Partial Differential Equations (PDE)

NeuralPDE.jl: Physics-Informed Neural Network (PINN) PDE Solvers

NeuralPDE.jl is a partial differential equation solver library which uses physics-informed neural networks (PINNs) to solve the equations. It uses the ModelingToolkit.jl symbolic PDESystem as its input and can handle a wide variety of equation types, including systems of partial differential equations, partial differential-algebraic equations, and integro-differential equations. Its benefit is its flexibility, and it can be used to easily generate surrogate solutions over entire parameter ranges. However, its downside is solver speed: PINN solvers tend to be a lot slower than other methods for solving PDEs.

MethodOflines.jl: Automated Finite Difference Method (FDM)

MethodOflines.jl is a partial differential equation solver library which automates the discretization of PDEs via the finite difference method. It uses the ModelingToolkit.jl symbolic PDESystem as its input, and generates AbstractSystems and SciMLProblems whose numerical solution gives the solution to the PDE.

FEniCS.jl: Wrappers for the Finite Element Method (FEM)

FEniCS.jl is a wrapper for the popular FEniCS finite element method library.

HighDimPDE.jl: High-dimensional PDE Solvers

HighDimPDE.jl is a partial differential equation solver library which implements algorithms that break down the curse of dimensionality to solve the equations. It implements deep-learning based and Picard-iteration based methods to approximately solve high-dimensional, nonlinear, non-local PDEs in up to 10,000 dimensions. Its cons are accuracy: high-dimensional solvers are stochastic, and might result in wrong solutions if the solver meta-parameters are not appropriate.

NeuralOperators.jl: (Fourier) Neural Operators and DeepONets for PDE Solving

NeuralOperators.jl is a library for operator learning based PDE solvers. This includes techniques like:

Fourier Neural Operators (FNO)
Deep Operator Networks (DeepONets)
Markov Neural Operators (MNO)

Currently, its connection to PDE solving must be specified manually, though an interface for ModelingToolkit PDESystems is in progress.

DiffEqOperators.jl: Operators for Finite Difference Method (FDM) Discretizations

DiffEqOperators.jl is a library for defining finite difference operators to easily perform manual FDM semi-discretizations of partial differential equations. This library is fairly incomplete and most cases should receive better performance using MethodOflines.jl.

Third-Party Libraries to Note

A more exhaustive list of Julia PDE packages can be found here: https://github.com/JuliaPDE/SurveyofPDEPackages

ApproxFun.jl: Automated Spectral Discretizations

ApproxFun.jl is a package for approximating functions in basis sets. One particular use case is with spectral basis sets, such as Chebyshev functions and Fourier decompositions, making it easy to represent spectral and pseudospectral discretizations of partial differential equations as ordinary differential equations for the SciML equation solvers.

Decapodes.jl: Discrete Exterior Calculus Applied to Partial and Ordinary Differential Equation Systems

Decapodes.jl is a computational physics framework based on the Discrete Exterior Calculus (DEC). It uses structure preserving discretization methods from the DEC to simulate multiphysics problems. Feature include solving PDEs on triangulated manifolds, a strongly typed equation representation that can help ensure correctness of simulations, and compositional methods for specifying multi-physics problems.

Ferrite.jl: Finite Element Toolbox for Julia

Ferrite.jl is a performant and extensible library which provides algorithms and data structures to develop finite element software. This library aims at users which need fine grained control over all algorithmic details, as for example often necessary in research when developing new grid-based PDE discretizations or other more advanced problem formulations for example found in continuum mechanics.