SimpleImplicitTauLeaping solver

Project.toml

@@ -4,6 +4,7 @@ authors = ["Chris Rackauckas <accounts@chrisrackauckas.com>"]
 version = "9.25.1"
 
 [deps]
+ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
 ArrayInterface = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
 DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 DiffEqBase = "2b5f629d-d688-5b77-993f-72d75c75574e"
@@ -17,6 +18,8 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RecursiveArrayTools = "731186ca-8d62-57ce-b412-fbd966d074cd"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462"
+Setfield = "efcf1570-3423-57d1-acb7-fd33fddbac46"
+SimpleNonlinearSolve = "727e6d20-b764-4bd8-a329-72de5adea6c7"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 SymbolicIndexingInterface = "2efcf032-c050-4f8e-a9bb-153293bab1f5"
 
@@ -54,6 +57,8 @@ RecursiveArrayTools = "3.35, 4"
 Reexport = "1.2"
 SafeTestsets = "0.1"
 SciMLBase = "2.115, 3.1"
+Setfield = "1"
+SimpleNonlinearSolve = "1, 2"
 StableRNGs = "1"
 StaticArrays = "1.9.8"
 Statistics = "1"
@@ -63,7 +68,6 @@ Test = "1"
 julia = "1.10"
 
 [extras]
-ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"
 Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
 ExplicitImports = "7d51a73a-1435-4ff3-83d9-f097790105c7"
 FastBroadcast = "7034ab61-46d4-4ed7-9d0f-46aef9175898"
@@ -78,4 +82,4 @@ StochasticDiffEq = "789caeaf-c7a9-5a7d-9973-96adeb23e2a0"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["ADTypes", "Aqua", "ExplicitImports", "FastBroadcast", "LinearAlgebra", "LinearSolve", "OrdinaryDiffEq", "Pkg", "SafeTestsets", "StableRNGs", "Statistics", "StochasticDiffEq", "Test"]
+test = ["Aqua", "ExplicitImports", "FastBroadcast", "LinearAlgebra", "LinearSolve", "OrdinaryDiffEq", "Pkg", "SafeTestsets", "StableRNGs", "Statistics", "StochasticDiffEq", "Test"]

src/JumpProcesses.jl

@@ -18,6 +18,9 @@ using StaticArrays: StaticArrays, SVector, setindex
 using Base.Threads: Threads
 using Base.FastMath: add_fast
 
+using SimpleNonlinearSolve: SimpleNonlinearSolve, SimpleNewtonRaphson
+using ADTypes: ADTypes, AutoFiniteDiff
+
 # Import functions we extend from Base
 import Base: size, getindex, setindex!, length, similar, show, merge!, merge
 
@@ -40,7 +43,7 @@ using DiffEqBase: DiffEqBase, CallbackSet, ContinuousCallback, DAEFunction,
                   ODESolution, ReturnCode, SDEFunction, SDEProblem, add_tstop!,
                   deleteat!, isinplace, remake, savevalues!, step!,
                   u_modified!
-using SciMLBase: SciMLBase, DEIntegrator
+using SciMLBase: SciMLBase, DEIntegrator, NonlinearProblem
 
 abstract type AbstractJump end
 abstract type AbstractMassActionJump <: AbstractJump end
@@ -131,7 +134,7 @@ export SSAStepper
 
 # leaping: 
 include("simple_regular_solve.jl")
-export SimpleTauLeaping, SimpleExplicitTauLeaping, EnsembleGPUKernel
+export SimpleTauLeaping, SimpleExplicitTauLeaping, SimpleImplicitTauLeaping, NewtonImplicitSolver, TrapezoidalImplicitSolver, EnsembleGPUKernel
 
 # spatial:
 include("spatial/spatial_massaction_jump.jl")

src/simple_regular_solve.jl

@@ -6,6 +6,18 @@ end
 
 SimpleExplicitTauLeaping(; epsilon = 0.05) = SimpleExplicitTauLeaping(epsilon)
 
