The PDE Definition Interface

struct PDEProblem{P, E, S} <: SciMLBase.AbstractPDEProblem

Concrete wrapper for PDE problems before discretization.

PDEProblem stores a symbolic or package-specific PDE problem object together with extrapolation and spatial-domain metadata used by discretization backends. SciMLBase owns only this lightweight wrapper; concrete PDE semantics and the conversion to ODE, nonlinear, optimization, or linear problems are supplied by downstream discretization packages through discretize and symbolic_discretize.

Fields

• prob::Any

• extrapolation::Any

• space::Any

discretize(sys, discretizer, args...; kwargs...)

Transform a symbolic or high-level problem description into a solver-ready problem.

Discretizer packages implement SciMLBase.discretize for the system/problem types and discretization algorithms they own. A method should return an AbstractSciMLProblem or another documented problem type accepted by the intended solver. Implementations should document the accepted input system type, the meaning of discretizer, the generated problem family, forwarded problem-construction keywords, and any metadata needed to map numerical solutions back to the original variables and domains.

Use symbolic_discretize when the caller needs a diagnostic or symbolic view of the discretized system rather than the solver-ready problem.

symbolic_discretize(sys, discretizer, args...; kwargs...)

Return a symbolic or diagnostic representation of a discretization.

Discretizer packages implement SciMLBase.symbolic_discretize when they can expose lowered equations, operators, grids, boundary-condition handling, or other intermediate artifacts without constructing only the final solver-ready problem. The result should correspond to the same mathematical discretization used by discretize, but may preserve additional information useful for inspection, debugging, code generation, or downstream transformations. Its type is package-defined and need not subtype a symbolic-system abstraction.

struct PDETimeSeriesSolution{T, N, uType, Disc, Sol, DType, tType, domType, ivType, dvType, P, A, IType, S} <: SciMLBase.AbstractPDETimeSeriesSolution{T, N, uType, Disc}

Solution to a PDE, solved from an ODEProblem generated by a discretizer.

PDETimeSeriesSolution is the concrete wrapper for time-dependent PDE discretizations whose metadata subtypes AbstractDiscretizationMetadata{Val(true)}. Discretizer packages usually define an outer constructor SciMLBase.PDETimeSeriesSolution(sol, metadata::D) for their metadata type D; the generic wrap_sol dispatch calls that constructor after the underlying ODE or time-dependent solve has finished.

Fields

• u: the solution to the PDE, as a dictionary of symbols to Arrays of values. The Arrays are of the same shape as the domain of the PDE. Time is always the first axis.
• original_sol: The original ODESolution that was used to generate this solution.
• t: the time points corresponding to the saved values of the ODE solution.
• ivdomain: The full list of domains for the independent variables. May be a grid for a discrete solution, or a vector/tuple of tuples for a continuous solution.
• ivs: The list of independent variables for the solution.
• dvs: The list of dependent variables for the solution.
• disc_data: Metadata about the discretization process and type.
• prob: The ODEProblem that was used to generate this solution.
• alg: The algorithm used to solve the ODEProblem.
• interp: Interpolations for the solution.
• retcode: the return code from the solver. Used to determine whether the solver solved successfully, whether it terminated early due to a user-defined callback, or whether it exited due to an error. For more details, see the return code documentation.
• stats: statistics of the solver, such as the number of function evaluations required.

struct PDENoTimeSolution{T, N, uType, Disc, Sol, domType, ivType, dvType, P, A, IType, S} <: SciMLBase.AbstractPDENoTimeSolution{T, N, uType, Disc}

Solution to a PDE, solved from a NonlinearProblem generated by a discretizer.

PDENoTimeSolution is the concrete wrapper for time-independent PDE discretizations whose metadata subtypes AbstractDiscretizationMetadata{Val(false)}. Discretizer packages usually define an outer constructor SciMLBase.PDENoTimeSolution(sol, metadata::D) for their metadata type D; the generic wrap_sol dispatch calls that constructor after the underlying nonlinear, linear, optimization, or stationary solve has finished.

