phlearn.phsystems

The phsystems subpackage is divided into two further subpackages, one for ODEs and one for PDEs. These implement pseudo-Hamiltonian systems and numerical integration of these to obtain data.

class phlearn.phsystems.AllenCahnSystem(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]), nu=0.001, init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

Implements a discretization of the Allen-Cahn equation with an optional external force, as described by

u_t - u_xx - u + u^3 = f(u,t,x)

on the pseudo-Hamiltonian formulation

du/dt = -grad[V(u)] + F(u,t,x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 1.0 - 1/100, 100)

The spatial discretization points.

nunumber, default -1.0

The parameter nu in the Cahn-Hilliard equation.

alphanumber, default 1.0

The parameter alpha in the Cahn-Hilliard equation.

munumber, default -0.001

The parameter mu in the Cahn-Hilliard equation.

init_samplercallable, default None

Function for sampling initial conditions. Callable taking a numpy random generator as input and returning an ndarray of shape same as x with initial conditions for the system. This sampler is used when calling CahnHilliardSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.BBMSystem(x=array([0., 0.2, 0.4, 0.6, 0.8, 1., 1.2, 1.4, 1.6, 1.8, 2., 2.2, 2.4, 2.6, 2.8, 3., 3.2, 3.4, 3.6, 3.8, 4., 4.2, 4.4, 4.6, 4.8, 5., 5.2, 5.4, 5.6, 5.8, 6., 6.2, 6.4, 6.6, 6.8, 7., 7.2, 7.4, 7.6, 7.8, 8., 8.2, 8.4, 8.6, 8.8, 9., 9.2, 9.4, 9.6, 9.8, 10., 10.2, 10.4, 10.6, 10.8, 11., 11.2, 11.4, 11.6, 11.8, 12., 12.2, 12.4, 12.6, 12.8, 13., 13.2, 13.4, 13.6, 13.8, 14., 14.2, 14.4, 14.6, 14.8, 15., 15.2, 15.4, 15.6, 15.8, 16., 16.2, 16.4, 16.6, 16.8, 17., 17.2, 17.4, 17.6, 17.8, 18., 18.2, 18.4, 18.6, 18.8, 19., 19.2, 19.4, 19.6, 19.8]), nu=0.0, init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

BBMSystem class, representing a discretization of the Benjamin-Bona-Mahony (BBM) equation.

Implements a discretization of the BBM equation with an optional viscosity term and external forces:

u_t - u_xxt + u_x + u u_x - nu u_xx = f(u,x,t)

on the pseudo-Hamiltonian formulation

(1-d^2/dx^2)(du/dt) = d/dx(grad[H(u)]) - grad[V(u)] + F(u, t, x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 20.0 - 0.2, 100)

The spatial discretization points

nunumber, default 0.0

The damping coefficient

init_samplercallable, default None

Function for sampling initial conditions. Callable taking a numpy random generator as input and returning an ndarray of shape same as x with initial conditions for the system. This sampler is used when calling BBMSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.CahnHilliardSystem(x=array([0., 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.2, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, 0.32, 0.33, 0.34, 0.35, 0.36, 0.37, 0.38, 0.39, 0.4, 0.41, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.5, 0.51, 0.52, 0.53, 0.54, 0.55, 0.56, 0.57, 0.58, 0.59, 0.6, 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.69, 0.7, 0.71, 0.72, 0.73, 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.8, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.89, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99]), nu=-1.0, alpha=1.0, mu=-0.001, init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

Implements a discretization of the Cahn-Hilliard equation with an optional external force, as described by

u_t - (nu u + alpha u^3 + mu u_xx)_xx = f(u,t,x)

on the pseudo-Hamiltonian formulation

du/dt = d^2/dx^2 grad[V(u)] + F(u,t,x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 1.0 - 1/100, 100)

The spatial discretization points.

nunumber, default -1.0

The parameter nu in the Cahn-Hilliard equation.

alphanumber, default 1.0

The parameter alpha in the Cahn-Hilliard equation.

munumber, default -0.001

The parameter mu in the Cahn-Hilliard equation.

init_samplercallable, default None

Function for sampling initial conditions. Callable taking a numpy random generator as input and returning an ndarray of shape same as x with initial conditions for the system. This sampler is used when calling CahnHilliardSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.HeatEquationSystem(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]), nu=1.0, init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

