Define a physics-informed problem

Collects everything a PINN needs: the domain, the names of the unknown functions, the equation residual, conditions, optional observations and the equation's parameters.

Usage

pinn_problem(
  domain,
  outputs,
  residual,
  conditions = list(),
  data = NULL,
  parms = list(),
  unknown = NULL,
  positive = TRUE,
  transform = NULL
)

Arguments

domain: A pinn_domain(), or a named list of ranges.
outputs: Character vector naming the unknown functions.
residual: Function returning the equation residual. See Writing the residual.
conditions: A condition or list of conditions from pinn_dirichlet(), pinn_neumann(), pinn_initial() or pinn_condition().
data: Optional data frame of observations, with one column per domain variable and one column for each observed output. Missing values (NA) are ignored, so outputs can be observed at different points.
parms: Named list or numeric vector of known constants.
unknown: Named numeric vector of initial guesses for parameters to estimate.
positive: Logical, recycled or named by unknown. Positive parameters are estimated on the log scale, which keeps them positive and usually helps convergence.
transform: Optional output transform. See Output transforms.

Value

An object of class pinn_problem.

Writing the residual

residual is an R function whose arguments are matched by name to:

the outputs (for example u),
the domain variables (for example x, t),
the parameters in parms and unknown, and
d, the derivative operator.

d(f, x) differentiates f with respect to the variable x; d(f, x, x) gives the second derivative and d(f, x, t) a mixed derivative. The function returns the expression that should equal zero, or a list of such expressions for a system of equations. For the viscous Burgers equation \(u_t + u u_x - \nu u_{xx} = 0\):

function(u, x, t, nu, d) d(u, t) + u * d(u, x) - nu * d(u, x, x)

All arguments are torch tensors with one row per point, so use arithmetic and maths functions (sin, exp, log, sqrt, ...) and avoid if, ifelse or functions that leave R's torch methods.

Output transforms

transform can enforce conditions exactly ("hard constraints"). It is called with the raw network outputs, domain variables and parameters (matched by name, as above) and returns a named list of modified outputs. For example, to make \(u(0, t) = u(1, t) = 0\) hold exactly: transform = function(u, x) list(u = x * (1 - x) * u).

Examples

# Heat equation u_t = a u_xx with u(x, 0) = sin(pi x) and u = 0 at both ends
heat <- pinn_problem(
  domain = pinn_domain(x = c(0, 1), t = c(0, 1)),
  outputs = "u",
  residual = function(u, x, t, a, d) d(u, t) - a * d(u, x, x),
  conditions = list(
    pinn_initial(function(x) sin(pi * x), time = "t"),
    pinn_dirichlet(where = list(x = "both"), value = 0)
  ),
  parms = list(a = 0.1)
)
heat
#> <pinn_problem>
#>   Domain:     x in [0, 1], t in [0, 1]
#>   Outputs:    u
#>   Parameters: a = 0.1
#>   Conditions:
#>     <pinn_condition: initial at t = lower, n = 100>
#>     <pinn_condition: dirichlet at x = both, n = 100>