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Kuramoto-Sivashinsky equation¤

In 1D:

\[ \frac{\partial u}{\partial t} + \frac{1}{2} \left(\frac{\partial u}{\partial x}\right)^2 + \frac{\partial^2 u}{\partial x^2} + \frac{\partial^4 u}{\partial x^4} = 0 \]

In higher dimensions:

\[ \frac{\partial u}{\partial t} + \frac{1}{2} \left \| \nabla u \right \|^2 + \nabla \cdot \nabla u + \nabla \cdot (\nabla \odot \nabla \odot \nabla) u = 0 \]

Uses the combustion format via the gradient norm that easily scales to higher dimensions.

exponax.stepper.KuramotoSivashinsky ¤

Bases: BaseStepper

Source code in exponax/stepper/_kuramoto_sivashinsky.py 
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class KuramotoSivashinsky(BaseStepper):
    gradient_norm_scale: float
    second_order_diffusivity: float
    fourth_order_diffusivity: float
    dealiasing_fraction: float

    def __init__(
        self,
        num_spatial_dims: int,
        domain_extent: float,
        num_points: int,
        dt: float,
        *,
        gradient_norm_scale: float = 1.0,
        second_order_diffusivity: float = 1.0,
        fourth_order_diffusivity: float = 1.0,
        dealiasing_fraction: float = 2 / 3,
        order: int = 2,
        num_circle_points: int = 16,
        circle_radius: float = 1.0,
    ):
        """
        Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Kuramoto-Sivashinsky
        equation on periodic boundary conditions. Uses the **combustion format**
        (or non-conservative format). Most deep learning papers in 1d considered
        the conservative format available as
        [`KuramotoSivashinskyConservative`](exponax/stepper/KuramotoSivashinskyConservative).

        In 1d, the KS equation is given by

        ```
            uₜ + b₂ 1/2 (uₓ)² + ν uₓₓ + μ uₓₓₓₓ = 0
        ```

        with `b₂` the gradient-norm coefficient, `ν` the diffusivity and `μ` the
        hyper viscosity. Note that both viscosity terms are on the left-hand
        side. As such for `ν, μ > 0`, the second-order term acts destabilizing
        (increases the energy of the system) and the fourth-order term acts
        stabilizing (decreases the energy of the system). A common configuration
        is `b₂ = ν = μ = 1` and the dynamics are only adapted using the
        `domain_extent`. For this, we espect the KS equation to experience
        spatio-temporal chaos roughly once `L > 60`.

        In this combustion (=non-conservative) format, the number of channels
        does **not** grow with the spatial dimension. A 2d KS still only has a
        single channel. In higher dimensions, the equation reads

        ```
            uₜ + b₂ 1/2 ‖ ∇u ‖₂² + ν (∇ ⋅ ∇) u + μ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))u = 0
        ```

        with `‖ ∇u ‖₂` the gradient norm, `∇ ⋅ ∇` effectively is the Laplace
        operator `Δ`. The fourth-order term generalizes to `((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))`
        which is **not** the same as `ΔΔ = Δ²` since the latter would mix
        spatially.

        **Arguments:**

        - `num_spatial_dims`: The number of spatial dimensions `d`.
        - `domain_extent`: The size of the domain `L`; in higher dimensions
            the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
        - `num_points`: The number of points `N` used to discretize the
            domain. This **includes** the left boundary point and **excludes**
            the right boundary point. In higher dimensions; the number of points
            in each dimension is the same. Hence, the total number of degrees of
            freedom is `Nᵈ`.
        - `dt`: The timestep size `Δt` between two consecutive states.
        - `gradient_norm_scale`: The gradient-norm coefficient `b₂`. Note
            that the gradient norm is already scaled by 1/2. This factor allows
            for further modification. Default: 1.0.
        - `second_order_diffusivity`: The diffusivity `ν` in the KS
            equation. The sign of this coefficient is interpreted as if the term
            was on the left-hand side. Hence it should have a positive value to
            act destabilizing. Default: 1.0.
        - `fourth_order_diffusivity`: The hyper viscosity `μ` in the KS
            equation. The sign of this coefficient is interpreted as if the term
            was on the left-hand side. Hence it should have a positive value to
            act stabilizing. Default: 1.0.
        - `order`: The order of the Exponential Time Differencing Runge
            Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
            solves the linear part of the equation. Use higher values for higher
            accuracy and stability. The default choice of `2` is a good
            compromise for single precision floats.
        - `dealiasing_fraction`: The fraction of the wavenumbers to keep
            before evaluating the nonlinearity. The default 2/3 corresponds to
            Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
            2/3.
        - `num_circle_points`: How many points to use in the complex contour
            integral method to compute the coefficients of the exponential time
            differencing Runge Kutta method. Default: 16.
        - `circle_radius`: The radius of the contour used to compute the
            coefficients of the exponential time differencing Runge Kutta
            method. Default: 1.0.