Implements a discretization of the heat equation with an optional external force,

u_t - u_xx = f(u,t,x),

on the pseudo-Hamiltonian formulation

du/dt = -grad[V(u)] + F(u,t,x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 1.0 - 1/100, 100)

The spatial discretization points.

nunumber, default -1.0

The parameter nu in the Cahn-Hilliard equation.

alphanumber, default 1.0

The parameter alpha in the Cahn-Hilliard equation.

munumber, default -0.001

The parameter mu in the Cahn-Hilliard equation.

init_samplercallable, default None

Function for sampling initial conditions. Callable taking a numpy random generator as input and returning an ndarray of shape same as x with initial conditions for the system. This sampler is used when calling CahnHilliardSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.KdVSystem(x=array([0., 0.2, 0.4, 0.6, 0.8, 1., 1.2, 1.4, 1.6, 1.8, 2., 2.2, 2.4, 2.6, 2.8, 3., 3.2, 3.4, 3.6, 3.8, 4., 4.2, 4.4, 4.6, 4.8, 5., 5.2, 5.4, 5.6, 5.8, 6., 6.2, 6.4, 6.6, 6.8, 7., 7.2, 7.4, 7.6, 7.8, 8., 8.2, 8.4, 8.6, 8.8, 9., 9.2, 9.4, 9.6, 9.8, 10., 10.2, 10.4, 10.6, 10.8, 11., 11.2, 11.4, 11.6, 11.8, 12., 12.2, 12.4, 12.6, 12.8, 13., 13.2, 13.4, 13.6, 13.8, 14., 14.2, 14.4, 14.6, 14.8, 15., 15.2, 15.4, 15.6, 15.8, 16., 16.2, 16.4, 16.6, 16.8, 17., 17.2, 17.4, 17.6, 17.8, 18., 18.2, 18.4, 18.6, 18.8, 19., 19.2, 19.4, 19.6, 19.8]), eta=6.0, gamma=1.0, nu=0.0, init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

Implements the a discretization of the KdV-Burgers equation with an optional external force,

u_t + eta u u_x - nu u_xx - gamma^2 u_xxx = f(u,x,t)

on the pseudo-Hamiltonian formulation

du/dt = d/dx(grad[H(u)]) - grad[V(u)] + F(u, t, x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 20.0 - 0.2, 100)

The spatial discretization points

etanumber, default 6.0

The parameter eta in the KdV equation

gammanumber, default 1.0

The parameter gamma in the KdV equation

nunumber, default 0.0

The damping coefficient

init_samplercallable, default None

Function for sampling initial conditions. Callabale taking a numpy random generator as input and returning an ndarray of shape same as x with inital conditions for the system. This sampler is used when calling KdVSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.MassSpringDamperSystem(mass=1.0, spring_constant=1.0, damping=0.3, **kwargs)

Bases: PseudoHamiltonianSystem

Implements a general forced and damped mass-spring system as a pseudo-Hamiltonian formulation:

   .
|  q |     |  0     1 |                   |      0     |
|  . |  =  |          | * grad[H(q, p)] + |            |
|  p |     | -1    -c |                   | f(q, p, t) |

where q is the position, p the momentum and c the damping coefficient.

Parameters
massnumber, default 1.0

Scalar mass

spring_constantnumber, default 1.0

Scalar spring coefficient

dampingnumber, default 0.3

Scalar damping coefficient. Corresponds to c.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianSystem constructor.

class phlearn.phsystems.PeronaMalikSystem(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]), init_sampler=None, **kwargs)

Bases: PseudoHamiltonianPDESystem

Implements a discretization of the Perona-Malik equation with an optional external force,

u_t + (u_x/(1+u_x^2))_x = f(u,t,x),

on the pseudo-Hamiltonian formulation

du/dt = -grad[V(u)] + F(u,t,x)

where u is a vector of the system states at the spatial points given by x, and t is time.

The system is by default integrated in time using the implicit midpoint method. The line ‘self.sample_trajectory = self.sample_trajectory_midpoint’ can be commented out to instead use the RK45 method of scipy.

