SciMLOperators.jl: Unified operator interface for Julia and SciML
SciMLOperators is a package for managing linear, nonlinear, time-dependent, and parameter dependent operators acting on vectors, (or column-vectors of matrices). We provide wrappers for matrix-free operators, fast tensor-product evaluations, pre-cached mutating evaluations, as well as Zygote-compatible non-mutating evaluations.
The lazily implemented operator algebra allows the user to update the operator state by passing in an update function that accepts arbitrary parameter objects. Further, our operators behave like AbstractMatrix types thanks to overloads defined for methods in Base, and LinearAlgebra.
Therefore, an AbstractSciMLOperator can be passed to LinearSolve.jl, or NonlinearSolve.jl as a linear or nonlinear operator, or to OrdinaryDiffEq.jl as an ODEFunction. Examples of usage within the SciML ecosystem are provided in the documentation.
Installation
To install SciMLOperators.jl, use the Julia package manager:
using Pkg
Pkg.add("SciMLOperators")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(v,u,p,t) has the same operation as update_coefficients(A, u, p, t); A * v, then A can be used in any place where a differential equation definition (u,p,t) -> A(u, u, p, t) is used without requiring the user or solver to do any extra work.
Another example is state-dependent mass matrices. M(u,p,t)*u' = f(u,p,t). When solving such an equation, the solver must understand how to "update M" during operations, and thus the ability to update the state of M is a required function in the interface. This is also required for the definition of Jacobians J(u,p,t) in order to be properly used with Krylov methods inside of ODE solves without reconstructing the matrix-free operator at each step.
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.
This means that LinearMaps.jl is fundamentally lacking and is incompatible with many of the tools in the SciML ecosystem, except for the specific cases where the matrix-free operator is a constant!
Features
- Matrix-free operators with
FunctionOperator - Fast tensor product evaluation with
TensorProductOperator - Lazy algebra: addition, subtraction, multiplication, inverse, adjoint, and transpose
- Couple fast methods for operator evaluation with inversion via
InvertibleOperator - One-line API to update operator state depending on arbitrary parameters.
- Mutating and nonmutating update behavior (Zygote compatible)
- One-line API for pre-caching operators for in-place operator evaluations
Contributing
Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page
Reproducibility
The documentation of this SciML package was built using these direct dependencies,
Status `~/work/SciMLOperators.jl/SciMLOperators.jl/docs/Project.toml`
[e30172f5] Documenter v1.13.0
[7a1cc6ca] FFTW v1.9.0
[c0aeaf25] SciMLOperators v1.3.1 `~/work/SciMLOperators.jl/SciMLOperators.jl`and using this machine and Julia version.
Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 7763 64-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)A more complete overview of all dependencies and their versions is also provided.
Status `~/work/SciMLOperators.jl/SciMLOperators.jl/docs/Manifest.toml`
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`You can also download the manifest file and the project file.