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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,7 +1,7 @@ | ||
| name = "NeuralPDE" | ||
| uuid = "315f7962-48a3-4962-8226-d0f33b1235f0" | ||
| authors = ["Chris Rackauckas <[email protected]>"] | ||
| version = "5.13.0" | ||
|
|
||
| [deps] | ||
| Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" | ||
|
|
@@ -34,25 +34,24 @@ Reexport = "189a3867-3050-52da-a836-e630ba90ab69" | |
| RuntimeGeneratedFunctions = "7e49a35a-f44a-4d26-94aa-eba1b4ca6b47" | ||
| SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462" | ||
| Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" | ||
| SymbolicUtils = "d1185830-fcd6-423d-90d6-eec64667417b" | ||
| Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" | ||
| UnPack = "3a884ed6-31ef-47d7-9d2a-63182c4928ed" | ||
| Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" | ||
|
|
||
| [compat] | ||
| Adapt = "4" | ||
| AdvancedHMC = "0.6" | ||
| Aqua = "0.8" | ||
| ArrayInterface = "7" | ||
| CUDA = "5.1" | ||
| ChainRulesCore = "1" | ||
| ComponentArrays = "0.15" | ||
| Cubature = "1.5" | ||
| DiffEqBase = "6" | ||
| DiffEqNoiseProcess = "5.1" | ||
| Distributions = "0.25" | ||
| DocStringExtensions = "0.9" | ||
| DomainSets = "0.6, 0.7" | ||
| Flux = "0.14" | ||
| ForwardDiff = "0.10" | ||
|
|
@@ -61,15 +60,15 @@ Integrals = "4" | |
| LineSearches = "7.2" | ||
| LinearAlgebra = "1" | ||
| LogDensityProblems = "2" | ||
| Lux = "0.5" | ||
| LuxCUDA = "0.3" | ||
| MCMCChains = "6" | ||
| ModelingToolkit = "8" | ||
| MonteCarloMeasurements = "1" | ||
| Optim = "1.7.8" | ||
| Optimization = "3" | ||
| OptimizationOptimJL = "0.2" | ||
| OptimizationOptimisers = "0.2" | ||
| OrdinaryDiffEq = "6" | ||
| Pkg = "1" | ||
| QuasiMonteCarlo = "0.3.2" | ||
|
|
@@ -79,12 +78,12 @@ RuntimeGeneratedFunctions = "0.5" | |
| SafeTestsets = "0.1" | ||
| SciMLBase = "2" | ||
| Statistics = "1" | ||
| SymbolicUtils = "1" | ||
| Symbolics = "5" | ||
| Test = "1" | ||
| UnPack = "1" | ||
| Zygote = "0.6" | ||
| MethodOfLines = "0.10.7" | ||
| julia = "1.6" | ||
|
|
||
| [extras] | ||
|
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@@ -98,6 +97,7 @@ OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" | |
| Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" | ||
| SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f" | ||
| Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40" | ||
| MethodOfLines = "94925ecb-adb7-4558-8ed8-f975c56a0bf4" | ||
|
|
||
| [targets] | ||
| test = ["Aqua", "Test", "CUDA", "SafeTestsets", "OptimizationOptimJL", "Pkg", "OrdinaryDiffEq", "LineSearches", "LuxCUDA", "Flux", "MethodOfLines"] | ||
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,117 @@ | ||
| ## Solving PDEs using Deep Galerkin Method | ||
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| ### Overview | ||
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| Deep Galerkin Method is a meshless deep learning algorithm to solve high dimensional PDEs. The algorithm does so by approximating the solution of a PDE with a neural network. The loss function of the network is defined in the similar spirit as PINNs, composed of PDE loss and boundary condition loss. | ||
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| In the following example, we demonstrate computing the loss function using Quasi-Random Sampling, a sampling technique that uses quasi-Monte Carlo sampling to generate low discrepancy random sequences in high dimensional spaces. | ||
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| ### Algorithm | ||
| The authors of DGM suggest a network composed of LSTM-type layers that works well for most of the parabolic and quasi-parabolic PDEs. | ||
|
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| ```math | ||
| \begin{align*} | ||
| S^1 &= \sigma_1(W^1 \vec{x} + b^1); \\ | ||
| Z^l &= \sigma_1(U^{z,l} \vec{x} + W^{z,l} S^l + b^{z,l}); \quad l = 1, \ldots, L; \\ | ||