Parameters
xnparray, default np.linspace(0, 1.0 - 1/100, 100)

The spatial discretization points.

nunumber, default -1.0

The parameter nu in the Cahn-Hilliard equation.

alphanumber, default 1.0

The parameter alpha in the Cahn-Hilliard equation.

munumber, default -0.001

The parameter mu in the Cahn-Hilliard equation.

init_samplercallable, default None

Function for sampling initial conditions. Callable taking a numpy random generator as input and returning an ndarray of shape same as x with initial conditions for the system. This sampler is used when calling CahnHilliardSystem.sample_trajectory if no initial condition is provided.

kwargsany, optional

Keyword arguments that are passed to PseudoHamiltonianPDESystem constructor.

class phlearn.phsystems.PseudoHamiltonianPDESystem(nstates, lhs_matrix=None, skewsymmetric_matrix=None, dissipation_matrix=None, hamiltonian=None, dissintegral=None, grad_hamiltonian=None, grad_dissintegral=None, hess_hamiltonian=None, hess_dissintegral=None, external_forces=None, jac_external_forces=None, controller=None, init_sampler=None)

Bases: object

Implements a spatially discretized pseudo-Hamiltonian PDE system of the form:

dx/dt = S*grad[H(x)] - R*grad[V(x)] + F(x, t, xspatial)

where x is the system state, A is a symmetric matrix, S is a skew-symmetric matrix, R is the symmetric dissipation matrix, H and V are discretized integrals of the systemm, F is the external force depending on state, time and space.

What is x here is usually u in the literature, and xspatial is x. We use x for the state to be consistent with the ODE case.

Parameters
nstatesint

Number of system states N.

skewsymmetric_matrix(N, N) ndarray or callable, default None

Corresponds to the S matrix. Must either be an ndarray, or callable taking an ndarray input of shape (nsamples, nstates) and returning an ndarray of shape (nsamples, nstates, nstates). If None, the system is assumed to be canonical, and the S matrix is set ot the skew-symmetric matrix [[0, I_n], [-I_n, 0]].

hamiltoniancallable, default None

The Hamiltonian H of the system. Callable taking a torch tensor input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, 1). If the gradient of the Hamiltonian is not provided, the gradient of this function will be computed by torch and used instead. If this is not provided, the grad_hamiltonian must be provided.

dissintegralcallable, default None

The dissipating integral V of the system. Callable taking a torch tensor input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, 1). If the gradient of the dissipating integral is not provided, the gradient of this function will be computed by torch and used instead. If this is not provided, the grad_dissintegral must be provided.

grad_hamiltoniancallable, default None

The gradient of the Hamiltonian H of the system. Callable taking an ndarray input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, nstates). If this is not provided, the hamiltonian must be provided.

grad_dissintegralcallable, default None

The gradient of the dissipating integral V of the system. Callable taking an ndarray input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, nstates). If this is not provided, the hamiltonian must be provided.

external_forcescallable, default None

The external forces affecting system. Callable taking two ndarrays as input, x and t, of shape (nsamples, nstates), (nsamples, 1), respectively and returning an ndarray of shape (nsamples, nstates).

controllerphlearn.control.PseudoHamiltonianController,
default None

Additional external forces set by a controller. Callable taking an ndarray x of shape (nstates,) and a scalar t as input and returning an ndarray of shape (nstates,). Note that this function should not take batch inputs, and that when calling PseudoHamiltonianSystem.sample_trajectory when a controller is provided, the Runge-Kutta 4 method will be used for integration in favor of Scipy’s solve_ivp.

init_samplercallable, default None

Function for sampling initial conditions. Callabale taking a numpy random generator as input and returning an ndarray of shape (nstates,) with inital conditions for the system. This sampler is used when calling PseudoHamiltonianSystem.sample_trajectory if no initial condition is provided.

Methods

sample_trajectory(t[, x0, noise_std, reference])

Samples a trajectory of the system at times t, found by using the solve_ivp solver for temporal integration, starting from the initial state x0.

sample_trajectory_midpoint(t[, x0, ...])

Samples a trajectory of the system at times t, found by using the implicit midpoint method for temporal integration, starting from the initial state x0.

seed(seed)

Set the internal random state.

time_derivative(integrator, *args, **kwargs)

See :py:meth:~`utils.derivatives.time_derivative`

x_dot(x, t[, u])

Computes the time derivative, the right hand side of the pseudo- Hamiltonian equation.

x_dot_jacobian(x, t[, u])

Computes the Jacobian of the right hand side of the pseudo- Hamiltonian equation.
sample_trajectory(t, x0=None, noise_std=0, reference=None)

Samples a trajectory of the system at times t, found by using the solve_ivp solver for temporal integration, starting from the initial state x0.

