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We often want to use numerical methods to evolve a system in time.

That is, for a set of differential equations, we can specify some parameters $\bar{\theta}$ and pass these into our numerical solver program $f$ and get an output $y(t) = f(\bar{\theta})$.

Does automatic differentiation still apply to these types of numerical systems? Is it possible with AD to compute $dy(t)/d\bar{\theta}$?

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    $\begingroup$ I think you need to be more specific what you want the function $f(\theta)$ to contain. Is it an ODE solver? A PDE solver? Just an algebraic expression? $\endgroup$ Sep 29, 2022 at 20:13
  • $\begingroup$ Have you looked into adjoint solvers such as cvodes? $\endgroup$ Oct 1, 2022 at 15:09

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I agree with the comment, that the kind of "solver" embodied by $f(\bar \theta)$ will guide you how to make it AD-aware. At one extreme, if it's just a 10 line function and you have the source code, it's probably not difficult to rewrite it to use dual/AD-numbers. If it's some large black-box nonlinear optimizer without any source, that's obviously a heavier lift.

It's worth pointing out that you can deduce the sensitivity of a solution operator $\mathbf A^{-1}$ in terms of the sensitivity of a forward operator $\mathbf A$, because the sensitivity of their product/composition (the identity operator) must be zero:

$d/d\theta \left[ \mathbf I \right] = 0$

$d/d\theta \left[ \mathbf A^{-1} \cdot \mathbf A \right] = 0$

$d/d\theta \left[ \mathbf A^{-1} \right] \cdot \mathbf A + \mathbf A^{-1} \cdot d/d\theta \left[ \mathbf A \right] = 0$

(solve for $d/d\theta \left[ \mathbf A^{-1} \right]$)

$d/d\theta \left[ \mathbf A^{-1} \right] = -\mathbf A^{-1} \cdot d/d\theta\left[\mathbf A\right] \cdot \mathbf A^{-1}$

(a composition of solve, multiply, solve, negate)

Here your $f$ is analogous to $\mathbf A^{-1}$, and $d/d\theta \left[\mathbf A\right]$ represents something new you'll have to write.

I apologize for introducing a different (linear-algebraic-like) symbol $\mathbf A$. But I spelled it this way on purpose, because this arises often in practice: you discretize some continuum model into a linear-algebraic one (like $\mathbf A$), then submit it to some black-box library (LAPACK, a krylov solver, whatever) which furnishes the solution operator $\mathbf A^{-1}$ in some opaque way. It's probably out of the question to rewrite LAPACK (or whatever) to use some dual/AD-number system, but if you wrote the discretization procedure to form $\mathbf A$, it's probably reasonable to extend it to compute $d/d\theta \left[ \mathbf A \right]$ too, then pursue this line of attack to find $d/d\theta \left[\mathbf A^{-1}\right]$

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tl;dr is that yes, automatic differentation does apply to solvers. But whether it's a good idea is a very deep topic.

Automatic differentiation of a "solver" is a subject with many details for doing it in the most effective form. For this reason, there are a lot of talks and courses that go into lots of depth on the topic. I recently gave a talk on some of the latest stuff in differentiable simulation with the American Statistical Association, and have some detailed notes on such adjoint derivations as part of the 18.337 Parallel Computing and Scientific Machine Learning graduate course at MIT. And there are entire organizations like my SciML Open Source Software Organization which work day-in and day-out on the development of new differentiable solvers.

I'll give a brief summary of all my materials here below.

Continuous vs Discrete Differentiation of Solvers

AD of a solver can be done in essentially two different ways: either directly performing automatic differentiation to the steps of the solver, or by defining higher level adjoint rules that will compute the derivative. In some cases these can be mathematically equivalent. For example, forward sensitivity analysis of an ODE $$u' = f(u,p,t)$$ follows by the chain rule:

$$\frac{d}{dp} \frac{du}{dt} = \frac{d}{dp} f(u,p,t) = \frac{df}{du} \frac{du}{dp} + \frac{\partial f}{\partial p}$$

Thus if you solve the extended system of equations:

$$u' = f(u,p,t)$$ $$s' = \frac{df}{du} s + \frac{\partial f}{\partial p}$$

then you get $s = \frac{du}{dp}$ as the solution to the new equations. So therefore, solve these bigger ODEs and you get the derivative of the solution with respect to parameters as the extra piece. One way to do "automatic differentation" is to add a derivative rule to the AD library that "if you see ODE solve, then replace the solve with this extended solve and take the latter part as the derivative". The other way of course is to simply do forward-mode automatic differentation of the ODE solver library steps itself. It turns out that in this case, if you work out the math the two are mathematically equivalent. Note that it's not computationally equivalent though since the AD process may SIMD the expressions in a different way, doing some constant folding and common subexpression elimination (CSE) in a way that's different from the handcoded version, and thus the performance can be very different even though it's computationally the same algorithm.

