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Usage with SciML and beyond

Why SciMLOperators?

Many functions, from linear solvers to differential equations, require the use of matrix-free operators to achieve maximum performance in many scenarios. SciMLOperators.jl defines the abstract interface for how operators in the SciML ecosystem are supposed to be defined. It gives the common set of functions and traits that solvers can rely on for properly performing their tasks. Along with that, SciMLOperators.jl provides definitions for the basic standard operators that are used as building blocks for most tasks, simplifying the use of operators while also demonstrating to users how such operators can be built and used in practice.

SciMLOperators.jl has the design that is required to be used in all scenarios of equation solvers. For example, Magnus integrators for differential equations require defining an operator $u' = A(t) u$, while Munthe-Kaas methods require defining operators of the form $u' = A(u) u$. Thus, the operators need some form of time and state dependence, which the solvers can update and query when they are non-constant (update_coefficients!). Additionally, the operators need the ability to act like “normal” functions for equation solvers. For example, if A(u,p,t) has the same operation as update_coefficients(A, u, p, t); A * u, then A can be used in any place where a differential equation definition f(u, p, t) is used without requiring the user or solver to do any extra work. Thus while previous good efforts for matrix-free operators have existed in the Julia ecosystem, such as LinearMaps.jl, those operator interfaces lack these aspects to actually be fully seamless with downstream equation solvers. This necessitates the definition and use of an extended operator interface with all of these properties, hence the AbstractSciMLOperator interface.

Some packages providing similar functionality are

Interoperability and extended Julia ecosystem

SciMLOperator.jl overloads the AbstractMatrix interface for AbstractSciMLOperators, allowing seamless compatibility with linear and nonlinear solvers. Further, due to the update functionality, AbstractSciMLOperators can represent an ODEFunction in OrdinaryDiffEq.jl, and downstream packages. See tutorials for examples of usage with OrdinaryDiffEq.jl, LinearSolve.jl, NonlinearSolve.jl.

Further, the nonmutating update functionality allows gradient propagation through AbstractSciMLOperators, and is compatible with automatic-differentiation libraries like Zygote.jl. An example of Zygote.jl usage with Lux.jl is also provided in the tutorials.

Please make an issue here if you come across an unexpected issue while using SciMLOperators.

We provide below a list of packages that make use of SciMLOperators. If you are using SciMLOperators in your work, feel free to create a PR and add your package to this list.