'`, then ReLU is used in place of tanh.

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        bias: If ``False``, then the layer does not use bias weights `b_ih` and `b_hh`.
            Default: ``True``
        nonlinearity: The non-linearity to use. Can be either ``'tanh'`` or ``'relu'``. Default: ``'tanh'``

    Inputs: input, hidden
        - **input**: tensor containing input features
        - **hidden**: tensor containing the initial hidden state
          Defaults to zero if not provided.

    Outputs: h'
        - **h'** of shape `(batch, hidden_size)`: tensor containing the next hidden state
          for each element in the batch

    Shape:
        - input: :math:`(N, H_{in})` or :math:`(H_{in})` tensor containing input features where
          :math:`H_{in}` = `input_size`.
        - hidden: :math:`(N, H_{out})` or :math:`(H_{out})` tensor containing the initial hidden
          state where :math:`H_{out}` = `hidden_size`. Defaults to zero if not provided.
        - output: :math:`(N, H_{out})` or :math:`(H_{out})` tensor containing the next hidden state.

    Attributes:
        weight_ih: the learnable input-hidden weights, of shape
            `(hidden_size, input_size)`
        weight_hh: the learnable hidden-hidden weights, of shape
            `(hidden_size, hidden_size)`
        bias_ih: the learnable input-hidden bias, of shape `(hidden_size)`
        bias_hh: the learnable hidden-hidden bias, of shape `(hidden_size)`

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    Examples::

        >>> rnn = nn.RNNCell(10, 20)
        >>> input = torch.randn(6, 3, 10)
        >>> hx = torch.randn(3, 20)
        >>> output = []
        >>> for i in range(6):
        ...     hx = rnn(input[i], hx)
        ...     output.append(hx)
    """

    __constants__ = ["input_size", "hidden_size", "bias", "nonlinearity"]
    nonlinearity: str

    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        bias: bool = True,
        nonlinearity: str = "tanh",
        device=None,
        dtype=None,
    ) -> None:
        factory_kwargs = {"device": device, "dtype": dtype}
        super().__init__(input_size, hidden_size, bias, num_chunks=1, **factory_kwargs)
        self.nonlinearity = nonlinearity

    def forward(self, input: Tensor, hx: Optional[Tensor] = None) -> Tensor:
        if input.dim() not in (1, 2):
            raise ValueError(
                f"RNNCell: Expected input to be 1D or 2D, got {input.dim()}D instead"
            )
        if hx is not None and hx.dim() not in (1, 2):
            raise ValueError(
                f"RNNCell: Expected hidden to be 1D or 2D, got {hx.dim()}D instead"
            )
        is_batched = input.dim() == 2
        if not is_batched:
            input = input.unsqueeze(0)

        if hx is None:
            hx = torch.zeros(
                input.size(0), self.hidden_size, dtype=input.dtype, device=input.device
            )
        else:
            hx = hx.unsqueeze(0) if not is_batched else hx

        if self.nonlinearity == "tanh":
            ret = _VF.rnn_tanh_cell(
                input,
                hx,
                self.weight_ih,
                self.weight_hh,
                self.bias_ih,
                self.bias_hh,
            )
        elif self.nonlinearity == "relu":
            ret = _VF.rnn_relu_cell(
                input,
                hx,
                self.weight_ih,
                self.weight_hh,
                self.bias_ih,
                self.bias_hh,
            )
        else:
            ret = input  # TODO: remove when jit supports exception flow
            raise RuntimeError(f"Unknown nonlinearity: {self.nonlinearity}")

        if not is_batched:
            ret = ret.squeeze(0)

        return ret


class LSTMCell(RNNCellBase):
    r"""A long short-term memory (LSTM) cell.

