diff --git a/docs/Project.toml b/docs/Project.toml index 9d78f3c..490f6d1 100644 --- a/docs/Project.toml +++ b/docs/Project.toml @@ -1,20 +1,27 @@ [deps] Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" +LineSearches = "d3d80556-e9d4-5f37-9878-2ab0fcc64255" Lux = "b2108857-7c20-44ae-9111-449ecde12c47" ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" ModelingToolkitNeuralNets = "f162e290-f571-43a6-83d9-22ecc16da15f" ModelingToolkitStandardLibrary = "16a59e39-deab-5bd0-87e4-056b12336739" Optimization = "7f7a1694-90dd-40f0-9382-eb1efda571ba" +OptimizationOptimJL = "36348300-93cb-4f02-beb5-3c3902f8871e" OptimizationOptimisers = "42dfb2eb-d2b4-4451-abcd-913932933ac1" -OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" +OrdinaryDiffEqTsit5 = "b1df2697-797e-41e3-8120-5422d3b24e4a" +OrdinaryDiffEqVerner = "79d7bb75-1356-48c1-b8c0-6832512096c2" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462" +SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1" SciMLStructures = "53ae85a6-f571-4167-b2af-e1d143709226" StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3" Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" SymbolicIndexingInterface = "2efcf032-c050-4f8e-a9bb-153293bab1f5" +SymbolicRegression = "8254be44-1295-4e6a-a16d-46603ac705cb" +Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" -[sources.ModelingToolkitNeuralNets] -path = ".." +[sources] +ModelingToolkitNeuralNets = {path = ".."} [compat] Documenter = "1.3" @@ -22,9 +29,8 @@ Lux = "1" ModelingToolkit = "10" ModelingToolkitNeuralNets = "2" ModelingToolkitStandardLibrary = "2.7" -Optimization = "3.24, 4.0" -OptimizationOptimisers = "0.2.1, 0.3" -OrdinaryDiffEq = "6.74" +Optimization = "4.0" +OptimizationOptimisers = "0.3" Plots = "1" SciMLStructures = "1.1.0" StableRNGs = "1" diff --git a/docs/make.jl b/docs/make.jl index 5aa376a..cbbeb87 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -20,9 +20,8 @@ makedocs(; linkcheck = true, pages = [ "Home" => "index.md", - "Tutorials" => [ - "Friction Model" => "friction.md" - ], + "Tutorials" => ["NeuralNetworkBlock" => "nnblock.md" + "Friction Model" => "friction.md"], "API" => "api.md" ] ) diff --git a/docs/src/api.md b/docs/src/api.md index 59d9f00..efeecac 100644 --- a/docs/src/api.md +++ b/docs/src/api.md @@ -3,4 +3,5 @@ ```@docs NeuralNetworkBlock SymbolicNeuralNetwork +multi_layer_feed_forward ``` diff --git a/docs/src/assets/Manifest.toml b/docs/src/assets/Manifest.toml new file mode 100644 index 0000000..6e7c6b5 --- /dev/null +++ b/docs/src/assets/Manifest.toml @@ -0,0 +1,3221 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = "7f80aa3f7724b4b3be0b6625fda77ab5e59135de" + +[[deps.ADTypes]] +git-tree-sha1 = "be7ae030256b8ef14a441726c4c37766b90b93a3" +uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" +version = "1.15.0" +weakdeps = ["ChainRulesCore", "ConstructionBase", "EnzymeCore"] + + 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b/docs/src/assets/Project.toml @@ -0,0 +1,38 @@ +[deps] +Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" +LineSearches = "d3d80556-e9d4-5f37-9878-2ab0fcc64255" +Lux = "b2108857-7c20-44ae-9111-449ecde12c47" +ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" +ModelingToolkitNeuralNets = "f162e290-f571-43a6-83d9-22ecc16da15f" +ModelingToolkitStandardLibrary = "16a59e39-deab-5bd0-87e4-056b12336739" +Optimization = "7f7a1694-90dd-40f0-9382-eb1efda571ba" +OptimizationOptimJL = "36348300-93cb-4f02-beb5-3c3902f8871e" +OptimizationOptimisers = "42dfb2eb-d2b4-4451-abcd-913932933ac1" +OrdinaryDiffEqTsit5 = "b1df2697-797e-41e3-8120-5422d3b24e4a" +OrdinaryDiffEqVerner = "79d7bb75-1356-48c1-b8c0-6832512096c2" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462" +SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1" +SciMLStructures = "53ae85a6-f571-4167-b2af-e1d143709226" +StableRNGs = "860ef19b-820b-49d6-a774-d7a799459cd3" +Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" +SymbolicIndexingInterface = "2efcf032-c050-4f8e-a9bb-153293bab1f5" +SymbolicRegression = "8254be44-1295-4e6a-a16d-46603ac705cb" +Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f" + +[sources] +ModelingToolkitNeuralNets = {path = ".."