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Fix differential equation tutorial not converging #592

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Fix differential equation tutorial not converging #592

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50 changes: 25 additions & 25 deletions tutorials/bayesian-differential-equations/index.qmd
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Original file line number Diff line number Diff line change
Expand Up @@ -79,7 +79,7 @@ To make the example more realistic we add random normally distributed noise to t

```{julia}
sol = solve(prob, Tsit5(); saveat=0.1)
odedata = Array(sol) + 0.8 * randn(size(Array(sol)))
odedata = Array(sol) + 0.5 * randn(size(Array(sol)))

# Plot simulation and noisy observations.
plot(sol; alpha=0.3)
Expand All @@ -98,18 +98,18 @@ Therefore, we write the Lotka-Volterra parameter estimation problem using the Tu
```{julia}
@model function fitlv(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.2, 0.5); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.5); lower=0, upper=2)
σ ~ InverseGamma(3, 2)
α ~ truncated(Normal(1.5, 0.2); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.1, 0.2); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.2); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.2); lower=0, upper=2)

# Simulate Lotka-Volterra model.
p = [α, β, γ, δ]
predicted = solve(prob, Tsit5(); p=p, saveat=0.1)

# Observations.
for i in 1:length(predicted)
for i in eachindex(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end

Expand Down Expand Up @@ -160,11 +160,11 @@ I.e., we fit the model only to the $y$ variable of the system without providing
```{julia}
@model function fitlv2(data::AbstractVector, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.2, 0.5); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.5); lower=0, upper=2)
σ ~ InverseGamma(3, 2)
α ~ truncated(Normal(1.5, 0.2); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.1, 0.2); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.2); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.2); lower=0, upper=2)

# Simulate Lotka-Volterra model but save only the second state of the system (predators).
p = [α, β, γ, δ]
Expand Down Expand Up @@ -260,18 +260,18 @@ Now we define the Turing model for the Lotka-Volterra model with delay and sampl
```{julia}
@model function fitlv_dde(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.2, 0.5); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.5); lower=0, upper=2)
σ ~ InverseGamma(3, 2)
α ~ truncated(Normal(1.5, 0.2); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.1, 0.2); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.2); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.2); lower=0, upper=2)

# Simulate Lotka-Volterra model.
p = [α, β, γ, δ]
predicted = solve(prob, MethodOfSteps(Tsit5()); p=p, saveat=0.1)

# Observations.
for i in 1:length(predicted)
for i in eachindex(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end
end
Expand Down Expand Up @@ -340,18 +340,18 @@ Here we will not choose a `sensealg` and let it use the default choice:
```{julia}
@model function fitlv_sensealg(data, prob)
# Prior distributions.
σ ~ InverseGamma(2, 3)
α ~ truncated(Normal(1.5, 0.5); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.2, 0.5); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.5); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.5); lower=0, upper=2)
σ ~ InverseGamma(3, 2)
α ~ truncated(Normal(1.5, 0.2); lower=0.5, upper=2.5)
β ~ truncated(Normal(1.1, 0.2); lower=0, upper=2)
γ ~ truncated(Normal(3.0, 0.2); lower=1, upper=4)
δ ~ truncated(Normal(1.0, 0.2); lower=0, upper=2)

# Simulate Lotka-Volterra model and use a specific algorithm for computing sensitivities.
p = [α, β, γ, δ]
predicted = solve(prob; p=p, saveat=0.1)

# Observations.
for i in 1:length(predicted)
for i in eachindex(predicted)
data[:, i] ~ MvNormal(predicted[i], σ^2 * I)
end

Expand All @@ -361,7 +361,7 @@ end;
model_sensealg = fitlv_sensealg(odedata, prob)

# Sample a single chain with 1000 samples using Zygote.
sample(model_sensealg, NUTS(;adtype=AutoZygote()), 1000; progress=false)
sample(model_sensealg, NUTS(; adtype=AutoZygote()), 1000; progress=false)
```

For more examples of adjoint usage on large parameter models, consult the [DiffEqFlux documentation](https://diffeqflux.sciml.ai/dev/).