Parameters
t(T, 1) ndarray

Times at which the trajectory is sampled.

x0(N,) ndarray, default None

Initial condition.

noise_stdnumber, default 0.

Standard deviation of Gaussian white noise added to the samples of the trajectory.

referencephlearn.control.Reference, default None

Not in use for now.

Returns
x(T, N) ndarray
dxdt(T, N) ndarray
t(T, 1) ndarray
us(T, N) ndarray
sample_trajectory_midpoint(t, x0=None, noise_std=0, reference=None)

Samples a trajectory of the system at times t, found by using the implicit midpoint method for temporal integration, starting from the initial state x0. Newton’s method is used for solving the system of nonlinear equations at each integration step.

Parameters
t(T, 1) ndarray

Times at which the trajectory is sampled.

x0(N,) ndarray, default None

Initial condition.

noise_stdnumber, default 0.

Standard deviation of Gaussian white noise added to the samples of the trajectory.

referencephlearn.control.Reference, default None

Not in use for now.

Returns
x(T, N) ndarray
dxdt(T, N) ndarray
t(T, 1) ndarray
us(T, N) ndarray
seed(seed)

Set the internal random state.

Parameters
seedint
time_derivative(integrator, *args, **kwargs)

See :py:meth:~`utils.derivatives.time_derivative`

x_dot(x, t, u=None)

Computes the time derivative, the right hand side of the pseudo- Hamiltonian equation.

Parameters
x(…, N) ndarray
t(…, 1) ndarray
u(…, N) ndarray or None, default None
Returns
(…, N) ndarray
x_dot_jacobian(x, t, u=None)

Computes the Jacobian of the right hand side of the pseudo- Hamiltonian equation.

Parameters
x(…, N) ndarray
t(…, 1) ndarray
u(…, N) ndarray or None, default None
Returns
(…, N) ndarray
class phlearn.phsystems.PseudoHamiltonianSystem(nstates, skewsymmetric_matrix=None, dissipation_matrix=None, hamiltonian=None, grad_hamiltonian=None, external_forces=None, controller=None, init_sampler=None)

Bases: object

Implements a pseudo-Hamiltonian system of the form:

dx/dt = (S(x) - R(x))*grad[H(x)] + F(x, t)

where x is the system state, S is the skew-symmetric interconnection matrix, R is the positive semi-definite dissipation matrix, H is the Hamiltonian of the systemm, F is the external force(s).

Parameters
nstatesint

Number of system states N.

skewsymmetric_matrix(N, N) ndarray or callable, default None

Corresponds to the S matrix. Must either be an ndarray, or callable taking an ndarray input of shape (nsamples, nstates) and returning an ndarray of shape (nsamples, nstates, nstates). If None, the system is assumed to be canonical, and the S matrix is set to be [[0, I_N], [-I_N, 0]].

dissipation_matrix(N, N) ndarray or callable, default None

Corresponds to the R matrix. Must either be an ndarray, or callable taking an ndarray input of shape (nsamples, nstates) and returning an ndarray of shape (nsamples, nstates, nstates). If None, the R matrix is set to be the zero matrix of shape (N, N).

hamiltoniancallable, default None

The Hamiltonian H of the system. Callable taking a torch tensor input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, 1). If the gradient of the Hamiltonian is not provided, the gradient of this function will be computed by torch and used instead. If this is not provided, the grad_hamiltonian must be provided.

grad_hamiltoniancallable, default None

The gradient of the Hamiltonian H of the system. Callable taking an ndarray input of shape (nsamples, nstates) and returning a torch tensor of shape (nsamples, nstates). If this is not provided, the hamiltonian must be provided.

external_forcescallable, default None

The external forces affecting system. Callable taking two ndarrays as input, x and t, of shape (nsamples, nstates), (nsamples, 1), respectively and returning an ndarray of shape (nsamples, nstates).

controllerphlearn.control.PseudoHamiltonianController,
default None

Additional external forces set by a controller. Callable taking an ndarray x of shape (nstates,) and a scalar t as input and returning an ndarray of shape (nstates,). Note that this function should not take batch inputs, and that when calling PseudoHamiltonianSystem.sample_trajectory when a controller is provided, the Runge-Kutta 4 method will be used for integration in favor of Scipy’s solve_ivp.

init_samplercallable, default None

Function for sampling initial conditions. Callabale taking a numpy random generator as input and returning an ndarray of shape (nstates,) with inital conditions for the system. This sampler is used when calling PseudoHamiltonianSystem.sample_trajectory if no initial condition is provided.