    .. math::

        \begin{array}{ll}
        i = \sigma(W_{ii} x + b_{ii} + W_{hi} h + b_{hi}) \\
        f = \sigma(W_{if} x + b_{if} + W_{hf} h + b_{hf}) \\
        g = \tanh(W_{ig} x + b_{ig} + W_{hg} h + b_{hg}) \\
        o = \sigma(W_{io} x + b_{io} + W_{ho} h + b_{ho}) \\
        c' = f \odot c + i \odot g \\
        h' = o \odot \tanh(c') \\
        \end{array}

    where :math:`\sigma` is the sigmoid function, and :math:`\odot` is the Hadamard product.

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        bias: If ``False``, then the layer does not use bias weights `b_ih` and
            `b_hh`. Default: ``True``

    Inputs: input, (h_0, c_0)
        - **input** of shape `(batch, input_size)` or `(input_size)`: tensor containing input features
        - **h_0** of shape `(batch, hidden_size)` or `(hidden_size)`: tensor containing the initial hidden state
        - **c_0** of shape `(batch, hidden_size)` or `(hidden_size)`: tensor containing the initial cell state

          If `(h_0, c_0)` is not provided, both **h_0** and **c_0** default to zero.

    Outputs: (h_1, c_1)
        - **h_1** of shape `(batch, hidden_size)` or `(hidden_size)`: tensor containing the next hidden state
        - **c_1** of shape `(batch, hidden_size)` or `(hidden_size)`: tensor containing the next cell state

    Attributes:
        weight_ih: the learnable input-hidden weights, of shape
            `(4*hidden_size, input_size)`
        weight_hh: the learnable hidden-hidden weights, of shape
            `(4*hidden_size, hidden_size)`
        bias_ih: the learnable input-hidden bias, of shape `(4*hidden_size)`
        bias_hh: the learnable hidden-hidden bias, of shape `(4*hidden_size)`

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    On certain ROCm devices, when using float16 inputs this module will use :ref:`different precision<fp16_on_mi200>` for backward.

    Examples::

        >>> rnn = nn.LSTMCell(10, 20)  # (input_size, hidden_size)
        >>> input = torch.randn(2, 3, 10)  # (time_steps, batch, input_size)
        >>> hx = torch.randn(3, 20)  # (batch, hidden_size)
        >>> cx = torch.randn(3, 20)
        >>> output = []
        >>> for i in range(input.size()[0]):
        ...     hx, cx = rnn(input[i], (hx, cx))
        ...     output.append(hx)
        >>> output = torch.stack(output, dim=0)
    """

    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        bias: bool = True,
        device=None,
        dtype=None,
    ) -> None:
        factory_kwargs = {"device": device, "dtype": dtype}
        super().__init__(input_size, hidden_size, bias, num_chunks=4, **factory_kwargs)

    def forward(
        self, input: Tensor, hx: Optional[Tuple[Tensor, Tensor]] = None
    ) -> Tuple[Tensor, Tensor]:
        if input.dim() not in (1, 2):
            raise ValueError(
                f"LSTMCell: Expected input to be 1D or 2D, got {input.dim()}D instead"
            )
        if hx is not None:
            for idx, value in enumerate(hx):
                if value.dim() not in (1, 2):
                    raise ValueError(
                        f"LSTMCell: Expected hx[{idx}] to be 1D or 2D, got {value.dim()}D instead"
                    )
        is_batched = input.dim() == 2
        if not is_batched:
            input = input.unsqueeze(0)

        if hx is None:
            zeros = torch.zeros(
                input.size(0), self.hidden_size, dtype=input.dtype, device=input.device
            )
            hx = (zeros, zeros)
        else:
            hx = (hx[0].unsqueeze(0), hx[1].unsqueeze(0)) if not is_batched else hx

        ret = _VF.lstm_cell(
            input,
            hx,
            self.weight_ih,
            self.weight_hh,
            self.bias_ih,
            self.bias_hh,
        )

        if not is_batched:
            ret = (ret[0].squeeze(0), ret[1].squeeze(0))
        return ret


class GRUCell(RNNCellBase):
    r"""A gated recurrent unit (GRU) cell.

