Hi,

I’m moving my first steps with NeuralPDE.jl and I came up with this issue. I want to solve a fairly simple PDE:

```
h(ω) = exp.(-(γ-1)*ω)
μω = λ*(ω-ωb)
σω = σ
eq = 1/100*Dτ(f(τ,ω)) ~ 0.5*σω^2*Dωω(f(τ,ω)) - μω*Dω(f(τ,ω)) - k*f(τ,ω) + h(ω)
```

where \omega\in[-2, 2] and \tau\in[0,1]. I managed to get a decent solution (just by visual comparison with the finite difference one, see figure below for a cross section at time \tau=1) by using GridTraining:

On the other hand, if I use QuadratureTraining, with say CubatureJLh (one of the suggested algos) I get a small loss function but the solution is nowhere near the real one (same cross section as before):

I tried different cubature methods and played around with some parameters with no luck.

Am I doing something obviously wrong? Is the problem particularly not suited to cubature? Any insight would be appreciated

Here’s some of the code I use to solve the problem if it might help:

```
τ_min= 0.
τ_max = 1.0
τ_span = (τ_min,τ_max)
ω_min = -2.
ω_max = 2.
dω = 0.05
dt_NN = 0.01
dt_BSDE = 0.01
μY = 0.03
σ = 0.03
λ = 0.1
ωb = μY/λ
γ = 1.5
k = 0.1
h(ω) = exp.(-(γ-1)*ω)
@parameters τ ω
@variables f(..)
Dω = Differential(ω)
Dωω = Differential(ω)^2
Dτ = Differential(τ)
μω = λ*(ω-ωb)
σω = σ
# PDE
eq = 1/100*Dτ(f(τ,ω)) ~ 0.5*σω^2*Dωω(f(τ,ω)) - μω*Dω(f(τ,ω)) - k*f(τ,ω) + h(ω)
# Initial and boundary conditions
bcs = [f(τ_min,ω) ~ h(ω),
Dω(f(τ,ω_max)) ~ 0,
Dω(f(τ,ω_min)) ~ 0]
# Space and time domains
domains = [τ ∈ Interval(τ_min,τ_max),
ω ∈ Interval(ω_min,ω_max)]
τs,ωs = [infimum(d.domain):dω:supremum(d.domain) for d in domains]
@named pde_system = PDESystem(eq,bcs,domains,[τ,ω],[f(τ, ω)])
# NN parameters
dim =2
hls = dim+50
# Neural network
chain = Chain(Dense(dim,hls,Flux.σ),
Dense(hls,hls,Flux.σ),
Dense(hls,hls,Flux.σ),
Dense(hls,1))
strategy = GridTraining([dω,dt_NN])
# strategy = QuadratureTraining(; quadrature_alg = CubatureJLp(),
# reltol = 1e-6, abstol = 1e-3,
# maxiters = 1_000)
discretization = PhysicsInformedNN(chain,
strategy)
prob = discretize(pde_system,discretization)
callback = function (p,l)
println("Current loss is: $l")
return false
end
res = Optimization.solve(prob,Flux.ADAM(0.01);callback = callback,maxiters=500)
prob = remake(prob,u0=res.u)
res = Optimization.solve(prob,Flux.ADAM(0.001);callback = callback,maxiters=2000)
prob = remake(prob,u0=res.u)
bfgs = OptimizationOptimJL.BFGS()
res = Optimization.solve(prob,bfgs ;callback = callback,maxiters=100)
prob = remake(prob,u0=res.u)
phi = discretization.phi
f_NN = [first(Array(phi([τ, ω], res.u))) for τ in τs for ω in ωs]
f_NN = reshape(f_NN,(length(ωs),length(τs)))
```