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https://github.com/SciML/NeuralPDE.jl

Physics-Informed Neural Networks (PINN) Solvers of (Partial) Differential Equations for Scientific Machine Learning (SciML) accelerated simulation
https://github.com/SciML/NeuralPDE.jl

differential-equations differentialequations machine-learning neural-differential-equations neural-network neural-networks ode ordinary-differential-equations partial-differential-equations pde pinn scientific-ai scientific-machine-learning scientific-ml sciml

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Physics-Informed Neural Networks (PINN) Solvers of (Partial) Differential Equations for Scientific Machine Learning (SciML) accelerated simulation

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README 

        
# NeuralPDE



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NeuralPDE.jl is a solver package which consists of neural network solvers for

partial differential equations using physics-informed neural networks (PINNs). This package utilizes

neural stochastic differential equations to solve PDEs at a greatly increased generality

compared with classical methods.



## Installation



Assuming that you already have Julia correctly installed, it suffices to install NeuralPDE.jl in the standard way, that is, by typing `] add NeuralPDE`. Note:

to exit the Pkg REPL-mode, just press Backspace or Ctrl + C.



## Tutorials and Documentation



For information on using the package,

[see the stable documentation](https://docs.sciml.ai/NeuralPDE/stable/). Use the

[in-development documentation](https://docs.sciml.ai/NeuralPDE/dev/) for the version of

the documentation, which contains the unreleased features.



## Features



  - Physics-Informed Neural Networks for ODE, SDE, RODE, and PDE solving

  - Ability to define extra loss functions to mix xDE solving with data fitting (scientific machine learning)

  - Automated construction of Physics-Informed loss functions from a high level symbolic interface

  - Sophisticated techniques like quadrature training strategies, adaptive loss functions, and neural adapters

    to accelerate training

  - Integrated logging suite for handling connections to TensorBoard

  - Handling of (partial) integro-differential equations and various stochastic equations

  - Specialized forms for solving `ODEProblem`s with neural networks

  - Compatibility with [Flux.jl](https://fluxml.ai/) and [Lux.jl](https://lux.csail.mit.edu/)

    for all of the GPU-powered machine learning layers available from those libraries.

  - Compatibility with [NeuralOperators.jl](https://docs.sciml.ai/NeuralOperators/stable/) for

    mixing DeepONets and other neural operators (Fourier Neural Operators, Graph Neural Operators,

    etc.) with physics-informed loss functions



## Example: Solving 2D Poisson Equation via Physics-Informed Neural Networks



```julia

using NeuralPDE, Lux, ModelingToolkit, Optimization, OptimizationOptimisers

import ModelingToolkit: Interval, infimum, supremum



@parameters x y

@variables u(..)

Dxx = Differential(x)^2

Dyy = Differential(y)^2



# 2D PDE

eq = Dxx(u(x, y)) + Dyy(u(x, y)) ~ -sin(pi * x) * sin(pi * y)



# Boundary conditions

bcs = [u(0, y) ~ 0.0, u(1, y) ~ 0,

    u(x, 0) ~ 0.0, u(x, 1) ~ 0]

# Space and time domains

domains = [x ∈ Interval(0.0, 1.0),

    y ∈ Interval(0.0, 1.0)]

# Discretization

dx = 0.1



# Neural network

dim = 2 # number of dimensions

chain = Lux.Chain(Dense(dim, 16, Lux.σ), Dense(16, 16, Lux.σ), Dense(16, 1))



discretization = PhysicsInformedNN(chain, QuadratureTraining())



@named pde_system = PDESystem(eq, bcs, domains, [x, y], [u(x, y)])

prob = discretize(pde_system, discretization)



callback = function (p, l)

    println("Current loss is: $l")

    return false

end



res = Optimization.solve(prob, ADAM(0.1); callback = callback, maxiters = 4000)

prob = remake(prob, u0 = res.minimizer)

res = Optimization.solve(prob, ADAM(0.01); callback = callback, maxiters = 2000)

phi = discretization.phi

```



And some analysis:



```julia

xs, ys = [infimum(d.domain):(dx / 10):supremum(d.domain) for d in domains]

analytic_sol_func(x, y) = (sin(pi * x) * sin(pi * y)) / (2pi^2)



u_predict = reshape([first(phi([x, y], res.minimizer)) for x in xs for y in ys],

    (length(xs), length(ys)))

u_real = reshape([analytic_sol_func(x, y) for x in xs for y in ys],

    (length(xs), length(ys)))

diff_u = abs.(u_predict .- u_real)



using Plots

p1 = plot(xs, ys, u_real, linetype = :contourf, title = "analytic");

p2 = plot(xs, ys, u_predict, linetype = :contourf, title = "predict");

p3 = plot(xs, ys, diff_u, linetype = :contourf, title = "error");

plot(p1, p2, p3)

```



![image](https://user-images.githubusercontent.com/12683885/90962648-2db35980-e4ba-11ea-8e58-f4f07c77bcb9.png)



### Citation



If you use NeuralPDE.jl in your research, please cite [this paper](https://arxiv.org/abs/2107.09443):



```bib

@article{zubov2021neuralpde,

  title={NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations},

  author={Zubov, Kirill and McCarthy, Zoe and Ma, Yingbo and Calisto, Francesco and Pagliarino, Valerio and Azeglio, Simone and Bottero, Luca and Luj{\'a}n, Emmanuel and Sulzer, Valentin and Bharambe, Ashutosh and others},

  journal={arXiv preprint arXiv:2107.09443},

  year={2021}

}

```