+# Define solver type hierarchy
+abstract type AbstractImplicitSolver end
+struct NewtonImplicitSolver <: AbstractImplicitSolver end
+struct TrapezoidalImplicitSolver <: AbstractImplicitSolver end
+
+struct SimpleImplicitTauLeaping{T <: AbstractFloat} <: DiffEqBase.DEAlgorithm
+    epsilon::T  # Error control parameter for tau selection
+    solver::AbstractImplicitSolver  # Solver type: Newton or Trapezoidal
+end
+
+SimpleImplicitTauLeaping(; epsilon=0.05, solver=NewtonImplicitSolver()) = SimpleImplicitTauLeaping(epsilon, solver)
+
 function validate_pure_leaping_inputs(jump_prob::JumpProblem, alg)
     if !(jump_prob.aggregator isa PureLeaping)
         @warn "When using $alg, please pass PureLeaping() as the aggregator to the \
@@ -69,6 +81,19 @@ function _process_saveat(saveat, tspan, save_start, save_end)
     return saveat_vec, _save_start, _save_end
 end
 
+function validate_pure_leaping_inputs(jump_prob::JumpProblem, alg::SimpleImplicitTauLeaping)
+    if !(jump_prob.aggregator isa PureLeaping)
+        @warn "When using $alg, please pass PureLeaping() as the aggregator to the \
+        JumpProblem, i.e. call JumpProblem(::DiscreteProblem, PureLeaping(),...). \
+        Passing $(jump_prob.aggregator) is deprecated and will be removed in the next breaking release."
+    end
+    isempty(jump_prob.jump_callback.continuous_callbacks) &&
+    isempty(jump_prob.jump_callback.discrete_callbacks) &&
+    isempty(jump_prob.constant_jumps) &&
+    isempty(jump_prob.variable_jumps) &&
+    jump_prob.massaction_jump !== nothing
+end
+
 function DiffEqBase.solve(jump_prob::JumpProblem, alg::SimpleTauLeaping;
         seed = nothing, dt = error("dt is required for SimpleTauLeaping."),
         saveat = nothing, save_start = nothing, save_end = nothing)
@@ -405,6 +430,291 @@ function DiffEqBase.solve(jump_prob::JumpProblem, alg::SimpleExplicitTauLeaping;
     return sol
 end
 