        **Notes:**

        - The KS equation enters a chaotic state if the domain extent is
            chosen large enough. In this chaotic attractor it can run
            indefinitely. It is in balancing state of the second-order term
            producing new energy, the nonlinearity transporting it into higher
            modes where the fourth-order term dissipates it.
        - If the domain extent is chosen large enough to eventually enter a
            chaotic state, the initial condition does not really matter. Since
            the KS "produces its own energy", the energy spectrum for the
            chaotic attractor is independent of the initial condition.
        - However, since the KS develops a certain spectrum based on the
            domain length, make sure to use enough discretization point to
            capture the highes occuring mode. For a domain extent of 60, this
            requires at least roughly 100 `num_points` in single precision
            floats.
        - For domain lengths smaller than the threshold to enter chaos, the
            KS equation, exhibits various other patterns like propagating waves,
            etc.
        - For higher dimensions (i.e., `num_spatial_dims > 1`), a chaotic
            state is already entered for smaller domain extents. For more
            details and the kind of dynamics that can occur see:
            https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0932

        **Good Values:**

        - For a simple spatio-temporal chaos in 1d, set
            `num_spatial_dims=1`, `domain_extent=60`, `num_points=100`,
            `dt=0.1`. The initial condition can be anything, important is that
            it is mean zero. The first 200-500 steps of the trajectory will be
            the transitional phase, after that the chaotic attractor is reached.
        """
        self.gradient_norm_scale = gradient_norm_scale
        self.second_order_diffusivity = second_order_diffusivity
        self.fourth_order_diffusivity = fourth_order_diffusivity
        self.dealiasing_fraction = dealiasing_fraction
        super().__init__(
            num_spatial_dims=num_spatial_dims,
            domain_extent=domain_extent,
            num_points=num_points,
            dt=dt,
            num_channels=1,
            order=order,
            num_circle_points=num_circle_points,
            circle_radius=circle_radius,
        )

    def _build_linear_operator(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> Complex[Array, "1 ... (N//2)+1"]:
        linear_operator = -self.second_order_diffusivity * build_laplace_operator(
            derivative_operator, order=2
        ) - self.fourth_order_diffusivity * build_laplace_operator(
            derivative_operator, order=4
        )
        return linear_operator

    def _build_nonlinear_fun(
        self,
        derivative_operator: Complex[Array, "D ... (N//2)+1"],
    ) -> GradientNormNonlinearFun:
        return GradientNormNonlinearFun(
            self.num_spatial_dims,
            self.num_points,
            derivative_operator=derivative_operator,
            dealiasing_fraction=self.dealiasing_fraction,
            zero_mode_fix=True,
            scale=self.gradient_norm_scale,
        )
__init__ ¤
__init__(
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    gradient_norm_scale: float = 1.0,
    second_order_diffusivity: float = 1.0,
    fourth_order_diffusivity: float = 1.0,
    dealiasing_fraction: float = 2 / 3,
    order: int = 2,
    num_circle_points: int = 16,
    circle_radius: float = 1.0
)

Timestepper for the d-dimensional (d ∈ {1, 2, 3}) Kuramoto-Sivashinsky equation on periodic boundary conditions. Uses the combustion format (or non-conservative format). Most deep learning papers in 1d considered the conservative format available as KuramotoSivashinskyConservative.

In 1d, the KS equation is given by

    uₜ + b₂ 1/2 (uₓ)² + ν uₓₓ + μ uₓₓₓₓ = 0

with b₂ the gradient-norm coefficient, ν the diffusivity and μ the hyper viscosity. Note that both viscosity terms are on the left-hand side. As such for ν, μ > 0, the second-order term acts destabilizing (increases the energy of the system) and the fourth-order term acts stabilizing (decreases the energy of the system). A common configuration is b₂ = ν = μ = 1 and the dynamics are only adapted using the domain_extent. For this, we espect the KS equation to experience spatio-temporal chaos roughly once L > 60.

In this combustion (=non-conservative) format, the number of channels does not grow with the spatial dimension. A 2d KS still only has a single channel. In higher dimensions, the equation reads

    uₜ + b₂ 1/2 ‖ ∇u ‖₂² + ν (∇ ⋅ ∇) u + μ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))u = 0

with ‖ ∇u ‖₂ the gradient norm, ∇ ⋅ ∇ effectively is the Laplace operator Δ. The fourth-order term generalizes to ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇)) which is not the same as ΔΔ = Δ² since the latter would mix spatially.