Methods

sample_trajectory(t[, x0, noise_std, reference])

Samples a trajectory of the system at times t.

seed(seed)

Set the internal random state.

set_controller(controller)

Set system controller.

time_derivative(integrator, *args, **kwargs)

See :py:meth:~`utils.derivatives.time_derivative`

x_dot(x, t[, u])

Computes the time derivative by the right hand side of the pseudo- Hamiltonian equation.
sample_trajectory(t, x0=None, noise_std=0, reference=None)

Samples a trajectory of the system at times t.

Parameters
t(T, 1) ndarray

Times at which the trajectory is sampled.

x0(N,) ndarray, default None

Initial condition.

noise_stdnumber, default 0.

Standard deviation of Gaussian white noise added to the samples of the trajectory.

referencephlearn.control.Reference, default None

If the system has a controller a reference object may be passed.

Returns
x(T, N) ndarray
dxdt(T, N) ndarray
t(T, 1) ndarray
us(T, N) ndarray
seed(seed)

Set the internal random state.

Parameters
seedint
set_controller(controller)

Set system controller.

time_derivative(integrator, *args, **kwargs)

See :py:meth:~`utils.derivatives.time_derivative`

x_dot(x, t, u=None)

Computes the time derivative by the right hand side of the pseudo- Hamiltonian equation.

Parameters
x(…, N) ndarray
t(…, 1) ndarray
u(…, N) ndarray or None, default None
Returns
(…, N) ndarray
class phlearn.phsystems.TankSystem(incidence_matrix=None, system_graph=None, npipes=None, ntanks=None, dissipation_pipes=None, J=1.0, A=1.0, rho=1.0, g=9.81, **kwargs)

Bases: PseudoHamiltonianSystem

Implements a pseudo-Hamiltonian version of a coupled tanks system:

.      | -R_p   B^T |
x   =  |            | * grad[H(x)] + F(x, t)
       |  -B     0  |

where the state x = [phi, mu], phi and mu being proportional to pipe flows and tank levels, respectively. The interconnection of the tanks is described by a directed graph where each vertex is a tank and each edge is a pipe between two tanks. The incidence matrix of this graph corresponds to the matrix B. The number of tanks is denoted by ntanks, the number of pipes by npipes. The number of states is denoted by nstates = npipes + ntanks.

External interaction can be specified for every state, but the most physically interpretable setting is to only have external interaction for the states corresponding to the tanks, which can be seen as volumetric flows into or out of the tanks.

Each tank is assumed to have a uniform cross-section w.r.t. it’s height.

The interconnection of tanks and pipes can either be specified by a directed graph or by an interconnection matrix S.

Parameters
incidence_matrix(N, N) ndarray, default None

Incidence matrix of the graph describing the tank system. Corresponds to the B matrix. Inferred from system_graph if system_graph is provided.

system_graphnetworkx.Graph, default None

networkx directed graph describing the interconnection of the tanks. The graph has ntanks vertices and npipes edges.

npipesint, default None

Number of pipes. Inferred from system_graph if system_graph is provided.

ntanksint, default None

Number of tanks. Inferred from system_graph if system_graph is provided.

dissipation_pipesndarray, default None

ndarray of size (npipes,) or (npipes, npipes), describing energy loss in the pipes. Corresponds to R_p. Defaults to zero if not provided.

Jnumber, default 1.

Scalar or ndarray of size (npipes,) of proportionality constants relating the change in volumetric flow through each pipe to the pressure drop and potential friction through each pipe. If scalar, the same constant is used for all pipes.

A(ntanks,) ndarray or number, default 1.

Scalar ndarray of size (ntanks,) of tank cross-section areas. If scalar, the same area is used for all tanks.

rhonumber, default 1.

Positive scalar. Density of liquid.

gnumber, default 9.81

Positive scalar. Gravitational acceleration.

kwargsany

Keyword arguments passed to PseudoHamiltonianSystem constructor.

