r"""Solves the viscous Burgers advection equation in 1D using a discrete PINN.

.. math::

    \partial_t u + \partial_x \frac {u^2}{2} - \sigma \partial_{xx} u
        & = f in \Omega \times (0, T) \\
    u
        & = g on \partial \Omega \times (0, T) \\
    u
        & = u_0 on \Omega \times {0}

where :math:`u: \partial \Omega \times (0, T) \to \mathbb{R}` is the unknown function,
:math:`\Omega \subset \mathbb{R}` is the spatial domain and
:math:`(0, T) \subset \mathbb{R}` is the time domain.
Homogeneous Dirichlet boundary conditions are prescribed, and the exact solution is
:math:`u(t, x) = \sin(2 \pi x) e^{-t}`.

The equation is solved on a segment domain; weak boundary conditions are used.
The natural gradient optimizer is used;
explicit and implicit time integrators are compared.
"""

import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt

from scimba_jax.domains.meshless_domains.domains_1d import Segment1D
from scimba_jax.nonlinear_approximation.approximation_spaces.approximation_spaces import (  # noqa: E501
    ApproximationSpace,
)
from scimba_jax.nonlinear_approximation.integration.monte_carlo import (
    DomainSampler,
    TensorizedSampler,
)
from scimba_jax.nonlinear_approximation.networks.mlp import MLP
from scimba_jax.nonlinear_approximation.numerical_solvers.discrete_pinns import (
    DiscretePINN,
)
from scimba_jax.physical_models.temporal_pde.viscous_burgers import ViscousBurgers
from scimba_jax.plots.plots_nd import plot_abstract_approx_spaces
from scimba_jax.time_discrete.butcher_tableau import (
    build_dirk_4_5_tableau,
    build_rk4_tableau,
)

N_COLLOC = 500
N_EPOCHS_INIT = 100
N_EPOCHS = 10

SIGMA = 1e-2

DOM_X = Segment1D((0.0, 1.0), is_main_domain=True)
DOM_T = (0.0, 1.0)
SAMPLER = TensorizedSampler([DomainSampler(DOM_X)], model_type="x")


def exact_sol(t: jnp.ndarray, x: jnp.ndarray) -> jnp.ndarray:
    """Computes the exact solution of the Burgers equation in 1D.

    Args:
        t: time values, shape (1,)
        x: spatial values, shape (1,)

    Returns:
        The exact solution of the Burgers equation in 1D, shape (1,)
    """
    return jnp.sin(2 * jnp.pi * x) * jnp.exp(-t)


def f_init(x: jnp.ndarray) -> jnp.ndarray:
    """Computes the initial condition of the Burgers equation in 1D.

    Args:
        x: spatial values, shape (1,)

    Returns:
        The initial condition of the Burgers equation in 1D, shape (1,)
    """
    return exact_sol(jnp.zeros_like(x), x)


def post_processing(approx: jnp.ndarray, x: jnp.ndarray) -> jnp.ndarray:
    """Post-processes the approximation of the Burgers equation in 1D.

    Args:
        approx: the approximation of the Burgers equation in 1D, shape (1,)
        x: spatial values, shape (1,)

    Returns:
        The post-processed approximation of the Burgers equation in 1D, shape (1,)
    """
    return approx * x * (1.0 - x)


def rhs(t: jnp.ndarray, x: jnp.ndarray, sigma: float) -> jnp.ndarray:
    """Computes the right-hand side of the Burgers equation in 1D.

    Args:
        t: time values, shape (1,)
        x: spatial values, shape (1,)
        sigma: the viscosity coefficient

    Returns:
        The right-hand side of the Burgers equation in 1D, shape (1,)
    """
    exp_neg_t = jnp.exp(-t)
    sin_x = jnp.sin(2 * jnp.pi * x)
    cos_x = jnp.cos(2 * jnp.pi * x)
    return (
        exp_neg_t
        * sin_x
        * (2 * jnp.pi * (cos_x * exp_neg_t + 2 * jnp.pi * sigma) - 1.0)
    )


# create the discrete PINN parameters

params = {
    "rk4": {
        "tableau": build_rk4_tableau(),
        "explicit_pde": ViscousBurgers(DOM_X, DOM_T, rhs, sigma=SIGMA),
        "implicit_pde": None,
    },
    "dirk_4_5": {
        "tableau": build_dirk_4_5_tableau(),
        "explicit_pde": None,
        "implicit_pde": ViscousBurgers(DOM_X, DOM_T, rhs, sigma=SIGMA),
    },
}

nt = 10
in_size = 1
out_size = 1

space = None
spaces = []


for method, param in params.items():
    butcher_tableau = param["tableau"]
    explicit_pde = param["explicit_pde"]
    implicit_pde = param["implicit_pde"]

    key = jax.random.PRNGKey(0)

    discrete_pinn = DiscretePINN(
        DOM_X, DOM_T, SAMPLER, out_size, nt, butcher_tableau, explicit_pde, implicit_pde
    )

    if space is None:
        # train the initial condition only for the first tableau
        nn = MLP(in_size=in_size, out_size=out_size, hidden_sizes=[12] * 3, key=key)
        space = ApproximationSpace(
            {"x": 1},
            [(nn, "scalar", None)],
            model_type="x",
            post_processing=post_processing,
        )

        print("Initializing the discrete PINN...")
        key, discrete_pinn = discrete_pinn.initialize(
            key, space, f_init, N_EPOCHS_INIT, N_COLLOC
        )
        space = discrete_pinn.space
        print("Initializing the discrete PINN... Done\n")

        plot_abstract_approx_spaces(
            [space], DOM_X, solution=f_init, error=f_init, title="initial condition"
        )

    print(f"\nSolving with the {method} method...")

    key, discrete_pinn = discrete_pinn.solve(key, space, N_EPOCHS, N_COLLOC)

    spaces.append(discrete_pinn.space)

    print(f"\nSolving with the {method} method... Done\n")

# %%

plot_abstract_approx_spaces(
    spaces,
    DOM_X,
    solution=lambda x: exact_sol(jnp.ones_like(x) * DOM_T[-1], x),
    error=lambda x: exact_sol(jnp.ones_like(x) * DOM_T[-1], x),
    title=f"solution at final time t={DOM_T[-1]}",
    titles=[f"solution with {method} method" for method in params.keys()],
)

plt.show()

# %%