# The SciML Common Interface for Julia Equation Solvers

The SciML common interface ties together the numerical solvers of the Julia package ecosystem into a single unified interface. It is designed for maximal efficiency and parallelism, while incorporating essential features for large-scale scientific machine learning such as differentiability, composability, and sparsity.

This documentation is made to pool together the docs of the various SciML libraries to paint the overarching picture, establish development norms, and document the shared/common functionality.

## Domains of SciML

The SciML common interface covers the following domains:

• Linear systems (LinearProblem)
• Direct methods for dense and sparse
• Iterative solvers with preconditioning
• Nonlinear Systems (NonlinearProblem)
• Systems of nonlinear equations
• Scalar bracketing systems
• Integrals (quadrature) (QuadratureProblem)
• Differential Equations
• Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations) (DiscreteProblem)
• Ordinary differential equations (ODEs) (ODEProblem)
• Split and Partitioned ODEs (Symplectic integrators, IMEX Methods) (SplitODEProblem)
• Stochastic ordinary differential equations (SODEs or SDEs) (SDEProblem)
• Stochastic differential-algebraic equations (SDAEs) (SDEProblem with mass matrices)
• Random differential equations (RODEs or RDEs) (RODEProblem)
• Differential algebraic equations (DAEs) (DAEProblem and ODEProblem with mass matrices)
• Delay differential equations (DDEs) (DDEProblem)
• Neutral, retarded, and algebraic delay differential equations (NDDEs, RDDEs, and DDAEs)
• Stochastic delay differential equations (SDDEs) (SDDEProblem)
• Experimental support for stochastic neutral, retarded, and algebraic delay differential equations (SNDDEs, SRDDEs, and SDDAEs)
• Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions) (DEProblems with callbacks)
• Optimization (OptimizationProblem)
• Nonlinear (constrained) optimization
• (Stochastic/Delay/Differential-Algebraic) Partial Differential Equations (PDESystem)
• Finite difference and finite volume methods
• Interfaces to finite element methods
• Physics-Informed Neural Networks (PINNs)
• Integro-Differential Equations
• Fractional Differential Equations

The SciML common interface also includes ModelingToolkit.jl for defining such systems symbolically, allowing for optimizations like automated generation of parallel code, symbolic simplification, and generation of sparsity patterns.

## Extended SciML Domain

In addition to the purely numerical representations of mathematical objects, there are also sets of problem types associated with common mathematical algorithms. These are:

• Data-driven modeling
• Discrete-time data-driven dynamical systems (DiscreteDataDrivenProblem)
• Continuous-time data-driven dynamical systems (ContinuousDataDrivenProblem)
• Symbolic regression (DirectDataDrivenProblem)
• Uncertainty quantification and expected values (ExpectationProblem)

## Inverse Problems, Parameter Estimation, and Structural Identification

We note that parameter estimation and inverse problems are solved directly on their constituant problem types using tools like DiffEqFlux.jl. Thus for example, there is no ODEInverseProblem, and instead ODEProblem is used to find the parameters p that solve the inverse problem.

## Common Interface High Level

The SciML interface is common as the usage of arguments is standardized across all of the problem domains. Underlying high level ideas include:

• All domains use the same interface of defining a SciMLProblem which is then solved via solve(prob,alg;kwargs), where alg is a SciMLAlgorithm. The keyword argument namings are standardized across the organization.
• SciMLProblems are generally defined by a SciMLFunction which can define extra details about a model function, such as its analytical Jacobian, its sparsity patterns and so on.
• There is an organization-wide method for defining linear and nonlinear solvers used within other solvers, giving maximum control of performance to the user.
• Types used within the packages are defined by the input types. For example, packages attempt to internally use the type of the initial condition as the type for the state within differential equation solvers.
• solve calls should be thread-safe and parallel-safe.
• init(prob,alg;kwargs) returns an iterator which allows for directly iterating over the solution process
• High performance is key. Any performance that is not at the top level is considered a bug and should be reported as such.
• All functions have an in-place and out-of-place form, where the in-place form is made to utilize mutation for high performance on large-scale problems and the out-of-place form is for compatibility with tooling like static arrays and some reverse-mode automatic differentiation systems.

## User-Facing Solver Libraries

• DifferentialEquations.jl
• Multi-package interface of high performance numerical solvers of differential equations
• ModelingToolkit.jl
• The symbolic modeling package which implements the SciML symbolic common interface.
• LinearSolve.jl
• Multi-package interface for specifying linear solvers (direct, sparse, and iterative), along with tools for caching and preconditioners for use in large-scale modeling.
• NonlinearSolve.jl
• High performance numerical solving of nonlinear systems.
• Multi-package interface for high performance, batched, and parallelized numerical quadrature.
• GalacticOptim.jl
• Multi-package interface for numerical solving of optimization problems.
• NeuralPDE.jl
• Physics-Informed Neural Network (PINN) package for transforming partial differential equations into optimization problems.
• DiffEqOperators.jl
• Automated finite difference method (FDM) package for transforming partial differential equations into nonlinear problems and ordinary differential equations.
• DiffEqFlux.jl
• High level package for scientific machine learning applications, such as neural and universal differential equations, solving of inverse problems, parameter estimation, nonlinear optimal control, and more.
• Multi-package interface for data-driven modeling, Koopman dynamic mode decomposition, symbolic regression/sparsification, and automated model discovery.
• DiffEqUncertainty.jl
• Extension to the dynamical modeling tools for performing uncertainty quantification and calculating expectations.

## Interface Implementation Libraries

• SciMLBase.jl
• The core package defining the interface which is consumed by the modeling and solver packages.
• DiffEqSensitivity.jl
• A package which pools together the definition of derivative overloads to define the common sensealg automatic differentiation interface.
• DiffEqNoiseProcess.jl
• A package which defines the stochastic AbstractNoiseProcess interface for the SciML ecosystem.
• RecursiveArrayTools.jl
• A package which defines the underlying AbstractVectorOfArray structure used as the output for all time series results.
• ArrayInterface.jl
• The package which defines the extended AbstractArray interface employed throughout the SciML ecosystem.

## Using-Facing Modeling Libraries

There are too many to name here and this will be populated when there is time!

## Flowchart Example for PDE-Constrained Optimal Control

The following example showcases how the pieces of the common interface connect to solve a problem that mixes inference, symbolics, and numerics.

## External Binding Libraries

• diffeqr
• Solving differential equations in R using DifferentialEquations.jl with ModelingToolkit for JIT compilation and GPU-acceleration
• diffeqpy
• Solving differential equations in Python using DifferentialEquations.jl

## Solver Libraries

There are too many to name here. Check out the SciML Organization Github Page for details.