| G^l &= \sigma_1(U^{g,l} \vec{x} + W^{g,l} S_l + b^{g,l}); \quad l = 1, \ldots, L; \\ | ||
| R^l &= \sigma_1(U^{r,l} \vec{x} + W^{r,l} S^l + b^{r,l}); \quad l = 1, \ldots, L; \\ | ||
| H^l &= \sigma_2(U^{h,l} \vec{x} + W^{h,l}(S^l \cdot R^l) + b^{h,l}); \quad l = 1, \ldots, L; \\ | ||
| S^{l+1} &= (1 - G^l) \cdot H^l + Z^l \cdot S^{l}; \quad l = 1, \ldots, L; \\ | ||
| f(t, x; \theta) &= \sigma_{out}(W S^{L+1} + b). | ||
| \end{align*} | ||
| ``` | ||
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| where $\vec{x}$ is the concatenated vector of $(t, x)$ and $L$ is the number of LSTM type layers in the network. | ||
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| ### Example | ||
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| Let's try to solve the following Burger's equation using Deep Galerkin Method for $\alpha = 0.05$ and compare our solution with the finite difference method: | ||
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| $$ | ||
| \partial_t u(t, x) + u(t, x) \partial_x u(t, x) - \alpha \partial_{xx} u(t, x) = 0 | ||
| $$ | ||
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| defined over | ||
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| $$ | ||
| t \in [0, 1], x \in [-1, 1] | ||
| $$ | ||
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| with boundary conditions | ||
| ```math | ||
| \begin{align*} | ||
| u(t, x) & = - sin(πx), \\ | ||
| u(t, -1) & = 0, \\ | ||
| u(t, 1) & = 0 | ||
| \end{align*} | ||
| ``` | ||
|
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| ### Copy- Pasteable code | ||
| ```@example dgm | ||
| using NeuralPDE | ||
| using ModelingToolkit, Optimization, OptimizationOptimisers | ||
| import Lux: tanh, identity | ||
| using Distributions | ||
| import ModelingToolkit: Interval, infimum, supremum | ||
| using MethodOfLines, OrdinaryDiffEq | ||
| @parameters x t | ||
| @variables u(..) | ||
| Dt = Differential(t) | ||
| Dx = Differential(x) | ||
| Dxx = Dx^2 | ||
| α = 0.05 | ||
| # Burger's equation | ||
| eq = Dt(u(t,x)) + u(t,x) * Dx(u(t,x)) - α * Dxx(u(t,x)) ~ 0 | ||
| # boundary conditions | ||
| bcs = [ | ||
| u(0.0, x) ~ - sin(π*x), | ||
| u(t, -1.0) ~ 0.0, | ||
| u(t, 1.0) ~ 0.0 | ||
| ] | ||
| domains = [t ∈ Interval(0.0, 1.0), x ∈ Interval(-1.0, 1.0)] | ||
| # MethodOfLines, for FD solution | ||
| dx = 0.01 | ||
| order = 2 | ||
| discretization = MOLFiniteDifference([x => dx], t, saveat = 0.01) | ||
| @named pde_system = PDESystem(eq, bcs, domains, [t, x], [u(t,x)]) | ||
| prob = discretize(pde_system, discretization) | ||
| sol= solve(prob, Tsit5()) | ||
| ts = sol[t] | ||
| xs = sol[x] | ||
| u_MOL = sol[u(t,x)] | ||
| # NeuralPDE, using Deep Galerkin Method | ||
| strategy = QuasiRandomTraining(256, minibatch= 32) | ||
| discretization = DeepGalerkin(2, 1, 50, 5, tanh, tanh, identity, strategy) | ||
| @named pde_system = PDESystem(eq, bcs, domains, [t, x], [u(t,x)]) | ||
| prob = discretize(pde_system, discretization) | ||
| global iter = 0 | ||
| callback = function (p, l) | ||
| global iter += 1 | ||
| if iter%20 == 0 | ||
| println("$iter => $l") | ||
| end | ||
| return false | ||
| end | ||
| res = Optimization.solve(prob, Adam(0.1); callback = callback, maxiters = 100) | ||
| prob = remake(prob, u0 = res.u) | ||
| res = Optimization.solve(prob, Adam(0.01); callback = callback, maxiters = 500) | ||
| phi = discretization.phi | ||
| u_predict= [first(phi([t, x], res.minimizer)) for t in ts, x in xs] | ||
| diff_u = abs.(u_predict .- u_MOL) | ||
| using Plots | ||
| p1 = plot(tgrid, xgrid, u_MOL', linetype = :contourf, title = "FD"); | ||
| p2 = plot(tgrid, xgrid, u_predict', linetype = :contourf, title = "predict"); | ||
| p3 = plot(tgrid, xgrid, diff_u', linetype = :contourf, title = "error"); | ||
| plot(p1, p2, p3) | ||
| ``` |
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