Methods

H_tanksystem

dH_tanksystem

pipeflows

tanklevels
H_tanksystem(x, t=None)
dH_tanksystem(x, t=None)
pipeflows(x)
tanklevels(x)
phlearn.phsystems.init_msdsystem()

Initialize a standard example of a damped mass-spring system affected by a sine force.

Returns
MassSpringDamperSystem
phlearn.phsystems.init_tanksystem(u=None)

Initialize standard tank system.

Parameters
uphlearn.control.PseudoHamiltonianController, defult None
Returns
TankSystem
phlearn.phsystems.init_tanksystem_leaky(nleaks=0)

Initialize tank system with a leaks.

Parameters
nleaksint, default 0

If 0, no leaks. If 1, there is a leak on the last tank. If 2, there is a leak on the last and tank number 2.

Returns
TankSystem
phlearn.phsystems.initial_condition_ac(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]))

Creates an initial condition sampler for the Allen-Cahn eqation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and 5.98 with step size 0.02.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_bbm(x=array([0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5., 5.5, 6., 6.5, 7., 7.5, 8., 8.5, 9., 9.5, 10., 10.5, 11., 11.5, 12., 12.5, 13., 13.5, 14., 14.5, 15., 15.5, 16., 16.5, 17., 17.5, 18., 18.5, 19., 19.5, 20., 20.5, 21., 21.5, 22., 22.5, 23., 23.5, 24., 24.5, 25., 25.5, 26., 26.5, 27., 27.5, 28., 28.5, 29., 29.5, 30., 30.5, 31., 31.5, 32., 32.5, 33., 33.5, 34., 34.5, 35., 35.5, 36., 36.5, 37., 37.5, 38., 38.5, 39., 39.5, 40., 40.5, 41., 41.5, 42., 42.5, 43., 43.5, 44., 44.5, 45., 45.5, 46., 46.5, 47., 47.5, 48., 48.5, 49., 49.5]))

Creates an initial condition sampler for the Benjamin-Bona-Mahony equation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and 49.5 with step size 0.5.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_ch(x=array([0., 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19, 0.2, 0.21, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, 0.32, 0.33, 0.34, 0.35, 0.36, 0.37, 0.38, 0.39, 0.4, 0.41, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.5, 0.51, 0.52, 0.53, 0.54, 0.55, 0.56, 0.57, 0.58, 0.59, 0.6, 0.61, 0.62, 0.63, 0.64, 0.65, 0.66, 0.67, 0.68, 0.69, 0.7, 0.71, 0.72, 0.73, 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.8, 0.81, 0.82, 0.83, 0.84, 0.85, 0.86, 0.87, 0.88, 0.89, 0.9, 0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99]))

Creates an initial condition sampler for the Cahn-Hilliard eqation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and .99 with step size 0.01.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_heat(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]))

Creates an initial condition sampler for the heat eqation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and 5.98 with step size 0.02.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_kdv(x=array([0., 0.2, 0.4, 0.6, 0.8, 1., 1.2, 1.4, 1.6, 1.8, 2., 2.2, 2.4, 2.6, 2.8, 3., 3.2, 3.4, 3.6, 3.8, 4., 4.2, 4.4, 4.6, 4.8, 5., 5.2, 5.4, 5.6, 5.8, 6., 6.2, 6.4, 6.6, 6.8, 7., 7.2, 7.4, 7.6, 7.8, 8., 8.2, 8.4, 8.6, 8.8, 9., 9.2, 9.4, 9.6, 9.8, 10., 10.2, 10.4, 10.6, 10.8, 11., 11.2, 11.4, 11.6, 11.8, 12., 12.2, 12.4, 12.6, 12.8, 13., 13.2, 13.4, 13.6, 13.8, 14., 14.2, 14.4, 14.6, 14.8, 15., 15.2, 15.4, 15.6, 15.8, 16., 16.2, 16.4, 16.6, 16.8, 17., 17.2, 17.4, 17.6, 17.8, 18., 18.2, 18.4, 18.6, 18.8, 19., 19.2, 19.4, 19.6, 19.8]), eta=6.0)

Creates an initial condition sampler for the KdV-Burgers equation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and 19.8 with step size 0.2.

