PINNs: grad-div system 2D

Solves a grad-div system in 2D with Dirichlet boundary conditions using a PINN.

\[\begin{split}\left\{\begin{array}{rl} -\nabla (\nabla \cdot u) + u & = f \text{ in } \Omega \times M \\ u & = g \text{ on } \partial \Omega \times M\end{array}\right.\end{split}\]

where \(u: \Omega \times M \to \mathbb{R}^2\) is the unknown function, \(\Omega \subset \mathbb{R}^2\) is the spatial domain and \(M \subset \mathbb{R}\) is the parametric domain and \(f: \Omega \times M \to \mathbb{R}^2\).

The equation is solved on a square domain with PINNs with energy natural gradient preconditioning and PINNs with Anagram preconditioning.

Weak and strong boundary conditions are used.

from typing import Callable, Tuple

import matplotlib.pyplot as plt
import torch

from scimba_torch.approximation_space.abstract_space import AbstractApproxSpace
from scimba_torch.approximation_space.nn_space import NNxSpace
from scimba_torch.domain.meshless_domain.domain_2d import Square2D
from scimba_torch.integration.monte_carlo import DomainSampler, TensorizedSampler
from scimba_torch.integration.monte_carlo_parameters import UniformParametricSampler
from scimba_torch.neural_nets.coordinates_based_nets.mlp import GenericMLP
from scimba_torch.numerical_solvers.elliptic_pde.pinns import (
    AnagramPinnsElliptic,
    NaturalGradientPinnsElliptic,
)
from scimba_torch.plots.plots_nd import plot_abstract_approx_spaces
from scimba_torch.utils.scimba_tensors import LabelTensor, MultiLabelTensor
from scimba_torch.utils.typing_protocols import VarArgCallable

torch.manual_seed(0)

class GradDiv2D:
    def __init__(
        self,
        space: AbstractApproxSpace,
        f: Callable,
        g: Callable,
        **kwargs,
    ):
        self.space = space
        self.f = f
        self.g = g

        self.linear = True
        self.residual_size = 2
        self.bc_residual_size = 2

    def grad(
        self,
        w: torch.Tensor | MultiLabelTensor,
        y: torch.Tensor | LabelTensor,
    ) -> torch.Tensor | Tuple[torch.Tensor, ...]:
        return self.space.grad(w, y)

    def rhs(
        self, w: MultiLabelTensor, x: LabelTensor, mu: LabelTensor
    ) -> Tuple[torch.Tensor]:
        return self.f(x, mu)

    def bc_rhs(
        self, w: MultiLabelTensor, x: LabelTensor, n: LabelTensor, mu: LabelTensor
    ) -> Tuple[torch.Tensor]:
        return self.g(x, mu)

    def operator(
        self, w: MultiLabelTensor, xs: LabelTensor, mu: LabelTensor
    ) -> Tuple[torch.Tensor]:
        x, y = xs.get_components()
        u, v = w.get_components()
        u_x, u_y = self.grad(u, xs)
        u_xx, u_xy = self.grad(u_x, xs)

        v_x, v_y = self.grad(v, xs)
        v_yx, v_yy = self.grad(v_y, xs)

        return u_xx + v_yx + u, u_xy + v_yy + v

    def restrict_to_component(self, i: int, func):
        return lambda *args: func(*args)[i : i + 1, ...]

    def functional_operator(
        self,
        func: VarArgCallable,
        x: torch.Tensor,
        mu: torch.Tensor,
        theta: torch.Tensor,
    ) -> torch.Tensor:
        uv_x = func(x, mu, theta)
        grad_u = self.restrict_to_component(0, torch.func.jacrev(func, 0))
        grad_v = self.restrict_to_component(1, torch.func.jacrev(func, 0))
        hessian_u = torch.func.jacrev(grad_u, 0)(x, mu, theta).squeeze()
        hessian_v = torch.func.jacrev(grad_v, 0)(x, mu, theta).squeeze()
        res = hessian_u[..., 0] + hessian_v[..., 1] + uv_x
        return res

