r"""Solves the heat equation in 1D using a discrete PINN.

.. math::

    \partial_t u - \partial_{xx} u & = f in \Omega \times (0, T) \\
    \partial_x 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.

The equation is solved on a segment domain; strong (homogeneous Dirichlet) boundary conditions and natural gradient preconditioning are used.

The equation is discretized in time using:
- an explicit Euler method, where the Laplacian is treated as an explicit operator, and
- an implicit Euler method, where the Laplacian is treated as an implicit operator.
"""

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.linear_approximation.time_integrators.butcher_tableau import (
    build_explicit_euler_tableau,
    build_implicit_euler_tableau,
)
from scimba_jax.nonlinear_approximation.approximation_spaces.approximation_spaces import (
    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.heat_equations import SemiDiscreteHeatND
from scimba_jax.plots.plots_nd import plot_abstract_approx_spaces

N_COLLOC = 900
N_BC_COLLOC = 2000
N_EPOCHS_INIT = 100
N_EPOCHS = 10

DOM_X = Segment1D((0.0, 1.0), is_main_domain=True)
DOM_T = (0.0, 0.2)
SAMPLER = TensorizedSampler([DomainSampler(DOM_X)], model_type="x")


def exact_sol(t: jnp.ndarray, x: jnp.ndarray) -> jnp.ndarray:
    return jnp.exp(-(t * jnp.pi**2)) * jnp.sin(jnp.pi * x)


def f_init(x: jnp.ndarray) -> jnp.ndarray:
    t = jnp.zeros_like(x)
    return exact_sol(t, x)


def post_processing(approx: jnp.ndarray, x: jnp.ndarray) -> jnp.ndarray:
    return approx * x * (1.0 - x)


# create the discrete PINN parameters

pde = SemiDiscreteHeatND(main_domain=DOM_X, time_domain=DOM_T, bc="strong")

params = {
    "explicit euler": {
        "tableau": build_explicit_euler_tableau(),
        "explicit_pde": pde,
        "implicit_pde": None,
    },
    "implicit euler": {
        "tableau": build_implicit_euler_tableau(),
        "explicit_pde": None,
        "implicit_pde": pde,
    },
}

nt = 15
in_size = 1
out_size = 1

space = None
spaces = []
errors_over_time = []


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,
        exact_solution=exact_sol,
    )

    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] * 2, key=key)
        space = ApproximationSpace(
            {"x": 1},
            [(nn, "scalar", None)],
            model_type="x",
            post_processing=post_processing,
        )

        print("Initializing the discrete PINN...")
        key, space = discrete_pinn.initialize(
            key, space, f_init, N_EPOCHS_INIT, N_COLLOC
        )
        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, space_ = discrete_pinn.solve(key, space, N_EPOCHS, N_COLLOC)

    spaces.append(space_)
    errors_over_time.append(discrete_pinn.errors_over_time)

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

# %%

fig, ax = plt.subplots(1, 2, figsize=(12, 5))
error_type = ["L2", "Linf"]
for i in range(2):
    for j, (key, param) in enumerate(params.items()):
        ax[i].semilogy(
            jnp.linspace(DOM_T[0], DOM_T[1], nt + 1),
            errors_over_time[j][:, i],
            label=key,
        )
    ax[i].set_xlabel("time")
    ax[i].set_ylabel(f"relative {error_type[i]} error")
    ax[i].set_title(f"Relative {error_type[i]} error vs time")
    ax[i].legend()
    ax[i].grid()

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()

# # %%

# %%