Fields

• u: the solution to the PDE, as a dictionary of symbols to Arrays of values. The Arrays are of the same shape as the domain of the PDE. Time is always the first axis.
• original_sol: The original NonlinearSolution that was used to generate this solution.
• ivdomain: The full list of domains for the independent variables. May be a grid for a discrete solution, or a vector/tuple of tuples for a continuous solution.
• ivs: The list of independent variables for the solution.
• dvs: The list of dependent variables for the solution.
• disc_data: Metadata about the discretization process and type.
• prob: The NonlinearProblem that was used to generate this solution.
• alg: The algorithm used to solve the NonlinearProblem.
• interp: Interpolations for the solution.
• retcode: The return code from the solver. Used to determine whether the solver solved successfully (sol.retcode === ReturnCode.Success), whether it terminated due to a user-defined callback (sol.retcode === ReturnCode.Terminated), or whether it exited due to an error. For more details, see the return code section of the ODEProblem.jl documentation.

wrap_sol(sol)
wrap_sol(sol, problem_type_or_metadata)

Return sol or wrap it in a higher-level SciML solution container.

Solvers call wrap_sol(sol) after constructing a low-level solution. When sol.prob is an AbstractSciMLProblem, the default implementation queries problem_type and dispatches to wrap_sol(sol, problem_type_or_metadata) when the result is not nothing. Problem-family packages extend the two-argument form when a generated solver solution should be returned as a more specific public solution type.

The fallback two-argument method returns sol unchanged. PDE discretizer packages extend the metadata path by defining constructors such as PDETimeSeriesSolution(sol, metadata::D) or PDENoTimeSolution(sol, metadata::D) for their concrete discretization metadata type.

wrap_sol(sol, metadata::AbstractDiscretizationMetadata)

Wrap a solver solution produced by a PDE discretizer in the appropriate PDE solution type.

Metadata with parameter Val(true) is routed to PDETimeSeriesSolution(sol, metadata). Metadata with parameter Val(false) is routed to PDENoTimeSolution(sol, metadata). Discretizer packages are responsible for implementing those outer constructors for their concrete metadata types and for filling the wrapper fields from the discretized solve.

abstract type AbstractDifferentiableDiscretizationMetadata{hasTime} <: SciMLBase.AbstractDiscretizationMetadata{hasTime}

Metadata of a discretizer whose wrapped solutions differentiate in reverse mode: sol[sym], sol[sym, inds...] and sol(args...; dv) get rules in the ChainRulesCore, Mooncake and Enzyme extensions that route the cotangent to sol.original_sol, whose rules the solver packages provide, through pde_index_cotangent and pde_call_cotangent, which the discretizer implements for its metadata type. The solutions of metadata that subtypes AbstractDiscretizationMetadata directly get none of these rules.

pde_index_cotangent(sol::AbstractPDESolution, sym, Δ)

Reverse-mode hook behind sol[sym] and sol[sym, inds...] (sol[sym][inds...] in the interface) for a solution whose metadata subtypes AbstractDifferentiableDiscretizationMetadata: how a cotangent Δ of the array sol[sym] goes back to the solver's solution sol.original_sol. It returns nothing when sym does not depend on the solve (a grid, for instance), and otherwise a named tuple (; u, observed):

• u: the cotangent of sol.original_sol.u, of the same shape, or nothing;
• observed: a collection of terms (f, k, δ) for the values the solution observes rather than stores, each the cotangent δ of f(sol.original_sol.u[k], p, sol.original_sol.t[k]) with p = parameter_values(sol.original_sol), or terms (f, δ) of f(sol.original_sol.u, p) when sol.original_sol is not a time series (an optimization solution, for instance).

The hook is plain code, with no AD types: the AD extensions pull the observed terms back with the AD system in use and hand the cotangent to the solver's rules for original_sol. Only the map from a field back to the solver's solution is the discretizer's, implemented here for its metadata type D. sym is whatever the solution was indexed with, and the metadata and the functions f are constants of the solve. The fallback errors, and a hook that cannot provide a cotangent throws as well, so that a solution never returns a wrong gradient silently.

pde_call_cotangent(sol::AbstractPDESolution, args, dv, Δ)

Reverse-mode hook behind sol(args...; dv): a tuple of the cotangent of sol.original_sol, in the form pde_index_cotangent returns it, and the cotangents of args, for a cotangent Δ of the evaluation at args. dv is the keyword as it was passed to the call, or nothing when it was not, and Δ then has the shape of what the call returns in that case (a collection over sol.dvs in the documented interface). The cotangents of args are one per argument: that of a coordinate, of the shape of the argument when it is an array, or an error when the discretizer cannot provide it, and nothing for an index or a colon. The fallback errors.