Direct Adjoint Sensitivities of Differential Equations

adjointsensitivities(sol,alg,g,t=nothing,dg=nothing; abstol=1e-6,reltol=1e-3, checkpoints=sol.t, corfuncanalytical=nothing, callback = nothing, sensealg=InterpolatingAdjoint(), kwargs...)

Adjoint sensitivity analysis is used to find the gradient of the solution with respect to some functional of the solution. In many cases this is used in an optimization problem to return the gradient with respect to some cost function. It is equivalent to "backpropagation" or reverse-mode automatic differentiation of a differential equation.

Using adjoint_sensitivities directly let's you do three things. One it can allow you to be more efficient, since the sensitivity calculation can be done directly on a cost function, avoiding the overhead of building the derivative of the full concretized solution. It can also allow you to be more efficient by directly controlling the forward solve that is then reversed over. Lastly, it allows one to define a continuous cost function on the continuous solution, instead of just at discrete data points.

Syntax

There are two forms. For discrete adjoints, the form is:

du0,dp = adjoint_sensitivities(sol,alg,dg,ts;sensealg=InterpolatingAdjoint(),
                               checkpoints=sol.t,kwargs...)

where alg is the ODE algorithm to solve the adjoint problem, dg is the jump function, sensealg is the sensitivity algorithm, and ts is the time points for data. dg is given by:

dg(out,u,p,t,i)

which is the in-place gradient of the cost functional g at time point ts[i] with u=u(t).

For continuous functionals, the form is:

du0,dp = adjoint_sensitivities(sol,alg,g,nothing,(dgdu,dgdp);sensealg=InterpolatingAdjoint(),
                               checkpoints=sol.t,,kwargs...)

for the cost functional

g(u,p,t)

with in-place gradient

dgdu(out,u,p,t)
dgdp(out,u,p,t)

If the gradient is omitted, i.e.

du0,dp = adjoint_sensitivities(sol,alg,g,nothing;kwargs...)

then we assume dgdp is zero and dgdu will be computed automatically using ForwardDiff or finite differencing, depending on the autodiff setting in the AbstractSensitivityAlgorithm. Note that the keyword arguments are passed to the internal ODE solver for solving the adjoint problem.

Example discrete adjoints on a cost function

In this example we will show solving for the adjoint sensitivities of a discrete cost functional. First let's solve the ODE and get a high quality continuous solution:

function f(du,u,p,t)
  du[1] = dx = p[1]*u[1] - p[2]*u[1]*u[2]
  du[2] = dy = -p[3]*u[2] + u[1]*u[2]
end

p = [1.5,1.0,3.0]
prob = ODEProblem(f,[1.0;1.0],(0.0,10.0),p)
sol = solve(prob,Vern9(),abstol=1e-10,reltol=1e-10)

Now let's calculate the sensitivity of the $\ell_2$ error against 1 at evenly spaced points in time, that is:

\[L(u,p,t)=\sum_{i=1}^{n}\frac{\Vert1-u(t_{i},p)\Vert^{2}}{2}\]

for $t_i = 0.5i$. This is the assumption that the data is data[i]=1.0. For this function, notice we have that:

\[\begin{aligned} dg_{1}&=1-u_{1} \\ dg_{2}&=1-u_{2} \\ & \quad \vdots \end{aligned}\]

and thus:

dg(out,u,p,t,i) = (out.=1.0.-u)

Also, we can omit dgdp, because the cost function doesn't dependent on p. If we had data, we'd just replace 1.0 with data[i]. To get the adjoint sensitivities, call:

ts = 0:0.5:10
res = adjoint_sensitivities(sol,Vern9(),dg,ts,abstol=1e-14,
                            reltol=1e-14)

This is super high accuracy. As always, there's a tradeoff between accuracy and computation time. We can check this almost exactly matches the autodifferentiation and numerical differentiation results:

using ForwardDiff,Calculus,Tracker
function G(p)
  tmp_prob = remake(prob,u0=convert.(eltype(p),prob.u0),p=p)
  sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14,saveat=ts,
              sensealg=SensitivityADPassThrough())
  A = convert(Array,sol)
  sum(((1 .- A).^2)./2)
end
G([1.5,1.0,3.0])
res2 = ForwardDiff.gradient(G,[1.5,1.0,3.0])
res3 = Calculus.gradient(G,[1.5,1.0,3.0])
res4 = Tracker.gradient(G,[1.5,1.0,3.0])
res5 = ReverseDiff.gradient(G,[1.5,1.0,3.0])

and see this gives the same values.

