SimpleImplicitTauLeaping solver

Project.toml

@@ -18,6 +18,7 @@ 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"
 UnPack = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"

src/JumpProcesses.jl

@@ -20,6 +20,8 @@ using Base.Threads: Threads, @threads
 using Base.FastMath: add_fast
 using Setfield: @set, @set!
 
+using SimpleNonlinearSolve
+
 # Import functions we extend from Base
 import Base: size, getindex, setindex!, length, similar, show, merge!, merge
 
@@ -129,7 +131,7 @@ export SSAStepper
 
 # leaping: 
 include("simple_regular_solve.jl")
-export SimpleTauLeaping, EnsembleGPUKernel
+export SimpleTauLeaping, SimpleImplicitTauLeaping, NewtonImplicitSolver, TrapezoidalImplicitSolver, EnsembleGPUKernel
 
 # spatial:
 include("spatial/spatial_massaction_jump.jl")

src/simple_regular_solve.jl

@@ -1,5 +1,17 @@
 struct SimpleTauLeaping <: DiffEqBase.DEAlgorithm end
 
+# 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 \
@@ -14,6 +26,19 @@ function validate_pure_leaping_inputs(jump_prob::JumpProblem, alg)
     jump_prob.regular_jump !== nothing    
 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."))
     validate_pure_leaping_inputs(jump_prob, alg) ||
@@ -61,6 +86,236 @@ function DiffEqBase.solve(jump_prob::JumpProblem, alg::SimpleTauLeaping;
         interp = DiffEqBase.ConstantInterpolation(t, u))
 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.
+    hor = zeros(Int, numjumps)
+    for j in 1:numjumps
+        order = sum(stoch for (spec_idx, stoch) in reactant_stoch[j]; init=0)
+        if order > 3
+            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).
+    max_hor = zeros(Int, numspecies)
+    max_stoich = zeros(Int, numspecies)
+    for j in 1:numjumps
+        for (spec_idx, stoch) in reactant_stoch[j]
+            if stoch > 0  # 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.
+    if max_hor[i] == 0  # No reactions involve species i as a reactant
+        return 1.0
+    elseif max_hor[i] == 1
+        return 1.0
+    elseif max_hor[i] == 2
+        if max_stoich[i] == 1
+            return 2.0
+        elseif max_stoich[i] == 2
+            return u[i] > 1 ? 2.0 + 1.0 / (u[i] - 1) : 2.0  # Fallback to 2.0 if x_i <= 1
+        end
+    elseif max_hor[i] == 3
+        if max_stoich[i] == 1
+            return 3.0
+        elseif max_stoich[i] == 2
+            return u[i] > 1 ? 1.5 * (2.0 + 1.0 / (u[i] - 1)) : 3.0  # Fallback to 3.0 if x_i <= 1
+        elseif max_stoich[i] == 3
+            return u[i] > 2 ? 3.0 + 1.0 / (u[i] - 1) + 2.0 / (u[i] - 2) : 3.0  # Fallback to 3.0 if x_i <= 2
+        end
+    end
+    return 1.0  # 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(==(0.0), rate_cache)  # Handle case where all rates are zero
+        return dtmin
+    end
+    tau = Inf
+    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, 1.0) # max(epsilon * x_i / g_i, 1)
+        mu_term = abs(mu) > 0 ? bound / abs(mu) : Inf # First term in equation (8)
+        sigma_term = sigma2 > 0 ? bound^2 / sigma2 : Inf # 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
+    for j in 1:numjumps
+        for spec_idx in 1:size(nu, 1)
+            if isa(solver, NewtonImplicitSolver)
+                resid[spec_idx] -= nu[spec_idx, j] * rate_cache[j] * tau  # Cao et al. (2004)
+            else  # TrapezoidalImplicitSolver
+                rate_current = similar(rate_cache)
+                rate(rate_current, u_current, p, t)
+                resid[spec_idx] -= nu[spec_idx, j] * 0.5 * (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{Float64}, 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 DiffEqBase.solve(jump_prob::JumpProblem, alg::SimpleImplicitTauLeaping; 
+        seed = nothing,
+        dtmin = 1e-10,
+        saveat = nothing)
+    validate_pure_leaping_inputs(jump_prob, alg) ||
+        error("SimpleImplicitTauLeaping can only be used with PureLeaping JumpProblem with a MassActionJump.")
+
+    @unpack 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 :
+        (out, u, p, t) -> begin
+            for j in 1:numjumps
+                out[j] = evalrxrate(u, j, maj)
+            end
+        end
+    c = rj !== nothing ? rj.c : nothing
+    u0 = copy(prob.u0)
+    tspan = prob.tspan
+    p = prob.p
+
+    u_current = copy(u0)
+    t_current = tspan[1]
+    usave = [copy(u0)]
+    tsave = [tspan[1]]
+    rate_cache = zeros(Float64, numjumps)
+    counts = zeros(Int64, numjumps)
+    du = similar(u0)
+    t_end = tspan[2]
+    epsilon = alg.epsilon
+    solver = alg.solver
+
+    nu = zeros(Int64, 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)
+
+    saveat_times = isnothing(saveat) ? Vector{Float64}() : 
+                   (saveat isa Number ? collect(range(tspan[1], tspan[2], step=saveat)) : collect(saveat))
+    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)
+        counts .= pois_rand.(rng, max.(rate_cache * tau, 0.0))
+        du .= zero(eltype(u_current))
+        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
+
+        if any(<(0), 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], 0)
+        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
+
+    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