} + +[compat] +Documenter = "1.3" +Lux = "1" +ModelingToolkit = "10" +ModelingToolkitNeuralNets = "2" +ModelingToolkitStandardLibrary = "2.7" +Optimization = "4.0" +OptimizationOptimisers = "0.3" +Plots = "1" +SciMLStructures = "1.1.0" +StableRNGs = "1" +Statistics = "1" +SymbolicIndexingInterface = "0.3.15" diff --git a/docs/src/friction.md b/docs/src/friction.md index 9ec5237..8cd5160 100644 --- a/docs/src/friction.md +++ b/docs/src/friction.md @@ -12,7 +12,7 @@ using ModelingToolkit import ModelingToolkit.t_nounits as t import ModelingToolkit.D_nounits as Dt using ModelingToolkitStandardLibrary.Blocks -using OrdinaryDiffEq +using OrdinaryDiffEqVerner using Optimization using OptimizationOptimisers: Adam using SciMLStructures diff --git a/docs/src/index.md b/docs/src/index.md index c14fff4..39c29e7 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -4,16 +4,23 @@ CurrentModule = ModelingToolkitNeuralNets # ModelingToolkitNeuralNets.jl -ModelingToolkitNeuralNets.jl is a package to create neural network components defined similar to MTKStandardLibrary components, to use them for solving Universal Differential Equations. It can be plugged to any part of the equations in an ODESystem using `RealInputArray` and `RealOutputArray` components which gives a lot of flexibility to add missing physics only to the relevant parts of the model. +[`ModelingToolkitNeuralNets`](https://github.com/SciML/ModelingToolkitNeuralNets.jl) is a package that allows one to embed neural networks +inside [`ModelingToolkit`](https://github.com/SciML/ModelingToolkit.jl) systems in order to formulate Universal Differential Equations. +The neural network is symbolically represented either as a block component or a callable parameter, so it can be added to any part of the equations in an `System` which gives a lot of flexibility for model discovery, such that we make changes only to the relevant parts of the model in order to +discover missing physics. ## Installation -To install ModelingToolkitNeuralNets.jl, use the Julia package manager: +To install [`ModelingToolkitNeuralNets`](https://github.com/SciML/ModelingToolkitNeuralNets.jl), use the Julia package manager: ```julia using Pkg Pkg.add("ModelingToolkitNeuralNets") ``` +or +```julia-repl +julia> ]add ModelingToolkitNeuralNets +``` ## Contributing diff --git a/docs/src/nnblock.md b/docs/src/nnblock.md new file mode 100644 index 0000000..9b1de65 --- /dev/null +++ b/docs/src/nnblock.md @@ -0,0 +1,302 @@ +# Component based Universal Differential Equations with ModelingToolkit + +[`ModelingToolkitNeuralNets`](https://github.com/SciML/ModelingToolkitNeuralNets.jl) provides 2 main interfaces for representing neural networks symbolically: + +* The [`NeuralNetworkBlock`](@ref), which represents the neural network as a block component +* The [`SymbolicNeuralNetwork`](@ref), which represents the neural network via callable parameters + +This tutorial will introduce the [`NeuralNetworkBlock`](@ref). This representation is useful in the context of hierarchical acausal component-based model. + +For such models we have a component representation that is converted to a a differential-algebraic equation (DAE) system, where the algebraic equations are given by the constraints and equalities between different component variables. +The process of going from the component representation to the full DAE system at the end is reffered to as [structural simplification](https://docs.sciml.ai/ModelingToolkit/stable/API/model_building/#System-simplification). +In order to formulate Universal Differential Equations (UDEs) in this context, we could operate eiter operate before the structural simplification step or after that, on the +resulting DAE system. We call these the component UDE formulation and the system UDE formulation. + +The advantage of