    # Dirichlet conditions
    def bc_operator(
        self, w: MultiLabelTensor, x: LabelTensor, n: LabelTensor, mu: LabelTensor
    ) -> Tuple[torch.Tensor]:
        u, v = w.get_components()
        return u, v

    def functional_operator_bc(
        self,
        func: VarArgCallable,
        x: torch.Tensor,
        n: torch.Tensor,
        mu: torch.Tensor,
        theta: torch.Tensor,
    ) -> torch.Tensor:
        return func(x, mu, theta)


def exact_solution(xs: LabelTensor, mu: LabelTensor) -> torch.Tensor:
    x, y = xs.get_components()
    alpha = mu.get_components()
    return torch.cat(
        (
            torch.sin(2.0 * torch.pi * x) * torch.sin(2.0 * torch.pi * y),
            alpha * torch.sin(2.0 * torch.pi * x) * torch.sin(2.0 * torch.pi * y),
        ),
        dim=-1,
    )


def f_rhs(xs: LabelTensor, mu: LabelTensor) -> Tuple[torch.Tensor]:
    x, y = xs.get_components()
    alpha = mu.get_components()

    PI = torch.pi
    cos_x = torch.cos(2.0 * PI * x)
    cos_y = torch.cos(2.0 * PI * y)
    sin_x = torch.sin(2.0 * PI * x)
    sin_y = torch.sin(2.0 * PI * y)

    f1 = (1 - 4 * PI**2) * sin_x * sin_y + 4 * PI**2 * alpha * cos_x * cos_y
    f2 = (1 - 4 * PI**2 * alpha) * sin_x * sin_y + 4 * PI**2 * cos_x * cos_y
    return f1, f2


def f_bc(xs: LabelTensor, mu: LabelTensor) -> Tuple[torch.Tensor]:
    x, _ = xs.get_components()
    return torch.zeros_like(x), torch.zeros_like(x)


bc_weight = 10.0

domain_mu = [(0.75, 0.75)]
domain_x = Square2D([(0.0, 1), (0.0, 1)], is_main_domain=True)

sampler = TensorizedSampler(
    [DomainSampler(domain_x), UniformParametricSampler(domain_mu)]
)

ENG preconditioned PINN, weak boundary conditions

space = NNxSpace(2, 1, GenericMLP, domain_x, sampler, layer_sizes=[64])
pde = GradDiv2D(space, f_rhs, f_bc)

pinns = NaturalGradientPinnsElliptic(
    pde,
    bc_type="weak",
    bc_weight=bc_weight,
    one_loss_by_equation=True,
    matrix_regularization=1e-6,
)

pinns.solve(epochs=100, n_collocation=3000, n_bc_collocation=1000, verbose=True)

activating optimizer ScimbaSGD
epoch: 0, best loss: 1.187e+03
epoch: 0,      loss: 1.187e+03
epoch: 6, best loss: 8.863e+02
epoch: 7, best loss: 4.695e+02
epoch: 8, best loss: 2.227e+02
epoch: 9, best loss: 1.975e+02
epoch: 10, best loss: 1.643e+02
epoch: 11, best loss: 1.305e+02
epoch: 12, best loss: 1.185e+02
epoch: 13, best loss: 5.498e+01
epoch: 14, best loss: 2.025e+01
epoch: 15, best loss: 5.500e+00
epoch: 16, best loss: 1.770e+00
epoch: 17, best loss: 8.329e-01
epoch: 18, best loss: 4.628e-01
epoch: 19, best loss: 3.801e-01
epoch: 20, best loss: 2.729e-01
epoch: 21, best loss: 1.524e-01
epoch: 22, best loss: 7.866e-02
epoch: 23, best loss: 1.793e-02
epoch: 24, best loss: 7.101e-03
epoch: 25, best loss: 4.004e-03
epoch: 26, best loss: 2.936e-03
epoch: 27, best loss: 2.475e-03
epoch: 28, best loss: 2.127e-03
epoch: 29, best loss: 2.055e-03
epoch: 30, best loss: 1.831e-03
epoch: 31, best loss: 1.632e-03
epoch: 32, best loss: 1.443e-03
epoch: 33, best loss: 1.277e-03
epoch: 34, best loss: 1.181e-03
epoch: 35, best loss: 1.104e-03
epoch: 36, best loss: 1.095e-03
epoch: 37, best loss: 1.081e-03
epoch: 38, best loss: 9.631e-04
epoch: 48, best loss: 8.895e-04
epoch: 51, best loss: 8.725e-04
epoch: 53, best loss: 8.534e-04
epoch: 67, best loss: 8.243e-04
epoch: 71, best loss: 8.083e-04
epoch: 77, best loss: 8.054e-04
epoch: 87, best loss: 7.843e-04
epoch: 90, best loss: 7.681e-04
Training done!
    Final loss value: 8.208e-04
     Best loss value: 7.681e-04