etafloat, optional

The parameter eta in the KdV-Burgers equation. The default is 6.0.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_pm(x=array([0., 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.22, 0.24, 0.26, 0.28, 0.3, 0.32, 0.34, 0.36, 0.38, 0.4, 0.42, 0.44, 0.46, 0.48, 0.5, 0.52, 0.54, 0.56, 0.58, 0.6, 0.62, 0.64, 0.66, 0.68, 0.7, 0.72, 0.74, 0.76, 0.78, 0.8, 0.82, 0.84, 0.86, 0.88, 0.9, 0.92, 0.94, 0.96, 0.98, 1., 1.02, 1.04, 1.06, 1.08, 1.1, 1.12, 1.14, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.32, 1.34, 1.36, 1.38, 1.4, 1.42, 1.44, 1.46, 1.48, 1.5, 1.52, 1.54, 1.56, 1.58, 1.6, 1.62, 1.64, 1.66, 1.68, 1.7, 1.72, 1.74, 1.76, 1.78, 1.8, 1.82, 1.84, 1.86, 1.88, 1.9, 1.92, 1.94, 1.96, 1.98, 2., 2.02, 2.04, 2.06, 2.08, 2.1, 2.12, 2.14, 2.16, 2.18, 2.2, 2.22, 2.24, 2.26, 2.28, 2.3, 2.32, 2.34, 2.36, 2.38, 2.4, 2.42, 2.44, 2.46, 2.48, 2.5, 2.52, 2.54, 2.56, 2.58, 2.6, 2.62, 2.64, 2.66, 2.68, 2.7, 2.72, 2.74, 2.76, 2.78, 2.8, 2.82, 2.84, 2.86, 2.88, 2.9, 2.92, 2.94, 2.96, 2.98, 3., 3.02, 3.04, 3.06, 3.08, 3.1, 3.12, 3.14, 3.16, 3.18, 3.2, 3.22, 3.24, 3.26, 3.28, 3.3, 3.32, 3.34, 3.36, 3.38, 3.4, 3.42, 3.44, 3.46, 3.48, 3.5, 3.52, 3.54, 3.56, 3.58, 3.6, 3.62, 3.64, 3.66, 3.68, 3.7, 3.72, 3.74, 3.76, 3.78, 3.8, 3.82, 3.84, 3.86, 3.88, 3.9, 3.92, 3.94, 3.96, 3.98, 4., 4.02, 4.04, 4.06, 4.08, 4.1, 4.12, 4.14, 4.16, 4.18, 4.2, 4.22, 4.24, 4.26, 4.28, 4.3, 4.32, 4.34, 4.36, 4.38, 4.4, 4.42, 4.44, 4.46, 4.48, 4.5, 4.52, 4.54, 4.56, 4.58, 4.6, 4.62, 4.64, 4.66, 4.68, 4.7, 4.72, 4.74, 4.76, 4.78, 4.8, 4.82, 4.84, 4.86, 4.88, 4.9, 4.92, 4.94, 4.96, 4.98, 5., 5.02, 5.04, 5.06, 5.08, 5.1, 5.12, 5.14, 5.16, 5.18, 5.2, 5.22, 5.24, 5.26, 5.28, 5.3, 5.32, 5.34, 5.36, 5.38, 5.4, 5.42, 5.44, 5.46, 5.48, 5.5, 5.52, 5.54, 5.56, 5.58, 5.6, 5.62, 5.64, 5.66, 5.68, 5.7, 5.72, 5.74, 5.76, 5.78, 5.8, 5.82, 5.84, 5.86, 5.88, 5.9, 5.92, 5.94, 5.96, 5.98]))

Creates an initial condition sampler for the Perona-Malik eqation.

Parameters
xnumpy.ndarray, optional

Spatial grid on which to create the initial conditions. The default is an equidistant grid between 0 and 5.98 with step size 0.02.

Returns
callable

A function that takes a numpy random generator as input and returns an initial state on the spatial grid x.

phlearn.phsystems.initial_condition_radial(r_min, r_max)

Creates an initial condition sampler that draws samples uniformly from the disk r_min <= x^Tx < r_max.

Returns
callable

Function taking a numpy random generator and returning an initial state of size 2.

phlearn.phsystems.zero_force(x, t=None)

A force term that is always zero.

Parameters
x(…, N) ndarray
t(…, 1) ndarray, default None
Returns
(…, N) ndarray

All zero ndarray