the component UDE formulation is that it allows us to represent the model +discovery process in the block component formulation, allowing us to potentially target +deeply nested parts of a model for the model discovery process. +As such, this allows us to maximally reuse the existing information and also to have a result that can be expressed in the same manner as the original model. + +In the following we will explore a simple example with a thermal model built with components +from the [`ModelingToolkitStandardLibrary`](https://docs.sciml.ai/ModelingToolkitStandardLibrary/stable/). + +We will first start with a model that we generate synthetic data from and then we will build a simpler model that we will use for model discovery and try to recover the dynamics from the original model. + +Our model represents a pot (`HeatCapacitor`) that is heated. In the complete model we consider that we have a plate that delays the heating of the pot. +In the simple model we just consider that the heat source is directly connected to the pot. + +Let's start with the definitions of the 2 models + +```@example potplate +using ModelingToolkitNeuralNets, Lux +using ModelingToolkitStandardLibrary.Thermal +using ModelingToolkitStandardLibrary.Blocks +using ModelingToolkit +using ModelingToolkit: t_nounits as t, D_nounits as D +using OrdinaryDiffEqTsit5 +using Plots +using StableRNGs +using SciMLBase + +input_f(t) = (1+sin(0.005 * t^2))/2 + +@mtkmodel PotWithPlate begin + @parameters begin + C1 = 1 + C2 = 15 + end + @components begin + input = Blocks.TimeVaryingFunction(f=input_f) + source = PrescribedHeatFlow(T_ref=373.15) + plate = HeatCapacitor(C=C1, T=273.15) + pot = HeatCapacitor(C=C2, T=273.15) + conduction = ThermalConductor(G=1) + air = ThermalConductor(G=0.1) + env = FixedTemperature(T=293.15) + Tsensor = TemperatureSensor() + end + @equations begin + connect(input.output, :u, source.Q_flow) + connect(source.port, plate.port) + connect(plate.port, conduction.port_a) + connect(conduction.port_b, pot.port) + connect(pot.port, air.port_a) + connect(air.port_b, env.port) + connect(pot.port, Tsensor.port) + end +end +@mtkmodel SimplePot begin + @parameters begin + C2 = 15 + end + @components begin + input = Blocks.TimeVaryingFunction(f=input_f) + source = PrescribedHeatFlow(T_ref=373.15) + pot = HeatCapacitor(C=C2, T=273.15) + air = ThermalConductor(G=0.1) + env = FixedTemperature(T=293.15) + Tsensor = TemperatureSensor() + end + @equations begin + connect(input.output, :u, source.Q_flow) + connect(source.port, pot.port) + connect(pot.port, Tsensor.port) + connect(pot.port, air.port_a) + connect(air.port_b, env.port) + end +end +@mtkcompile sys1 = PotWithPlate() +@mtkcompile sys2 = SimplePot() + +## solve and plot the temperature of the pot in the 2 systems + +prob1 = ODEProblem(sys1, Pair[], (0, 100.0)) +sol1 = solve(prob1, Tsit5(), reltol=1e-6) +prob2 = ODEProblem(sys2, Pair[], (0, 100.0)) +sol2 = solve(prob2, Tsit5(), reltol=1e-6) +plot(sol1, idxs=sys1.pot.T, label="pot.T in original system") +plot!(sol2, idxs=sys1.pot.T, label="pot.T in simplified system") +``` + +If we take a closer look at the 2 models, the original system has 2 unknowns, + +```@example potplate +unknowns(sys1) +``` + +while the simplified system only has 1 unknown +```@example potplate +unknowns(sys2) +``` + +With the component UDE approach, we want to add a new component in the model that would +be connected between the source and the pot such that we can use it to discover the missing physics. +To this end, we will build a `ThermalNN` component that adds a new state in the system (`x`) +that is prescribed by the output of the neural network, but it also incorporates physics knoweledge +about the system by formulating the component as something that outputs a heat flow rate based +on some scaled input temperatures. + +The system in this example is quite simple, so we will use a very small neural network with 1 hidden layer +of size 4. For more complex dynamics, a