+function compute_hor(reactant_stoch, numjumps)
+    # Compute the highest order of reaction (HOR) for each reaction j, as per Cao et al. (2006), Section IV.
+    # HOR is the sum of stoichiometric coefficients of reactants in reaction j.
+    # Extract the element type from reactant_stoch to avoid hardcoding type assumptions.
+    stoch_type = eltype(first(first(reactant_stoch)))
+    hor = zeros(stoch_type, numjumps)
+    max_order = 3 * one(stoch_type)  # Maximum supported reaction order (type-aware)
+    for j in 1:numjumps
+        order = sum(stoch for (spec_idx, stoch) in reactant_stoch[j]; init=zero(stoch_type))
+        if order > max_order
+            error("Reaction $j has order $order, which is not supported (maximum order is 3).")
+        end
+        hor[j] = order
+    end
+    return hor
+end
+
+function precompute_reaction_conditions(reactant_stoch, hor, numspecies, numjumps)
+    # Precompute reaction conditions for each species i, including:
+    # - max_hor: the highest order of reaction (HOR) where species i is a reactant.
+    # - max_stoich: the maximum stoichiometry (nu_ij) in reactions with max_hor.
+    # Used to optimize compute_gi, as per Cao et al. (2006), Section IV, equation (27).
+    hor_type = eltype(hor)
+    max_hor = zeros(hor_type, numspecies)
+    max_stoich = zeros(hor_type, numspecies)
+    stoch_type = eltype(first(first(reactant_stoch)))
+    zero_stoch = zero(stoch_type)
+    for j in 1:numjumps
+        for (spec_idx, stoch) in reactant_stoch[j]
+            if stoch > zero_stoch  # Species is a reactant
+                if hor[j] > max_hor[spec_idx]
+                    max_hor[spec_idx] = hor[j]
+                    max_stoich[spec_idx] = stoch
+                elseif hor[j] == max_hor[spec_idx]
+                    max_stoich[spec_idx] = max(max_stoich[spec_idx], stoch)
+                end
+            end
+        end
+    end
+    return max_hor, max_stoich
+end
+
+function compute_gi(u, max_hor, max_stoich, i, t)
+    # Compute g_i for species i to bound the relative change in propensity functions,
+    # as per Cao et al. (2006), Section IV, equation (27).
+    # g_i is determined by the highest order of reaction (HOR) and maximum stoichiometry (nu_ij) where species i is a reactant:
+    # - HOR = 1 (first-order, e.g., S_i -> products): g_i = 1
+    # - HOR = 2 (second-order):
+    #   - nu_ij = 1 (e.g., S_i + S_k -> products): g_i = 2
+    #   - nu_ij = 2 (e.g., 2S_i -> products): g_i = 2 + 1/(x_i - 1)
+    # - HOR = 3 (third-order):
+    #   - nu_ij = 1 (e.g., S_i + S_k + S_m -> products): g_i = 3
+    #   - nu_ij = 2 (e.g., 2S_i + S_k -> products): g_i = (3/2) * (2 + 1/(x_i - 1))
+    #   - nu_ij = 3 (e.g., 3S_i -> products): g_i = 3 + 1/(x_i - 1) + 2/(x_i - 2)
+    # Uses precomputed max_hor and max_stoich to reduce work to O(num_species) per timestep.
+    one_max_hor = one(1 / one(eltype(u)))
+    hor_type = eltype(max_hor)
+    zero_hor = zero(hor_type)
+    one_hor = one(hor_type)
+    two_hor = one_hor + one_hor
+    three_hor = two_hor + one_hor
+
+    if max_hor[i] == zero_hor  # No reactions involve species i as a reactant
+        return one_max_hor
+    elseif max_hor[i] == one_hor
+        return one_max_hor
+    elseif max_hor[i] == two_hor
+        if max_stoich[i] == one_hor
+            return 2 * one_max_hor
+        else  # max_stoich[i] == 2
+            return u[i] > one_max_hor ? 
+                   2 * one_max_hor + one_max_hor / (u[i] - one_max_hor) : 2 * one_max_hor  # Fallback to 2 if x_i <= 1
+        end
+    elseif max_hor[i] == three_hor
+        if max_stoich[i] == one_hor
+            return 3 * one_max_hor
+        elseif max_stoich[i] == two_hor
+            return u[i] > one_max_hor ? 
+                   (3 * one_max_hor / 2) * (2 * one_max_hor + one_max_hor / (u[i] - one_max_hor)) : 3 * one_max_hor  # Fallback to 3 if x_i <= 1
+        else  # max_stoich[i] == 3
+            return u[i] > 2 * one_max_hor ? 
