NeuralPDE

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. Use the in-development documentation 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 ODEProblems with neural networks
Compatibility with Flux.jl and Lux.jl for all of the GPU-powered machine learning layers available from those libraries.
Compatibility with NeuralOperators.jl 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

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:

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)

Citation

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