larger network would be required. +The inputs to out neural network will use scaled temperature values, such that we don't input numbers +that would be too large for the chosen activation function. The chosen activation function will +always output positive numbers for positive inputs, so this also makes physical sense for out component. + +```@example potplate +@mtkmodel ThermalNN begin + begin + n_input = 2 + n_output = 1 + chain = multi_layer_feed_forward(; n_input, n_output, depth=1, width=4, activation=Lux.swish) + end + @components begin + port_a = HeatPort() + port_b = HeatPort() + nn = NeuralNetworkBlock(; n_input, n_output, chain, rng=StableRNG(1337)) + end + @parameters begin + T0 = 273.15 + T_range = 10 + C1 = 1 + end + @variables begin + dT(t), [guess = 0.0] + Q_flow(t), [guess = 0.0] + x(t) = T0 + end + @equations begin + dT ~ port_a.T - port_b.T + port_a.Q_flow ~ Q_flow + C1*D(x) ~ Q_flow - nn.outputs[1] + port_a.T ~ x + nn.outputs[1] + port_b.Q_flow ~ 0 + nn.inputs[1] ~ (x - T0) / T_range + nn.inputs[2] ~ (port_b.T - T0) / T_range + end +end + +@mtkmodel NeuralPot begin + @parameters begin + C2 = 15 + end + @components begin + input = Blocks.TimeVaryingFunction(f=input_f) + source = PrescribedHeatFlow(T_ref=373.15) + pot = HeatCapacitor(C=C2, T=273.15) + air = ThermalConductor(G=0.1) + env = FixedTemperature(T=293.15) + Tsensor = TemperatureSensor() + thermal_nn = ThermalNN() + end + @equations begin + connect(input.output, :u, source.Q_flow) + connect(pot.port, Tsensor.port) + connect(pot.port, air.port_a) + connect(air.port_b, env.port) + connect(source.port, thermal_nn.port_a) + connect(thermal_nn.port_b, pot.port) + end +end + +@named model = NeuralPot() +sys3 = mtkcompile(model) + +# Let's check that we can succesfully simulate the system in the +# initial state +prob3 = ODEProblem(sys3, Pair[], (0, 100.0)) +sol3 = solve(prob3, Tsit5(), abstol=1e-6, reltol=1e-6) +@assert SciMLBase.successful_retcode(sol3) +``` + +Now that we have the system with the embedded neural network, we can start training the network. +The training will be formulated as an optimization problem where we will minimize the mean absolute squared distance +between the predictions of the new system and the data obtained from the original system. +In order to gain some insight into the training process we will also add a callback that will plot various quantities +in the system versus their equivalents in the original system. In a more realistic scenarion we would not have access +to the original system, but we could still monitor how well we fit the traning data and the system predictions. + +```@example potplate +using SymbolicIndexingInterface +using Optimization +using OptimizationOptimJL +using LineSearches +using Statistics +using SciMLSensitivity +import Zygote + +tp = Symbolics.scalarize(sys3.thermal_nn.nn.p) +x0 = prob3.ps[tp] + +oop_update = setsym_oop(prob3, tp); + +plot_cb = (opt_state, loss) -> begin + opt_state.iter % 1000 ≠ 0 && return false + @info "step $(opt_state.iter), loss: $loss" + + (new_u0, new_p) = oop_update(prob3, opt_state.u) + new_prob = remake(prob3, u0=new_u0, p=new_p) + sol = solve(new_prob, Tsit5(), abstol=1e-8, reltol=1e-8) + + plt = plot(sol, layout=(2,3), idxs=[ + sys3.thermal_nn.nn.inputs[1], sys3.thermal_nn.x, + sys3.thermal_nn.nn.outputs[1], sys3.thermal_nn.port_b.T, + sys3.pot.T, sys3.pot.port.Q_flow], + size=(950,800)) + plot!(plt, sol1, idxs=[ + (sys1.conduction.port_a.T-273.15)/10, sys1.conduction.port_a.T, + sys1.conduction.port_a.Q_flow, sys1.conduction.port_b.T, + sys1.pot.T, sys1.pot.port.Q_flow]) + display(plt) + false +end + +function cost(x, opt_ps) + prob, oop_update, data, ts, get_T = opt_ps + + u0, p = oop_update(prob, x) + new_prob = remake(prob; u0, p) + + new_sol = solve(new_prob, Tsit5(), saveat=ts, abstol=1e-8, reltol=1e-8, verbose=false, sensealg=GaussAdjoint()) + + !SciMLBase.successful_retcode(new_sol) && return Inf + + mean(abs2.(get_T(new_sol) .