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| Original file line number | Diff line number | Diff line change |
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| # Using `ahmc_bayesian_pinn_pde` with the `BayesianPINN` Discretizer for the 1-D Burgers' Equation | ||
| # Using `ahmc_bayesian_pinn_pde` with the `BayesianPINN` Discretizer for the Kuramoto–Sivashinsky equation | ||
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| Let's consider the Burgers' equation: | ||
| Consider the Kuramoto–Sivashinsky equation: | ||
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| ```math | ||
| \begin{gather*} | ||
| ∂_t u + u ∂_x u - (0.01 / \pi) ∂_x^2 u = 0 \, , \quad x \in [-1, 1], t \in [0, 1] \, , \\ | ||
| u(0, x) = - \sin(\pi x) \, , \\ | ||
| u(t, -1) = u(t, 1) = 0 \, , | ||
| \end{gather*} | ||
| ∂_t u(x, t) + u(x, t) ∂_x u(x, t) + \alpha ∂^2_x u(x, t) + \beta ∂^3_x u(x, t) + \gamma ∂^4_x u(x, t) = 0 \, , | ||
| ``` | ||
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| with Bayesian Physics-Informed Neural Networks. Here is an example of using `BayesianPINN` discretization with `ahmc_bayesian_pinn_pde` : | ||
| where $\alpha = \gamma = 1$ and $\beta = 4$. The exact solution is: | ||
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| ```math | ||
| u_e(x, t) = 11 + 15 \tanh \theta - 15 \tanh^2 \theta - 15 \tanh^3 \theta \, , | ||
| ``` | ||
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| where $\theta = t - x/2$ and with initial and boundary conditions: | ||
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| ```math | ||
| \begin{align*} | ||
| u( x, 0) &= u_e( x, 0) \, ,\\ | ||
| u( 10, t) &= u_e( 10, t) \, ,\\ | ||
| u(-10, t) &= u_e(-10, t) \, ,\\ | ||
| ∂_x u( 10, t) &= ∂_x u_e( 10, t) \, ,\\ | ||
| ∂_x u(-10, t) &= ∂_x u_e(-10, t) \, . | ||
| \end{align*} | ||
| ``` | ||
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| With Bayesian Physics-Informed Neural Networks, here is an example of using `BayesianPINN` discretization with `ahmc_bayesian_pinn_pde` : | ||
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| ```@example low_level_2 | ||
| using NeuralPDE, Lux, ModelingToolkit | ||
| import ModelingToolkit: Interval, infimum, supremum | ||
| using NeuralPDE, Flux, Lux, ModelingToolkit, LinearAlgebra, AdvancedHMC | ||
| import ModelingToolkit: Interval, infimum, supremum, Distributions | ||
| using Plots, MonteCarloMeasurements | ||
| @parameters t, x | ||
| @parameters x, t, α | ||
| @variables u(..) | ||
| Dt = Differential(t) | ||
| Dx = Differential(x) | ||
| Dxx = Differential(x)^2 | ||
| Dx2 = Differential(x)^2 | ||
| Dx3 = Differential(x)^3 | ||
| Dx4 = Differential(x)^4 | ||
| # α = 1 | ||
| β = 4 | ||
| γ = 1 | ||
| eq = Dt(u(x, t)) + u(x, t) * Dx(u(x, t)) + α * Dx2(u(x, t)) + β * Dx3(u(x, t)) + γ * Dx4(u(x, t)) ~ 0 | ||
| #2D PDE | ||
| eq = Dt(u(t, x)) + u(t, x) * Dx(u(t, x)) - (0.01 / pi) * Dxx(u(t, x)) ~ 0 | ||
| u_analytic(x, t; z = -x / 2 + t) = 11 + 15 * tanh(z) - 15 * tanh(z)^2 - 15 * tanh(z)^3 | ||
| du(x, t; z = -x / 2 + t) = 15 / 2 * (tanh(z) + 1) * (3 * tanh(z) - 1) * sech(z)^2 | ||
| # Initial and boundary conditions | ||
| bcs = [u(0, x) ~ -sin(pi * x), | ||
| u(t, -1) ~ 0.0, | ||
| u(t, 1) ~ 0.0, | ||
| u(t, -1) ~ u(t, 1)] | ||
| bcs = [u(x, 0) ~ u_analytic(x, 0), | ||
| u(-10, t) ~ u_analytic(-10, t), | ||
| u(10, t) ~ u_analytic(10, t), | ||
| Dx(u(-10, t)) ~ du(-10, t), | ||
| Dx(u(10, t)) ~ du(10, t)] | ||
| # Space and time domains | ||
| domains = [t ∈ Interval(0.0, 1.0), | ||
| x ∈ Interval(-1.0, 1.0)] | ||
| domains = [x ∈ Interval(-10.0, 10.0), | ||
| t ∈ Interval(0.0, 1.0)] | ||
| # Discretization | ||