Anagram preconditioned PINN, weak boundary conditions

space2 = NNxSpace(2, 1, GenericMLP, domain_x, sampler, layer_sizes=[64])
pde2 = GradDiv2D(space2, f_rhs, f_bc)

pinns2 = AnagramPinnsElliptic(
    pde2,
    bc_type="weak",
    bc_weight=bc_weight,
    one_loss_by_equation=True,
    svd_threshold=5e-2,
)

pinns2.solve(epochs=100, n_collocation=3000, n_bc_collocation=1000, verbose=True)

activating optimizer ScimbaSGD
epoch: 0, best loss: 1.164e+03
epoch: 0,      loss: 1.164e+03
epoch: 18, best loss: 8.480e+02
epoch: 19, best loss: 5.002e+02
epoch: 20, best loss: 4.685e+02
epoch: 21, best loss: 3.914e+02
epoch: 22, best loss: 2.957e+02
epoch: 23, best loss: 2.436e+02
epoch: 24, best loss: 1.911e+02
epoch: 25, best loss: 1.452e+02
epoch: 26, best loss: 3.642e+01
epoch: 27, best loss: 1.812e+01
epoch: 28, best loss: 7.986e+00
epoch: 29, best loss: 5.778e+00
epoch: 30, best loss: 5.474e+00
epoch: 31, best loss: 4.938e+00
epoch: 32, best loss: 3.282e+00
epoch: 33, best loss: 2.551e+00
epoch: 34, best loss: 1.865e+00
epoch: 35, best loss: 1.016e+00
epoch: 36, best loss: 5.133e-01
epoch: 37, best loss: 3.869e-01
epoch: 38, best loss: 1.882e-01
epoch: 39, best loss: 9.455e-02
epoch: 40, best loss: 6.518e-02
epoch: 41, best loss: 5.510e-02
epoch: 42, best loss: 4.507e-02
epoch: 43, best loss: 3.249e-02
epoch: 44, best loss: 3.083e-02
epoch: 45, best loss: 2.796e-02
epoch: 46, best loss: 2.543e-02
epoch: 47, best loss: 2.086e-02
epoch: 48, best loss: 1.754e-02
epoch: 49, best loss: 1.462e-02
epoch: 50, best loss: 1.296e-02
epoch: 51, best loss: 1.105e-02
epoch: 52, best loss: 9.611e-03
epoch: 53, best loss: 8.886e-03
epoch: 54, best loss: 8.526e-03
epoch: 55, best loss: 8.218e-03
epoch: 56, best loss: 7.773e-03
epoch: 57, best loss: 6.944e-03
epoch: 58, best loss: 6.787e-03
epoch: 60, best loss: 6.568e-03
epoch: 61, best loss: 6.526e-03
epoch: 62, best loss: 6.138e-03
epoch: 64, best loss: 6.100e-03
epoch: 66, best loss: 5.991e-03
epoch: 69, best loss: 5.854e-03
epoch: 72, best loss: 5.827e-03
epoch: 74, best loss: 5.713e-03
epoch: 76, best loss: 5.616e-03
epoch: 87, best loss: 5.584e-03
epoch: 96, best loss: 5.575e-03
epoch: 99, best loss: 5.538e-03
Training done!
    Final loss value: 5.548e-03
     Best loss value: 5.538e-03