+                   3 * one_max_hor + one_max_hor / (u[i] - one_max_hor) + 2 * one_max_hor / (u[i] - 2 * one_max_hor) : 3 * one_max_hor  # Fallback to 3 if x_i <= 2
+        end
+    end
+    return one_max_hor  # Default case
+end
+
+function compute_tau(u, rate_cache, nu, hor, p, t, epsilon, rate, dtmin, max_hor, max_stoich, numjumps)
+    # Compute the tau-leaping step-size using equation (20) from Cao et al. (2006):
+    # tau = min_{i in I_rs} { max(epsilon * x_i / g_i, 1) / |mu_i(x)|, max(epsilon * x_i / g_i, 1)^2 / sigma_i^2(x) }
+    # where mu_i(x) and sigma_i^2(x) are defined in equations (9a) and (9b):
+    # mu_i(x) = sum_j nu_ij * a_j(x), sigma_i^2(x) = sum_j nu_ij^2 * a_j(x)
+    # I_rs is the set of reactant species (assumed to be all species here, as critical reactions are not specified).
+    rate(rate_cache, u, p, t)
+    if all(<=(0), rate_cache)  # Handle case where all rates are zero or negative
+        return dtmin
+    end
+    tau = typemax(typeof(t))
+    for i in 1:length(u)
+        mu = zero(eltype(u))
+        sigma2 = zero(eltype(u))
+        for j in 1:size(nu, 2)
+            mu += nu[i, j] * rate_cache[j] # Equation (9a)
+            sigma2 += nu[i, j]^2 * rate_cache[j] # Equation (9b)
+        end
+        gi = compute_gi(u, max_hor, max_stoich, i, t)
+        bound = max(epsilon * u[i] / gi, one(eltype(u))) # max(epsilon * x_i / g_i, 1)
+        mu_term = abs(mu) > 0 ? bound / abs(mu) : typemax(typeof(t)) # First term in equation (8)
+        sigma_term = sigma2 > 0 ? bound^2 / sigma2 : typemax(typeof(t)) # Second term in equation (8)
+        tau = min(tau, mu_term, sigma_term) # Equation (8)
+    end
+    return max(tau, dtmin)
+end
+
+# Define residual for implicit equation
+# Newton: u_new = u_current + sum_j nu_j * a_j(u_new) * tau (Cao et al., 2004)
+# Trapezoidal: u_new = u_current + sum_j nu_j * (a_j(u_current) + a_j(u_new))/2 * tau
+function implicit_equation!(resid, u_new, params)
+    u_current, rate_cache, nu, p, t, tau, rate, numjumps, solver = params
+    rate(rate_cache, u_new, p, t + tau)
+    resid .= u_new .- u_current
+    if isa(solver, NewtonImplicitSolver)
+        for j in 1:numjumps
+            for spec_idx in 1:size(nu, 1)
+                resid[spec_idx] -= nu[spec_idx, j] * rate_cache[j] * tau  # Cao et al. (2004)
+            end
+        end
+    else  # TrapezoidalImplicitSolver
+        rate_current = similar(rate_cache)
+        rate(rate_current, u_current, p, t)
+        half = one(eltype(rate_cache)) / 2
+        for j in 1:numjumps
+            for spec_idx in 1:size(nu, 1)
+                resid[spec_idx] -= nu[spec_idx, j] * half * (rate_cache[j] + rate_current[j]) * tau
+            end
+        end
+    end
+    resid .= max.(resid, -u_new)  # Ensure non-negative solution
+end
+
+# Solve implicit equation using SimpleNonlinearSolve
+function solve_implicit(u_current, rate_cache, nu, p, t, tau, rate, numjumps, solver)
+    u_new = convert(Vector{float(eltype(u_current))}, u_current)
+    prob = NonlinearProblem(implicit_equation!, u_new, (u_current, rate_cache, nu, p, t, tau, rate, numjumps, solver))
+    sol = solve(prob, SimpleNewtonRaphson(autodiff=AutoFiniteDiff()); abstol=1e-6, reltol=1e-6)
+    return sol.u, sol.retcode == ReturnCode.Success
+end
+
+# Function to generate a mass action rate function
+function massaction_rate(maj, numjumps)
+    return (out, u, p, t) -> begin
+        for j in 1:numjumps
+            out[j] = evalrxrate(u, j, maj)
+        end
+    end
+end
+
+function simple_implicit_tau_leaping_loop!(
+        prob, alg, u_current, t_current, t_end, p, rng,
+        rate, nu, hor, max_hor, max_stoich, numjumps, epsilon,
+        dtmin, saveat_times, usave, tsave, du, counts, rate_cache,
+        maj, solver, save_end)
+    save_idx = 1
+
+    while t_current < t_end
+        rate(rate_cache, u_current, p, t_current)
+        tau = compute_tau(u_current, rate_cache, nu, hor, p, t_current, epsilon, rate, dtmin, max_hor, max_stoich, numjumps)
+        tau = min(tau, t_end - t_current)
+        if !isempty(saveat_times) && save_idx <= length(saveat_times) && t_current + tau > saveat_times[save_idx]
