SciMLOperators.jl: Unified operator interface for `SciML.ai` and beyond

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")

Examples

Let M, D, F be matrix-based, diagonal-matrix-based, and function-based SciMLOperators respectively.

N = 4
f = (u, p, t) -> u .* u

M = MatrixOperator(rand(N, N))
D = DiagonalOperator(rand(N))
F = FunctionOperator(f, zeros(N), zeros(N))

Then, the following codes just work.

L1 = 2M + 3F + LinearAlgebra.I + rand(N, N)
L2 = D * F * M'
L3 = kron(M, D, F)
L4 = M \ D
L5 = [M; D]' * [M F; F D] * [F; D]

Each L# can be applied to AbstractVectors of appropriate sizes:

p = nothing # parameter struct
t = 0.0     # time

u = rand(N)
v = L1(u, p, t) # == L1 * u

u_kron = rand(N^3)
v_kron = L3(u_kron, p, t) # == L3 * u_kron

For mutating operator evaluations, call cache_operator to generate an in-place cache, so the operation is nonallocating.

α, β = rand(2)

# allocate cache
L2 = cache_operator(L2, u)
L4 = cache_operator(L4, u)

# allocation-free evaluation
L2(v, u, p, t) # == mul!(v, L2, u)
L4(v, u, p, t, α, β) # == mul!(v, L4, u, α, β)

The calling signature L(u, p, t), for out-of-place evaluations, is equivalent to L * u, and the in-place evaluation L(v, u, p, t, args...) is equivalent to LinearAlgebra.mul!(v, L, u, args...), where the arguments p, t are passed to L to update its state. More details are provided in the operator update section below. While overloads to Base.* and LinearAlgebra.mul! are available, where a SciMLOperator behaves like an AbstractMatrix, we recommend sticking with the L(u, p, t), L(v, u, p, t), L(v, u, p, t, α, β) calling signatures as the latter internally update the operator state.

The (u, p, t) calling signature is standardized over the SciML ecosystem and is flexible enough to support use cases such as time-evolution in ODEs, as well as sensitivity computation with respect to the parameter object p.

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 v1.9.0
  [7a1cc6ca] FFTW v1.8.1
  [c0aeaf25] SciMLOperators v0.3.13 `~/work/SciMLOperators.jl/SciMLOperators.jl`

and using this machine and Julia version.

Julia Version 1.11.4
Commit 8561cc3d68d (2025-03-10 11:36 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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