plot_abstract_approx_spaces(
    pinns.space,
    domain_x,
    domain_mu,
    loss=pinns.losses,
    error=exact_solution,
    draw_contours=True,
    n_drawn_contours=20,
    title="solving GradDiv2D with ENG preconditioned PINN",
)

plt.show()

plot_abstract_approx_spaces(
    pinns2.space,
    domain_x,
    domain_mu,
    loss=pinns2.losses,
    error=exact_solution,
    draw_contours=True,
    n_drawn_contours=20,
    title="solving GradDiv2D with Anagram preconditioned PINN",
)

plt.show()

Strong boundary conditions

def post_processing(inputs: torch.Tensor, xs: LabelTensor, mu: LabelTensor):
    x, y = xs.get_components()
    # _ = mu.get_components()
    phi = x * (x - 1.0) * y * (y - 1.0)
    return inputs * phi


def functional_post_processing(
    func, x: torch.Tensor, mu: torch.Tensor, theta: torch.Tensor
) -> torch.Tensor:
    phi = x[0] * (x[0] - 1.0) * x[1] * (x[1] - 1.0)
    return func(x, mu, theta) * phi

ENG preconditioned PINN, strong boundary conditions

space3 = NNxSpace(2, 1, GenericMLP, domain_x, sampler, layer_sizes=[64], post_processing=post_processing)
pde3 = GradDiv2D(space3, f_rhs, f_bc)

pinns3 = NaturalGradientPinnsElliptic(
    pde3,
    bc_type="strong",
    one_loss_by_equation=True,
    matrix_regularization=1e-6,
    functional_post_processing=functional_post_processing,
)

pinns3.solve(epochs=50, n_collocation=3000, verbose=True)

activating optimizer ScimbaSGD
epoch: 0, best loss: 1.186e+03
epoch: 0,      loss: 1.186e+03
epoch: 4, best loss: 9.399e+02
epoch: 5, best loss: 9.278e+02
epoch: 6, best loss: 8.645e+02
epoch: 7, best loss: 8.380e+02
epoch: 8, best loss: 6.795e+02
epoch: 9, best loss: 5.968e+02
epoch: 10, best loss: 4.763e+02
epoch: 11, best loss: 3.300e+02
epoch: 12, best loss: 2.229e+02
epoch: 13, best loss: 9.942e+01
epoch: 14, best loss: 9.582e+01
epoch: 15, best loss: 7.084e+01
epoch: 16, best loss: 5.207e+01
epoch: 17, best loss: 2.797e+01
epoch: 18, best loss: 1.751e+01
epoch: 19, best loss: 1.431e+01
epoch: 20, best loss: 5.647e+00
epoch: 21, best loss: 3.861e+00
epoch: 22, best loss: 2.183e+00
epoch: 23, best loss: 1.075e+00
epoch: 24, best loss: 4.906e-01
epoch: 25, best loss: 2.642e-01
epoch: 26, best loss: 1.743e-01
epoch: 27, best loss: 1.200e-01
epoch: 28, best loss: 8.177e-02
epoch: 29, best loss: 2.851e-02
epoch: 30, best loss: 2.349e-02
epoch: 31, best loss: 1.335e-02
epoch: 32, best loss: 6.307e-03
epoch: 33, best loss: 2.736e-03
epoch: 34, best loss: 2.411e-03
epoch: 35, best loss: 2.402e-03
epoch: 36, best loss: 2.231e-03
epoch: 37, best loss: 2.115e-03
epoch: 39, best loss: 2.065e-03
epoch: 41, best loss: 1.985e-03
epoch: 42, best loss: 1.910e-03
epoch: 45, best loss: 1.892e-03
epoch: 46, best loss: 1.823e-03
epoch: 47, best loss: 1.729e-03
Training done!
    Final loss value: 1.725e-03
     Best loss value: 1.729e-03