    .. math::

        \begin{array}{ll}
        r = \sigma(W_{ir} x + b_{ir} + W_{hr} h + b_{hr}) \\
        z = \sigma(W_{iz} x + b_{iz} + W_{hz} h + b_{hz}) \\
        n = \tanh(W_{in} x + b_{in} + r \odot (W_{hn} h + b_{hn})) \\
        h' = (1 - z) \odot n + z \odot h
        \end{array}

    where :math:`\sigma` is the sigmoid function, and :math:`\odot` is the Hadamard product.

    Args:
        input_size: The number of expected features in the input `x`
        hidden_size: The number of features in the hidden state `h`
        bias: If ``False``, then the layer does not use bias weights `b_ih` and
            `b_hh`. Default: ``True``

    Inputs: input, hidden
        - **input** : tensor containing input features
        - **hidden** : tensor containing the initial hidden
          state for each element in the batch.
          Defaults to zero if not provided.

    Outputs: h'
        - **h'** : tensor containing the next hidden state
          for each element in the batch

    Shape:
        - input: :math:`(N, H_{in})` or :math:`(H_{in})` tensor containing input features where
          :math:`H_{in}` = `input_size`.
        - hidden: :math:`(N, H_{out})` or :math:`(H_{out})` tensor containing the initial hidden
          state where :math:`H_{out}` = `hidden_size`. Defaults to zero if not provided.
        - output: :math:`(N, H_{out})` or :math:`(H_{out})` tensor containing the next hidden state.

    Attributes:
        weight_ih: the learnable input-hidden weights, of shape
            `(3*hidden_size, input_size)`
        weight_hh: the learnable hidden-hidden weights, of shape
            `(3*hidden_size, hidden_size)`
        bias_ih: the learnable input-hidden bias, of shape `(3*hidden_size)`
        bias_hh: the learnable hidden-hidden bias, of shape `(3*hidden_size)`

    .. note::
        All the weights and biases are initialized from :math:`\mathcal{U}(-\sqrt{k}, \sqrt{k})`
        where :math:`k = \frac{1}{\text{hidden\_size}}`

    On certain ROCm devices, when using float16 inputs this module will use :ref:`different precision<fp16_on_mi200>` for backward.

    Examples::

        >>> rnn = nn.GRUCell(10, 20)
        >>> input = torch.randn(6, 3, 10)
        >>> hx = torch.randn(3, 20)
        >>> output = []
        >>> for i in range(6):
        ...     hx = rnn(input[i], hx)
        ...     output.append(hx)
    """

    def __init__(
        self,
        input_size: int,
        hidden_size: int,
        bias: bool = True,
        device=None,
        dtype=None,
    ) -> None:
        factory_kwargs = {"device": device, "dtype": dtype}
        super().__init__(input_size, hidden_size, bias, num_chunks=3, **factory_kwargs)

    def forward(self, input: Tensor, hx: Optional[Tensor] = None) -> Tensor:
        if input.dim() not in (1, 2):
            raise ValueError(
                f"GRUCell: Expected input to be 1D or 2D, got {input.dim()}D instead"
            )
        if hx is not None and hx.dim() not in (1, 2):
            raise ValueError(
                f"GRUCell: Expected hidden to be 1D or 2D, got {hx.dim()}D instead"
            )
        is_batched = input.dim() == 2
        if not is_batched:
            input = input.unsqueeze(0)

        if hx is None:
            hx = torch.zeros(
                input.size(0), self.hidden_size, dtype=input.dtype, device=input.device
            )
        else:
            hx = hx.unsqueeze(0) if not is_batched else hx

        ret = _VF.gru_cell(
            input,
            hx,
            self.weight_ih,
            self.weight_hh,
            self.bias_ih,
            self.bias_hh,
        )

        if not is_batched:
            ret = ret.squeeze(0)

        return ret