Example controlling adjoint method choices and checkpointing

In the previous examples, all calculations were done using the interpolating method. This maximizes speed but at a cost of requiring a dense sol. If it is not possible to hold a dense forward solution in memory, then one can use checkpointing. For example:

ts = [0.0,0.2,0.5,0.7]
sol = solve(prob,Vern9(),saveat=ts)

Creates a non-dense solution with checkpoints at [0.0,0.2,0.5,0.7]. Now we can do:

res = adjoint_sensitivities(sol,Vern9(),dg,ts,
                            sensealg=InterpolatingAdjoint(checkpointing=true))

When grabbing a Jacobian value during the backwards solution, it will no longer interpolate to get the value. Instead, it will start a forward solution at the nearest checkpoint to build local interpolants in a way that conserves memory. By default the checkpoints are at sol.t, but we can override this:

res = adjoint_sensitivities(sol,Vern9(),dg,ts,
                            sensealg=InterpolatingAdjoint(checkpointing=true),
                            checkpoints = [0.0,0.5])

Example continuous adjoints on an energy functional

In this case we'd like to calculate the adjoint sensitivity of the scalar energy functional:

\[G(u,p)=\int_{0}^{T}\frac{\sum_{i=1}^{n}u_{i}^{2}(t)}{2}dt\]

which is:

g(u,p,t) = (sum(u).^2) ./ 2

Notice that the gradient of this function with respect to the state u is:

function dg(out,u,p,t)
  out[1]= u[1] + u[2]
  out[2]= u[1] + u[2]
end

To get the adjoint sensitivities, we call:

res = adjoint_sensitivities(sol,Vern9(),g,nothing,dg,abstol=1e-8,
                                 reltol=1e-8,iabstol=1e-8,ireltol=1e-8)

Notice that we can check this against autodifferentiation and numerical differentiation as follows:

using QuadGK
function G(p)
  tmp_prob = remake(prob,p=p)
  sol = solve(tmp_prob,Vern9(),abstol=1e-14,reltol=1e-14)
  res,err = quadgk((t)-> (sum(sol(t)).^2)./2,0.0,10.0,atol=1e-14,rtol=1e-10)
  res
end
res2 = ForwardDiff.gradient(G,[1.5,1.0,3.0])
res3 = Calculus.gradient(G,[1.5,1.0,3.0])

H = secondordersensitivities(loss,prob,alg,args...; sensealg=ForwardDiffOverAdjoint(InterpolatingAdjoint(autojacvec=ReverseDiffVJP())), kwargs...)

Second order sensitivity analysis is used for the fast calculation of Hessian matrices.

Example second order sensitivity analysis calculation

using DiffEqSensitivity, OrdinaryDiffEq, ForwardDiff
using Test

function lotka!(du,u,p,t)
  du[1] = dx = p[1]*u[1] - p[2]*u[1]*u[2]
  du[2] = dy = -p[3]*u[2] + p[4]*u[1]*u[2]
end

p = [1.5,1.0,3.0,1.0]; u0 = [1.0;1.0]
prob = ODEProblem(lotka!,u0,(0.0,10.0),p)
loss(sol) = sum(sol)
v = ones(4)

H  = second_order_sensitivities(loss,prob,Vern9(),saveat=0.1,abstol=1e-12,reltol=1e-12)

Arguments

The arguments for this function match adjoint_sensitivities. The only notable difference is sensealg which requires a second order sensitivity algorithm, of which currently the only choice is ForwardDiffOverAdjoint which uses forward-over-reverse to mix a forward-mode sensitivity analysis with an adjoint sensitivity analysis for a faster computation than either double forward or double reverse. ForwardDiffOverAdjoint's positional argument just accepts a first order sensitivity algorithm.

Hv = secondordersensitivity_product(loss,v,prob,alg,args...; sensealg=ForwardDiffOverAdjoint(InterpolatingAdjoint(autojacvec=ReverseDiffVJP())), kwargs...)

Second order sensitivity analysis product is used for the fast calculation of Hessian-vector products $Hv$ without requiring the construction of the Hessian matrix.

Example second order sensitivity analysis calculation

using DiffEqSensitivity, OrdinaryDiffEq, ForwardDiff
using Test

function lotka!(du,u,p,t)
  du[1] = dx = p[1]*u[1] - p[2]*u[1]*u[2]
  du[2] = dy = -p[3]*u[2] + p[4]*u[1]*u[2]
end

p = [1.5,1.0,3.0,1.0]; u0 = [1.0;1.0]
prob = ODEProblem(lotka!,u0,(0.0,10.0),p)
loss(sol) = sum(sol)
v = ones(4)

Hv = second_order_sensitivity_product(loss,v,prob,Vern9(),saveat=0.1,abstol=1e-12,reltol=1e-12)

Arguments

The arguments for this function match adjoint_sensitivities. The only notable difference is sensealg which requires a second order sensitivity algorithm, of which currently the only choice is ForwardDiffOverAdjoint which uses forward-over-reverse to mix a forward-mode sensitivity analysis with an adjoint sensitivity analysis for a faster computation than either double forward or double reverse. ForwardDiffOverAdjoint's positional argument just accepts a first order sensitivity algorithm.