Anagram preconditioned PINN, strong boundary conditions

space4 = NNxSpace(2, 1, GenericMLP, domain_x, sampler, layer_sizes=[64], post_processing=post_processing)
pde4 = GradDiv2D(space4, f_rhs, f_bc)

pinns4 = AnagramPinnsElliptic(
    pde4,
    bc_type="strong",
    one_loss_by_equation=True,
    svd_threshold=5e-3,
    functional_post_processing=functional_post_processing,
)

pinns4.solve(epochs=200, n_collocation=3000, verbose=True)

activating optimizer ScimbaSGD
epoch: 0, best loss: 1.185e+03
epoch: 0,      loss: 1.185e+03
epoch: 1, best loss: 1.181e+03
epoch: 2, best loss: 1.176e+03
epoch: 3, best loss: 1.158e+03
epoch: 4, best loss: 1.142e+03
epoch: 5, best loss: 1.129e+03
epoch: 8, best loss: 9.615e+02
epoch: 9, best loss: 7.664e+02
epoch: 10, best loss: 7.253e+02
epoch: 11, best loss: 6.925e+02
epoch: 12, best loss: 6.380e+02
epoch: 13, best loss: 5.574e+02
epoch: 14, best loss: 5.205e+02
epoch: 15, best loss: 4.750e+02
epoch: 16, best loss: 2.980e+02
epoch: 17, best loss: 2.883e+02
epoch: 18, best loss: 2.780e+02
epoch: 19, best loss: 2.135e+02
epoch: 20, best loss: 1.766e+02
epoch: 21, best loss: 1.584e+02
epoch: 22, best loss: 1.250e+02
epoch: 23, best loss: 1.148e+02
epoch: 24, best loss: 1.030e+02
epoch: 25, best loss: 6.275e+01
epoch: 26, best loss: 5.841e+01
epoch: 27, best loss: 5.044e+01
epoch: 28, best loss: 3.358e+01
epoch: 29, best loss: 3.195e+01
epoch: 30, best loss: 1.905e+01
epoch: 31, best loss: 1.816e+01
epoch: 32, best loss: 1.607e+01
epoch: 33, best loss: 1.584e+01
epoch: 34, best loss: 1.178e+01
epoch: 36, best loss: 1.028e+01
epoch: 37, best loss: 6.866e+00
epoch: 38, best loss: 5.576e+00
epoch: 39, best loss: 4.982e+00
epoch: 40, best loss: 4.402e+00
epoch: 41, best loss: 3.635e+00
epoch: 42, best loss: 3.549e+00
epoch: 43, best loss: 2.289e+00
epoch: 44, best loss: 1.437e+00
epoch: 45, best loss: 1.376e+00
epoch: 46, best loss: 1.355e+00
epoch: 47, best loss: 1.329e+00
epoch: 48, best loss: 1.159e+00
epoch: 49, best loss: 1.126e+00
epoch: 50, best loss: 1.022e+00
epoch: 51, best loss: 7.298e-01
epoch: 52, best loss: 6.850e-01
epoch: 53, best loss: 6.290e-01
epoch: 54, best loss: 5.523e-01
epoch: 55, best loss: 4.639e-01
epoch: 56, best loss: 4.345e-01
epoch: 57, best loss: 3.530e-01
epoch: 59, best loss: 3.016e-01
epoch: 60, best loss: 2.946e-01
epoch: 61, best loss: 2.807e-01
epoch: 62, best loss: 2.586e-01
epoch: 63, best loss: 2.558e-01
epoch: 64, best loss: 1.854e-01
epoch: 65, best loss: 1.511e-01
epoch: 66, best loss: 1.425e-01
epoch: 67, best loss: 1.259e-01
epoch: 69, best loss: 1.211e-01
epoch: 70, best loss: 1.192e-01
epoch: 71, best loss: 8.800e-02
epoch: 72, best loss: 8.199e-02
epoch: 73, best loss: 7.392e-02
epoch: 74, best loss: 7.158e-02