| dx = 0.05 | ||
| dx = 0.4; | ||
| dt = 0.2; | ||
| # Function to compute analytical solution at a specific point (x, t) | ||
| function u_analytic_point(x, t) | ||
| z = -x / 2 + t | ||
| return 11 + 15 * tanh(z) - 15 * tanh(z)^2 - 15 * tanh(z)^3 | ||
| end | ||
| # Function to generate the dataset matrix | ||
| function generate_dataset_matrix(domains, dx, dt) | ||
| x_values = -10:dx:10 | ||
| t_values = 0.0:dt:1.0 | ||
| dataset = [] | ||
| for t in t_values | ||
| for x in x_values | ||
| u_value = u_analytic_point(x, t) | ||
| push!(dataset, [u_value, x, t]) | ||
| end | ||
| end | ||
| return vcat([data' for data in dataset]...) | ||
| end | ||
| datasetpde = [generate_dataset_matrix(domains, dx, dt)] | ||
| plot(datasetpde[1][:, 2], datasetpde[1][:, 1], title="Dataset from Analytical Solution") | ||
| # Add noise to dataset | ||
| datasetpde[1][:, 1] = datasetpde[1][:, 1] .+ randn(size(datasetpde[1][:, 1])) .* 5 / 100 .* | ||
| datasetpde[1][:, 1] | ||
| plot!(datasetpde[1][:, 2], datasetpde[1][:, 1]) | ||
| # Neural network | ||
| chain = Lux.Chain(Lux.Dense(2, 10, Lux.σ), Lux.Dense(10, 10, Lux.σ), Lux.Dense(10, 1)) | ||
| strategy = NeuralPDE.GridTraining([dx,dx]) | ||
| chain = Lux.Chain(Lux.Dense(2, 8, Lux.tanh), | ||
| Lux.Dense(8, 8, Lux.tanh), | ||
| Lux.Dense(8, 1)) | ||
| discretization = NeuralPDE.BayesianPINN([chain], strategy) | ||
| discretization = NeuralPDE.BayesianPINN([chain], | ||
| GridTraining([dx, dt]), param_estim = true, dataset = [datasetpde, nothing]) | ||
| @named pde_system = PDESystem(eq, bcs, domains, [x, t], [u(x, t)]) | ||
| @named pde_system = PDESystem(eq, | ||
| bcs, | ||
| domains, | ||
| [x, t], | ||
| [u(x, t)], | ||
| [α], | ||
| defaults = Dict([α => 0.5])) | ||
| sol1 = ahmc_bayesian_pinn_pde(pde_system, | ||
| discretization; | ||
| draw_samples = 100, | ||
| bcstd = [0.01, 0.03, 0.03, 0.01], | ||
| phystd = [0.01], | ||
| draw_samples = 100, Kernel = AdvancedHMC.NUTS(0.8), | ||
| bcstd = [0.2, 0.2, 0.2, 0.2, 0.2], | ||
| phystd = [1.0], l2std = [0.05], param = [Distributions.LogNormal(0.5, 2)], | ||
| priorsNNw = (0.0, 10.0), | ||
| saveats = [1 / 100.0, 1 / 100.0],progress=true) | ||
| saveats = [1 / 100.0, 1 / 100.0], progress = true) | ||
| ``` | ||
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| And some analysis: | ||
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| ```@example low_level | ||
| using Plots | ||
| ts, xs = [infimum(d.domain):0.01:supremum(d.domain) for d in domains] | ||
| u_predict_contourf = reshape([first(phi([t, x], res.u)) for t in ts for x in xs], | ||
| length(xs), length(ts)) | ||
| plot(ts, xs, u_predict_contourf, linetype = :contourf, title = "predict") | ||
| u_predict = [[first(phi([t, x], res.u)) for x in xs] for t in ts] | ||
| p1 = plot(xs, u_predict[3], title = "t = 0.1"); | ||
| p2 = plot(xs, u_predict[11], title = "t = 0.5"); | ||
| p3 = plot(xs, u_predict[end], title = "t = 1"); | ||
| ```@example low_level_2 | ||
| phi = discretization.phi[1] | ||
| xs, ts = [infimum(d.domain):dx:supremum(d.domain) for (d, dx) in zip(domains, [dx / 10, dt])] | ||
| u_predict = [[first(pmean(phi([x, t], sol1.estimated_nn_params[1]))) for x in xs] | ||
| for t in ts] | ||
| u_real = [[u_analytic(x, t) for x in xs] for t in ts] | ||
| diff_u = [[abs(u_analytic(x, t) - first(pmean(phi([x, t], sol1.estimated_nn_params[1])))) | ||
| for x in xs] | ||
| for t in ts] | ||
| p1 = plot(xs, u_predict, title = "predict") | ||
| p2 = plot(xs, u_real, title = "analytic") | ||
| p3 = plot(xs, diff_u, title = "error") | ||
| plot(p1, p2, p3) | ||
| ``` | ||
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