+            tau = saveat_times[save_idx] - t_current
+        end
+
+        u_new_float, converged = solve_implicit(u_current, rate_cache, nu, p, t_current, tau, rate, numjumps, solver)
+        if !converged
+            tau /= 2
+            continue
+        end
+
+        rate(rate_cache, u_new_float, p, t_current + tau)
+        zero_rate = zero(eltype(rate_cache))
+        counts .= pois_rand.(rng, max.(rate_cache * tau, zero_rate))
+        du .= zero(eltype(du))
+        for j in 1:numjumps
+            for (spec_idx, stoch) in maj.net_stoch[j]
+                du[spec_idx] += stoch * counts[j]
+            end
+        end
+        u_new = u_current + du
+
+        zero_pop = zero(eltype(u_new))
+        if any(<(zero_pop), u_new)
+            # Halve tau to avoid negative populations, as per Cao et al. (2006), Section 3.3
+            tau /= 2
+            continue
+        end
+        # Ensure non-negativity, as per Cao et al. (2006), Section 3.3
+        for i in eachindex(u_new)
+            u_new[i] = max(u_new[i], zero_pop)
+        end
+        t_new = t_current + tau
+
+        if isempty(saveat_times) || (save_idx <= length(saveat_times) && t_new >= saveat_times[save_idx])
+            push!(usave, copy(u_new))
+            push!(tsave, t_new)
+            if !isempty(saveat_times) && t_new >= saveat_times[save_idx]
+                save_idx += 1
+            end
+        end
+
+        u_current = u_new
+        t_current = t_new
+    end
+
+    # Save endpoint if requested and not already saved
+    if save_end && (isempty(tsave) || tsave[end] != t_end)
+        push!(usave, copy(u_current))
+        push!(tsave, t_end)
+    end
+end
+
+function DiffEqBase.solve(jump_prob::JumpProblem, alg::SimpleImplicitTauLeaping; 
+        seed = nothing,
+        dtmin = nothing,
+        saveat = nothing, save_start = nothing, save_end = nothing)
+    validate_pure_leaping_inputs(jump_prob, alg) ||
+        error("SimpleImplicitTauLeaping can only be used with PureLeaping JumpProblem with a MassActionJump.")
+
+    (; prob, rng) = jump_prob
+    (seed !== nothing) && seed!(rng, seed)
+
+    maj = jump_prob.massaction_jump
+    numjumps = get_num_majumps(maj)
+    rj = jump_prob.regular_jump
+    # Extract rates
+    rate = rj !== nothing ? rj.rate : massaction_rate(maj, numjumps)
+    c = rj !== nothing ? rj.c : nothing
+    u0 = copy(prob.u0)
+    tspan = prob.tspan
+    p = prob.p
+
+    if dtmin === nothing
+        dtmin = 1e-10 * one(typeof(tspan[2]))
+    end
+
+    saveat_times, save_start, save_end = _process_saveat(saveat, tspan, save_start, save_end)
+
+    # Initialize current state and saved history
+    u_current = copy(u0)
+    t_current = tspan[1]
+    if save_start
+        usave = [copy(u0)]
+        tsave = [tspan[1]]
+    else
+        usave = typeof(u0)[]
+        tsave = typeof(tspan[1])[]
+    end
+    rate_cache = zeros(float(eltype(u0)), numjumps)
+    counts = zero(rate_cache)
+    du = similar(u0)
+    t_end = tspan[2]
+    epsilon = alg.epsilon
+    solver = alg.solver
+
+    nu = zeros(float(eltype(u0)), length(u0), numjumps)
+    for j in 1:numjumps
+        for (spec_idx, stoch) in maj.net_stoch[j]
+            nu[spec_idx, j] = stoch
+        end
+    end
+    reactant_stoch = maj.reactant_stoch
+    hor = compute_hor(reactant_stoch, numjumps)
+    max_hor, max_stoich = precompute_reaction_conditions(reactant_stoch, hor, length(u0), numjumps)
+
+    simple_implicit_tau_leaping_loop!(
+        prob, alg, u_current, t_current, t_end, p, rng,
+        rate, nu, hor, max_hor, max_stoich, numjumps, epsilon,
+        dtmin, saveat_times, usave, tsave, du, counts, rate_cache,
+        maj, solver, save_end)
+
+    sol = DiffEqBase.build_solution(prob, alg, tsave, usave,
+        calculate_error=false,
+        interp=DiffEqBase.ConstantInterpolation(tsave, usave))
+    return sol
+end
+
 struct EnsembleGPUKernel{Backend} <: SciMLBase.EnsembleAlgorithm
     backend::Backend
     cpu_offload::Float64