Arguments:

  • num_spatial_dims: The number of spatial dimensions d.
  • domain_extent: The size of the domain L; in higher dimensions the domain is assumed to be a scaled hypercube Ω = (0, L)ᵈ.
  • num_points: The number of points N used to discretize the domain. This includes the left boundary point and excludes the right boundary point. In higher dimensions; the number of points in each dimension is the same. Hence, the total number of degrees of freedom is Nᵈ.
  • dt: The timestep size Δt between two consecutive states.
  • gradient_norm_scale: The gradient-norm coefficient b₂. Note that the gradient norm is already scaled by 1/2. This factor allows for further modification. Default: 1.0.
  • second_order_diffusivity: The diffusivity ν in the KS equation. The sign of this coefficient is interpreted as if the term was on the left-hand side. Hence it should have a positive value to act destabilizing. Default: 1.0.
  • fourth_order_diffusivity: The hyper viscosity μ in the KS equation. The sign of this coefficient is interpreted as if the term was on the left-hand side. Hence it should have a positive value to act stabilizing. Default: 1.0.
  • order: The order of the Exponential Time Differencing Runge Kutta method. Must be one of {0, 1, 2, 3, 4}. The option 0 only solves the linear part of the equation. Use higher values for higher accuracy and stability. The default choice of 2 is a good compromise for single precision floats.
  • dealiasing_fraction: The fraction of the wavenumbers to keep before evaluating the nonlinearity. The default 2/3 corresponds to Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default: 2/3.
  • num_circle_points: How many points to use in the complex contour integral method to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 16.
  • circle_radius: The radius of the contour used to compute the coefficients of the exponential time differencing Runge Kutta method. Default: 1.0.

Notes:

  • The KS equation enters a chaotic state if the domain extent is chosen large enough. In this chaotic attractor it can run indefinitely. It is in balancing state of the second-order term producing new energy, the nonlinearity transporting it into higher modes where the fourth-order term dissipates it.
  • If the domain extent is chosen large enough to eventually enter a chaotic state, the initial condition does not really matter. Since the KS "produces its own energy", the energy spectrum for the chaotic attractor is independent of the initial condition.
  • However, since the KS develops a certain spectrum based on the domain length, make sure to use enough discretization point to capture the highes occuring mode. For a domain extent of 60, this requires at least roughly 100 num_points in single precision floats.
  • For domain lengths smaller than the threshold to enter chaos, the KS equation, exhibits various other patterns like propagating waves, etc.
  • For higher dimensions (i.e., num_spatial_dims > 1), a chaotic state is already entered for smaller domain extents. For more details and the kind of dynamics that can occur see: https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0932

Good Values:

  • For a simple spatio-temporal chaos in 1d, set num_spatial_dims=1, domain_extent=60, num_points=100, dt=0.1. The initial condition can be anything, important is that it is mean zero. The first 200-500 steps of the trajectory will be the transitional phase, after that the chaotic attractor is reached.
Source code in exponax/stepper/_kuramoto_sivashinsky.py 
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def __init__(
    self,
    num_spatial_dims: int,
    domain_extent: float,
    num_points: int,
    dt: float,
    *,
    gradient_norm_scale: float = 1.0,
    second_order_diffusivity: float = 1.0,
    fourth_order_diffusivity: float = 1.0,
    dealiasing_fraction: float = 2 / 3,
    order: int = 2,
    num_circle_points: int = 16,
    circle_radius: float = 1.0,
):
    """
    Timestepper for the d-dimensional (`d ∈ {1, 2, 3}`) Kuramoto-Sivashinsky
    equation on periodic boundary conditions. Uses the **combustion format**
    (or non-conservative format). Most deep learning papers in 1d considered
    the conservative format available as
    [`KuramotoSivashinskyConservative`](exponax/stepper/KuramotoSivashinskyConservative).

    In 1d, the KS equation is given by

    ```
        uₜ + b₂ 1/2 (uₓ)² + ν uₓₓ + μ uₓₓₓₓ = 0
    ```

    with `b₂` the gradient-norm coefficient, `ν` the diffusivity and `μ` the
    hyper viscosity. Note that both viscosity terms are on the left-hand
    side. As such for `ν, μ > 0`, the second-order term acts destabilizing
    (increases the energy of the system) and the fourth-order term acts
    stabilizing (decreases the energy of the system). A common configuration
    is `b₂ = ν = μ = 1` and the dynamics are only adapted using the
    `domain_extent`. For this, we espect the KS equation to experience
    spatio-temporal chaos roughly once `L > 60`.

    In this combustion (=non-conservative) format, the number of channels
    does **not** grow with the spatial dimension. A 2d KS still only has a
    single channel. In higher dimensions, the equation reads

    ```
        uₜ + b₂ 1/2 ‖ ∇u ‖₂² + ν (∇ ⋅ ∇) u + μ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))u = 0
    ```

    with `‖ ∇u ‖₂` the gradient norm, `∇ ⋅ ∇` effectively is the Laplace
    operator `Δ`. The fourth-order term generalizes to `((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))`
    which is **not** the same as `ΔΔ = Δ²` since the latter would mix
    spatially.