- data)) +end + +of = OptimizationFunction(cost, AutoForwardDiff()) + +data = sol1[sys1.pot.T] +get_T = getsym(prob3, sys3.pot.T) +opt_ps = (prob3, oop_update, data, sol1.t, get_T); + +op = OptimizationProblem(of, x0, opt_ps,) + +res = solve(op, Adam(); maxiters=10_000, callback=plot_cb) +op2 = OptimizationProblem(of, res.u, opt_ps) +res2 = solve(op2, LBFGS(linesearch=BackTracking()); maxiters=2000, callback=plot_cb) + +(new_u0, new_p) = oop_update(prob3, res2.u) +new_prob1 = remake(prob3, u0=new_u0, p=new_p) +new_sol1 = solve(new_prob1, Tsit5(), abstol=1e-6, reltol=1e-6) + +plt = plot(new_sol1, layout=(2,3), idxs=[ + sys3.thermal_nn.nn.inputs[1], sys3.thermal_nn.x, + sys3.thermal_nn.nn.outputs[1], sys3.thermal_nn.port_b.T, + sys3.pot.T, sys3.pot.port.Q_flow], + size=(950,800)) +plot!(plt, sol1, idxs=[ + (sys1.conduction.port_a.T-273.15)/10, sys1.conduction.port_a.T, + sys1.conduction.port_a.Q_flow, sys1.conduction.port_b.T, + sys1.pot.T, sys1.pot.port.Q_flow], ls=:dash) +``` + +As we can see from the final plot, the neural network fits very well and not only the training data fits, but also the rest of the +predictions of the system match the original system. Let us also compare against the predictions of the incomplete system: + +```@example potplate +plot(sol1, label=["original sys: pot T" "original sys: plate T"], lw=3) +plot!(sol3; idxs=[sys3.pot.T], label="untrained UDE", lw=2.5) +plot!(sol2; idxs=[sys2.pot.T], label="incomplete sys: pot T", lw=2.5) +plot!(new_sol1; idxs=[sys3.pot.T, sys3.thermal_nn.x], label="trained UDE", ls=:dash, lw=2.5) +``` + +Now that our neural network is trained, we can go a step further and use [`SymbolicRegression.jl`](https://github.com/MilesCranmer/SymbolicRegression.jl) to find +a symbolic expression for the function represented by the neural network. + +```@example potplate +using SymbolicRegression + +lux_model = new_sol1.ps[sys3.thermal_nn.nn.lux_model] +nn_p = new_sol1.ps[sys3.thermal_nn.nn.p] +T = new_sol1.ps[sys3.thermal_nn.nn.T] + +sr_input = reduce(hcat, new_sol1[sys3.thermal_nn.nn.inputs]) +sr_output = LuxCore.stateless_apply(lux_model, sr_input, convert(T, nn_p)) + +equation_search(sr_input, sr_output) +``` + +Looking at the results above, we see that we have a term linear in the difference between its inputs, `x₁ - x₂`, which were +the scaled temperature values. This also makes sense physically, as the heat flow rate through the missing components in our +system is proportional with the temperature difference between the ports. +Having the symbolic expression for the UDE results thus helps us understand the missing physics better. diff --git a/src/utils.jl b/src/utils.jl index a576ea6..6f35d12 100644 --- a/src/utils.jl +++ b/src/utils.jl @@ -1,6 +1,22 @@ -function multi_layer_feed_forward(input_length, output_length; width::Int = 5, - depth::Int = 1, activation = tanh) - Lux.Chain(Lux.Dense(input_length, width, activation), - [Lux.Dense(width, width, activation) for _ in 1:(depth)]..., - Lux.Dense(width, output_length)) +""" + multi_layer_feed_forward(; n_input, n_output, width::Int = 4, + depth::Int = 1, activation = tanh, use_bias = true, initial_scaling_factor = 1e-8) + +Create a Lux.jl `Chain` for use in [`NeuralNetworkBlock`](@ref)s. The weights of the last layer +are multipled by the `initial_scaling_factor` in order to make the initial contribution +of the network small and thus help with acheiving a stable starting position for the training. +""" +function multi_layer_feed_forward(; n_input, n_output, width::Int = 4, + depth::Int = 1, activation = tanh, use_bias = true, initial_scaling_factor = 1e-8) + Lux.Chain( + Lux.Dense(n_input, width, activation; use_bias), + [Lux.Dense(width, width, activation; use_bias) for _ in 1:(depth)]..., + Lux.Dense(width, n_output; + init_weight = (rng, a...) -> initial_scaling_factor * + Lux.kaiming_uniform(rng, a...), use_bias) + ) +end + +function multi_layer_feed_forward(n_input, n_output; kwargs...) + multi_layer_feed_forward(; n_input, n_output, kwargs...) end