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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_scale: float
    fourth_order_scale: 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_scale: float = 1.0,
        fourth_order_scale: 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
        [`exponax.stepper.KuramotoSivashinskyConservative`][].

        In 1d, the KS equation is given by

        ```
            uₜ + b₂ 1/2 (uₓ)² + ψ₁ uₓₓ + ψ₂ uₓₓₓₓ = 0
        ```

        with `b₂` the gradient-norm coefficient, `ψ₁` the second-order scale and
        `ψ₂` the fourth-order. If the latter two terms were on the right-hand
        side, they could be interpreted as diffusivity and hyper-diffusivity,
        respectively. Here, 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 = ψ₁
        = ψ₂ = 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 + ψ₁ν (∇ ⋅ ∇) u + ψ₂ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))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_scale`: The "diffusivity" `ψ₁` in the KS equation.
        - `fourth_order_diffusivity`: The "hyper-diffusivity" `ψ₂` in the KS
            equation.
        - `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_scale = second_order_scale
        self.fourth_order_scale = fourth_order_scale
        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"]:
        # Minuses are required to move the terms to the right-hand side
        linear_operator = -self.second_order_scale * build_laplace_operator(
            derivative_operator, order=2
        ) - self.fourth_order_scale * 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_scale: float = 1.0,
    fourth_order_scale: 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 exponax.stepper.KuramotoSivashinskyConservative.

In 1d, the KS equation is given by

    uₜ + b₂ 1/2 (uₓ)² + ψ₁ uₓₓ + ψ₂ uₓₓₓₓ = 0

with b₂ the gradient-norm coefficient, ψ₁ the second-order scale and ψ₂ the fourth-order. If the latter two terms were on the right-hand side, they could be interpreted as diffusivity and hyper-diffusivity, respectively. Here, 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 = ψ₁ = ψ₂ = 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 + ψ₁ν (∇ ⋅ ∇) u + ψ₂ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))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_scale: The "diffusivity" ψ₁ in the KS equation.
  • fourth_order_diffusivity: The "hyper-diffusivity" ψ₂ in the KS equation.
  • 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_scale: float = 1.0,
    fourth_order_scale: 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
    [`exponax.stepper.KuramotoSivashinskyConservative`][].

    In 1d, the KS equation is given by

    ```
        uₜ + b₂ 1/2 (uₓ)² + ψ₁ uₓₓ + ψ₂ uₓₓₓₓ = 0
    ```

    with `b₂` the gradient-norm coefficient, `ψ₁` the second-order scale and
    `ψ₂` the fourth-order. If the latter two terms were on the right-hand
    side, they could be interpreted as diffusivity and hyper-diffusivity,
    respectively. Here, 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 = ψ₁
    = ψ₂ = 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 + ψ₁ν (∇ ⋅ ∇) u + ψ₂ ((∇ ⊗ ∇) ⋅ (∇ ⊗ ∇))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_scale`: The "diffusivity" `ψ₁` in the KS equation.
    - `fourth_order_diffusivity`: The "hyper-diffusivity" `ψ₂` in the KS
        equation.
    - `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_scale = second_order_scale
    self.fourth_order_scale = fourth_order_scale
    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"]

Perform one step of the time integration for a single state.

Arguments:

  • u: The state vector, shape (C, ..., N,).

Returns:

  • u_next: The state vector after one step, shape (C, ..., N,).

Tip

Use this call method together with exponax.rollout to efficiently produce temporal trajectories.

Info

For batched operation, use jax.vmap on this function.

Source code in exponax/_base_stepper.py 
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def __call__(
    self,
    u: Float[Array, "C ... N"],
) -> Float[Array, "C ... N"]:
    """
    Perform one step of the time integration for a single state.

    **Arguments:**

    - `u`: The state vector, shape `(C, ..., N,)`.

    **Returns:**

    - `u_next`: The state vector after one step, shape `(C, ..., N,)`.

    !!! tip
        Use this call method together with `exponax.rollout` to efficiently
        produce temporal trajectories.

    !!! info
        For batched operation, use `jax.vmap` on this function.
    """
    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)