求问这里积分怎么搞？
using Flux
println("Integro Differential equation tests")
using DiffEqFlux
println("Starting Soon!")
using ModelingToolkit
using DiffEqBase
using Test, NeuralPDE
println("Starting Soon!")
using GalacticOptim
using Optim
using Quadrature,Cubature, Cuba
using QuasiMonteCarlo
using SciMLBase
import ModelingToolkit: Interval, infimum, supremum
using DomainSets, Plots
using Random
Random.seed!(100)
cb = function (p,l)
println("Current loss is: $l")
return false
end
#simple multidimensitonal integral test
println("simple multidimensitonal integral test")
@parameters x,y
@variables u(..)
Dx = Differential(x)
Dy = Differential(y)
Ix = Integral((x,y) in DomainSets.UnitSquare())
eq = Ix(u(x,y)) ~ 1/3
bcs = [u(0., 0.) ~ 1, Dx(u(x,y)) ~ -2*x , Dy(u(x ,y)) ~ -2*y ]
domains = [x ∈ Interval(0.0,1.00), y ∈ Interval(0.0,1.00)]
chain = Chain(Dense(2,15,Flux.σ),Dense(15,1))
initθ = Float64.(DiffEqFlux.initial_params(chain))
strategy_ = NeuralPDE.GridTraining(0.1)
discretization = NeuralPDE.PhysicsInformedNN(chain,
strategy_;
init_params = nothing,
phi = nothing,
derivative = nothing,
)
@named pde_system = PDESystem(eq,bcs,domains,[x,y],[u(x, y)])
prob = NeuralPDE.discretize(pde_system,discretization)
res = GalacticOptim.solve(prob, BFGS(); cb = cb, maxiters=100)
phi = discretization.phi
xs = 0.00:0.01:1.00
ys = 0.00:0.01:1.00
u_real = collect(1 - x^2 - y^2 for y in ys, x in xs);
u_predict = collect(Array(phi([x,y], res.minimizer))[1] for y in ys, x in xs);
@test Flux.mse(u_real, u_predict) < 0.001
error_ = u_predict .- u_real
p1 = plot(xs,ys,u_real,linetype=:contourf,label = "analytic")
p2 = plot(xs,ys,u_predict,linetype=:contourf,label = "predict")
p3 = plot(xs,ys,error_,linetype=:contourf,label = "error")
plot(p1,p2,p3)
如何搞出来对y 的积分 比如∫ u dy