bump in Burger's equation's solution

[feiyu-chen96]

When solving Burger's equation, a ' bump' always appear in the residual of the solution. The cause is unknown.

def test_burgers():

    nu = 1
    a = 2  # a parameter for the special case
    T = 0.1

    burgers = lambda u, x, t: diff(u, t) + u * diff(u, x) - nu * diff(u, x, order=2)

    ibvp = IBVP1D(
        x_min=0, x_min_val=lambda t: 0,
        x_max=1, x_max_val=lambda t: 0,
        t_min=0, t_min_val=lambda x: 2 * nu * np.pi * torch.sin(np.pi * x) / (a + torch.cos(np.pi * x))
    )
    net = FCNN(n_input_units=2, n_hidden_units=32, n_hidden_layers=1)

    solution_neural_net_burgers, _ = solve2D(
        pde=burgers, condition=ibvp, xy_min=[0, 0], xy_max=[1, T],
        net=net, max_epochs=300,
        train_generator=ExampleGenerator2D([50, 50], [0, 0], [1, T], method='equally-spaced-noisy'),
        batch_size=64,
        monitor=Monitor2D(check_every=10, xy_min=[0, 0], xy_max=[1, T])
    )

    def solution_analytical_burgers(x, t):
        numer = 2 * nu * np.pi * np.exp(-np.pi ** 2 * nu * t) * np.sin(np.pi * x)
        denom = a + np.exp(-np.pi ** 2 * nu * t) * np.cos(np.pi * x)
        return numer / denom

    xs = np.linspace(0, 1, 101)
    ts = np.linspace(0, T, 101)
    xx, tt = np.meshgrid(xs, ts)
    make_animation(solution_neural_net_burgers, xs, ts) # test animation
    sol_ana = solution_analytical_burgers(xx, tt)
    sol_net = solution_neural_net_burgers(xx, tt, as_type='np')
    assert isclose(sol_net, sol_ana, atol=0.1).all()
    print('Burgers test passed.')