    **Arguments:**

    - `num_spatial_dims`: The number of spatial dimensions `d`.
    - `domain_extent`: The size of the domain `L`; in higher dimensions
        the domain is assumed to be a scaled hypercube `Ω = (0, L)ᵈ`.
    - `num_points`: The number of points `N` used to discretize the
        domain. This **includes** the left boundary point and **excludes**
        the right boundary point. In higher dimensions; the number of points
        in each dimension is the same. Hence, the total number of degrees of
        freedom is `Nᵈ`.
    - `dt`: The timestep size `Δt` between two consecutive states.
    - `gradient_norm_scale`: The gradient-norm coefficient `b₂`. Note
        that the gradient norm is already scaled by 1/2. This factor allows
        for further modification. Default: 1.0.
    - `second_order_diffusivity`: The diffusivity `ν` in the KS
        equation. The sign of this coefficient is interpreted as if the term
        was on the left-hand side. Hence it should have a positive value to
        act destabilizing. Default: 1.0.
    - `fourth_order_diffusivity`: The hyper viscosity `μ` in the KS
        equation. The sign of this coefficient is interpreted as if the term
        was on the left-hand side. Hence it should have a positive value to
        act stabilizing. Default: 1.0.
    - `order`: The order of the Exponential Time Differencing Runge
        Kutta method. Must be one of {0, 1, 2, 3, 4}. The option `0` only
        solves the linear part of the equation. Use higher values for higher
        accuracy and stability. The default choice of `2` is a good
        compromise for single precision floats.
    - `dealiasing_fraction`: The fraction of the wavenumbers to keep
        before evaluating the nonlinearity. The default 2/3 corresponds to
        Orszag's 2/3 rule. To fully eliminate aliasing, use 1/2. Default:
        2/3.
    - `num_circle_points`: How many points to use in the complex contour
        integral method to compute the coefficients of the exponential time
        differencing Runge Kutta method. Default: 16.
    - `circle_radius`: The radius of the contour used to compute the
        coefficients of the exponential time differencing Runge Kutta
        method. Default: 1.0.

    **Notes:**

    - The KS equation enters a chaotic state if the domain extent is
        chosen large enough. In this chaotic attractor it can run
        indefinitely. It is in balancing state of the second-order term
        producing new energy, the nonlinearity transporting it into higher
        modes where the fourth-order term dissipates it.
    - If the domain extent is chosen large enough to eventually enter a
        chaotic state, the initial condition does not really matter. Since
        the KS "produces its own energy", the energy spectrum for the
        chaotic attractor is independent of the initial condition.
    - However, since the KS develops a certain spectrum based on the
        domain length, make sure to use enough discretization point to
        capture the highes occuring mode. For a domain extent of 60, this
        requires at least roughly 100 `num_points` in single precision
        floats.
    - For domain lengths smaller than the threshold to enter chaos, the
        KS equation, exhibits various other patterns like propagating waves,
        etc.
    - For higher dimensions (i.e., `num_spatial_dims > 1`), a chaotic
        state is already entered for smaller domain extents. For more
        details and the kind of dynamics that can occur see:
        https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0932

    **Good Values:**

    - For a simple spatio-temporal chaos in 1d, set
        `num_spatial_dims=1`, `domain_extent=60`, `num_points=100`,
        `dt=0.1`. The initial condition can be anything, important is that
        it is mean zero. The first 200-500 steps of the trajectory will be
        the transitional phase, after that the chaotic attractor is reached.
    """
    self.gradient_norm_scale = gradient_norm_scale
    self.second_order_diffusivity = second_order_diffusivity
    self.fourth_order_diffusivity = fourth_order_diffusivity
    self.dealiasing_fraction = dealiasing_fraction
    super().__init__(
        num_spatial_dims=num_spatial_dims,
        domain_extent=domain_extent,
        num_points=num_points,
        dt=dt,
        num_channels=1,
        order=order,
        num_circle_points=num_circle_points,
        circle_radius=circle_radius,
    )
__call__ ¤
__call__(
    u: Float[Array, "C ... N"]
) -> Float[Array, "C ... N"]

Performs a check

Source code in exponax/_base_stepper.py 
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def __call__(
    self,
    u: Float[Array, "C ... N"],
) -> Float[Array, "C ... N"]:
    """
    Performs a check
    """
    expected_shape = (self.num_channels,) + spatial_shape(
        self.num_spatial_dims, self.num_points
    )
    if u.shape != expected_shape:
        raise ValueError(
            f"Expected shape {expected_shape}, got {u.shape}. For batched operation use `jax.vmap` on this function."
        )
    return self.step(u)