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Boundary, initial and custom conditions

Source: R/conditions.R
conditions.Rd

Conditions are enforced as soft penalties on points sampled where they apply. Each condition fixes one or more domain variables (where) and samples the remaining variables over their full range.

Usage

pinn_dirichlet(
  where,
  value = 0,
  output = NULL,
  n = 100,
  weight = 1,
  name = NULL
)

pinn_neumann(
  where,
  along,
  value = 0,
  output = NULL,
  n = 100,
  weight = 1,
  name = NULL
)

pinn_initial(
  value,
  time = "t",
  output = NULL,
  n = 100,
  weight = 1,
  name = NULL
)

pinn_condition(where, fn, n = 100, weight = 1, name = NULL)

Arguments

where

Named list giving the fixed value(s) of one or more domain variables. See Locations.

value

Target value. See Values.

output

Name(s) of the output(s) the condition applies to. The default NULL applies it to every output.

n

Number of points used for this condition.

weight

Relative weight of this condition in the loss.

name

Optional label used in printed output.

along

Name of the domain variable to differentiate with respect to.

time

Name of the time variable for pinn_initial().

fn

For pinn_condition(), a function returning the residual (a tensor, or a list of tensors) that should be zero. Arguments are matched by name to outputs, domain variables, parameters and the derivative operator d (see pinn_problem()).

Value

An object of class pinn_condition.

Details

  • pinn_dirichlet() fixes the value of the solution, \(u = g\).

  • pinn_neumann() fixes a first derivative, \(\partial u / \partial v = g\).

  • pinn_initial() is a Dirichlet condition at the lower end of the time variable.

  • pinn_condition() takes any residual function, for Robin, periodic or other conditions.

Locations

Each element of where is a number or one of the strings "lower", "upper" or "both", which refer to the ends of that variable's range. where = list(x = "both") places points on both ends of x. When several variables are fixed, every combination of their values is used: where = list(x = "both", y = "both") places points on the four corners.

Values

value can be a number, a numeric vector named by output, or a function. A function's arguments are matched by name to the domain variables, outputs and parameters, so value = function(x) sin(pi * x) uses the spatial variable x. Inside these functions the arguments are torch tensors, so use arithmetic and maths functions (sin, exp, log, ...) rather than control flow such as if or ifelse.

Examples

# u = 0 on both ends of x
pinn_dirichlet(where = list(x = "both"), value = 0)
#> <pinn_condition: dirichlet at x = both, n = 100>

# u(x, 0) = sin(pi x)
pinn_initial(function(x) sin(pi * x), time = "t")
#> <pinn_condition: initial at t = lower, n = 100>

# insulated right boundary: du/dx = 0 at x = 1
pinn_neumann(where = list(x = "upper"), along = "x")
#> <pinn_condition: neumann at x = upper, n = 100>

# Robin condition du/dx + 2 u = 0 at x = 1, as a custom residual
pinn_condition(where = list(x = "upper"),
               fn = function(u, x, d) d(u, x) + 2 * u)
#> <pinn_condition: custom at x = upper, n = 100>