epoch: 75, best loss: 6.658e-02
epoch: 76, best loss: 6.580e-02
epoch: 77, best loss: 5.822e-02
epoch: 78, best loss: 5.423e-02
epoch: 80, best loss: 4.673e-02
epoch: 81, best loss: 4.062e-02
epoch: 83, best loss: 3.784e-02
epoch: 84, best loss: 3.435e-02
epoch: 85, best loss: 3.382e-02
epoch: 86, best loss: 3.200e-02
epoch: 87, best loss: 3.148e-02
epoch: 88, best loss: 2.772e-02
epoch: 89, best loss: 2.737e-02
epoch: 90, best loss: 2.422e-02
epoch: 91, best loss: 2.170e-02
epoch: 94, best loss: 2.165e-02
epoch: 95, best loss: 1.999e-02
epoch: 97, best loss: 1.938e-02
epoch: 98, best loss: 1.860e-02
epoch: 99, best loss: 1.769e-02
epoch: 100, best loss: 1.599e-02
epoch: 100,      loss: 1.599e-02
epoch: 104, best loss: 1.480e-02
epoch: 105, best loss: 1.457e-02
epoch: 107, best loss: 1.432e-02
epoch: 109, best loss: 1.331e-02
epoch: 112, best loss: 1.321e-02
epoch: 113, best loss: 1.275e-02
epoch: 114, best loss: 1.255e-02
epoch: 115, best loss: 1.245e-02
epoch: 116, best loss: 1.142e-02
epoch: 117, best loss: 1.121e-02
epoch: 118, best loss: 1.025e-02
epoch: 121, best loss: 1.018e-02
epoch: 122, best loss: 1.004e-02
epoch: 124, best loss: 9.550e-03
epoch: 125, best loss: 9.125e-03
epoch: 127, best loss: 8.791e-03
epoch: 128, best loss: 8.469e-03
epoch: 132, best loss: 8.021e-03
epoch: 133, best loss: 7.959e-03
epoch: 136, best loss: 7.822e-03
epoch: 137, best loss: 7.396e-03
epoch: 138, best loss: 7.377e-03
epoch: 141, best loss: 7.252e-03
epoch: 142, best loss: 7.245e-03
epoch: 143, best loss: 6.758e-03
epoch: 145, best loss: 6.507e-03
epoch: 147, best loss: 6.289e-03
epoch: 150, best loss: 5.832e-03
epoch: 154, best loss: 5.368e-03
epoch: 156, best loss: 5.345e-03
epoch: 159, best loss: 5.289e-03
epoch: 162, best loss: 4.416e-03
epoch: 166, best loss: 4.062e-03
epoch: 171, best loss: 3.685e-03
epoch: 172, best loss: 3.560e-03
epoch: 177, best loss: 3.558e-03
epoch: 179, best loss: 3.070e-03
epoch: 183, best loss: 2.968e-03
epoch: 186, best loss: 2.894e-03
epoch: 189, best loss: 2.884e-03
epoch: 191, best loss: 2.715e-03
epoch: 196, best loss: 2.713e-03
epoch: 199, best loss: 2.700e-03
Training done!
    Final loss value: 2.602e-03
     Best loss value: 2.700e-03

plot_abstract_approx_spaces(
    pinns3.space,
    domain_x,
    domain_mu,
    loss=pinns3.losses,
    error=exact_solution,
    draw_contours=True,
    n_drawn_contours=20,
    title="solving GradDiv2D with ENG preconditioned PINN",
)

plt.show()

plot_abstract_approx_spaces(
    pinns4.space,
    domain_x,
    domain_mu,
    loss=pinns4.losses,
    error=exact_solution,
    draw_contours=True,
    n_drawn_contours=20,
    title="solving GradDiv2D with Anagram preconditioned PINN",
)

plt.show()