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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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NeuralPDE

NeuralPDE.jl is a solver package which consists of neural network solvers forpartial differential equations using physics-informed neural networks (PINNs). This package utilizesneural stochastic differential equations to solve PDEs at a greatly increased generalitycompared 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 pressBackspace orCtrl +C.

Tutorials and Documentation

For information on using the package,see the stable documentation. Use thein-development documentation for the version ofthe 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 adaptersto accelerate training
Integrated logging suite for handling connections to TensorBoard
Handling of (partial) integro-differential equations and various stochastic equations
Specialized forms for solvingODEProblems with neural networks
Compatibility withFlux.jl andLux.jlfor all of the GPU-powered machine learning layers available from those libraries.
Compatibility withNeuralOperators.jl formixing 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

using NeuralPDE, Lux, ModelingToolkit, Optimization, OptimizationOptimisersimport DomainSets: Interval, infimum, supremum@parameters x y@variablesu(..)Dxx=Differential(x)^2Dyy=Differential(y)^2# 2D PDEeq=Dxx(u(x, y))+Dyy(u(x, y))~-sin(pi* x)*sin(pi* y)# Boundary conditionsbcs= [u(0, y)~0.0,u(1, y)~0,u(x,0)~0.0,u(x,1)~0]# Space and time domainsdomains= [x∈Interval(0.0,1.0),    y∈Interval(0.0,1.0)]# Discretizationdx=0.1# Neural networkdim=2# number of dimensionschain= 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")returnfalseendres= 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:

xs, ys= [infimum(d.domain):(dx/10):supremum(d.domain)for din domains]analytic_sol_func(x, y)= (sin(pi* x)*sin(pi* y))/ (2pi^2)u_predict=reshape([first(phi([x, y], res.minimizer))for xin xsfor yin ys],    (length(xs),length(ys)))u_real=reshape([analytic_sol_func(x, y)for xin xsfor yin ys],    (length(xs),length(ys)))diff_u=abs.(u_predict.- u_real)using Plotsp1=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)

Citation

If you use NeuralPDE.jl in your research, please citethis paper:

@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}}