However, there are cases where the "analytical" way of writing the derivative is not equivalent to its automatic differentiation counterpart. For example, the adjoint method is a different way to get $\frac{du}{dp}$ values in $\mathcal{O}(n+p)$ time (instead of the $\mathcal{O}(np)$ time of the forward sensitivities above) by solving an ODE forward and some related ODE backwards (for a full derivation and description, see the lecture notes or the recorded video). If you were to do reverse-mode automatic differentiation of the solver, you do not get a mathematically equivalent algorithm. For example, if the solver for the ODE was Euler's method, reverse-mode AD would be mathematically equivalent to solving the forward ODE with Euler's method and the reverse ODE with something like implicit Euler (where part of the implicit equation is solved exactly using a cached value from the forward solve).

So What is Better, Continuous Derivative Rules or Discrete Derivatives of the Solver?

Like any complex question, it depends. We had a manuscript which looked at this in quite some detail (and a biologically-oriented follow-up), and can boil it down to a few basic notes:

  • Forward-mode outperforms reverse-mode / adjoint techniques when the equations are "sufficiently small". For modern implementations this seems to be at around 100.
  • For forward-mode cases, "good" automatic differentiation libraries can make use of structure between the primal and derivative constructions to better CSE/SIMD the generated code for the derivative term, thus forward-mode AD of the solver can be much faster than forward sensitivity analysis even though the two are mathematically the same operation.
  • For reverse-mode cases, the continuous adjoints seem to be faster with current implementations.

But that last bit then has many caveats to put on it. For one, there seems to be a trade-off between performance and stability here. This is noted in the appendix of the paper "Universal Differential Equations for Scientific Machine Learning, which states:

Previous research has shown that the discrete adjoint approach is more stable than continuous adjoints in some cases [53, 47, 94, 95, 96, 97] while continuous adjoints have been demonstrated to be more stable in others [98, 95] and can reduce spurious oscillations [99, 100, 101]. This trade-off between discrete and continuous adjoint approaches has been demonstrated on some equations as a trade-off between stability and computational efficiency [102, 103, 104, 105, 106, 107, 108, 109, 110]. Care has to be taken as the stability of an adjoint approach can be dependent on the chosen discretization method [111, 112, 113, 114, 115]

with the references pointing to those in the manuscript.

This is discussed in even more detail in the manuscript Stiff Neural Ordinary Differential Equations which showcases how there are many ways to implement "the adjoint method", and they can have major differences in stability, essentially trading off memory or performance for improved stability properties.

Special Case: Implicit Equations

The above discussion shows that there are good reasons to differentiate solvers directly, and good reasons to instead write derivative rules for solvers which use forward/adjoint equations. For time series equations, this always has a trade-off. There is an important special case here though that for methods which iterate to convergence, automatic differentiation of the solver is essentially never a good idea. The reason is because the implicit function theorem gives that the derivative of the solution is directly defined at the solution point. For example, for solving $f(x,p) = 0$, if $x^\ast$ is the value of $x$ which satisfies the equation, then $\frac{d x^\ast}{dp} = ...$. In other words, Newton's method might take $n$ steps, and thus automatic differentation will need to differentiate $f$ at least $n$ times. But if you use the implicit function theorem result, then you only need to differentiate it once!

Note of course a similar performance vs stability trade-off does apply here. Since this derivation assumes you have $x^\ast$ such that $f(x^\ast,p) = 0$ exactly, but you don't. Newton's method from the solve will give you something that satisfies the equation to tolerance, so maybe $f(x^\ast,p) \approx 10^{-8}$, which means that the derivative expression is also only approximate, and this then induces an error in the gradient etc. Thus direct differentiation of Newton's method can be more accurate, and you need to worry about tolerance here if the gradients seem sufficiently off.

This does lead to some counter-intuitive results. For example, we had a paper where we exploited this to note that differentiating and ODE solve which goes to infinity (steady state) is faster than a "long ODE", since steady states have a similar implicit definition. It's quicker to go to infinity than it is to go to 1000, who would've thought? Math is fun.

Does Differentiation of Solver Internals Make Sense or Have a Meaning?

"ODE solvers" have all sorts of things in there, like adaptivity parameters and heuristics. One of the things that happens when you do automatic differentiation of the solver is that you aren't just differentiating the solver's states and parameters, but the process will differentiate everything. It turns out that AD of a solver can thus be useful in some tricky ways which put this to use. For example, at ICML we had a paper which regularized the parameters of a neural ODE by the sum of the computed error estimates of the adaptivity heuristics. This would then push the learned equation towards an area of parameter space where the adaptivity gives the largest time steps possible, and thus the learned equation is the "fastest possible learned equation that fits the time series". Such a trick is only possible if you are doing automatic differentiation of the solver since you'd need to differentiate the solver's internals in order to have access to those values in the loss function! This just shows one of many ways in which AD's "extra information" which analytical continuous derivative definitions don't have could potentially be useful for some applications.

Automatic Differentiation in Continuous Sensitivity Methods

Finally, I want to note that even when you attempt to avoid automatic differentation of the solver by using continuous sensitivity methods, it turns out that the optimal way to build the extended equations is to use automatic differentiation!

Summary: there are many good reasons to do automatic differentiation of solvers, but there are also many good reasons to use some analytical derivative techniques. But even if you do analytical derivative techniques, you still want to automatic differentiate something in order to do it optimally! For example, let's return to the forward sensitivity equations:

$$u' = f(u,p,t)$$ $$s' = \frac{df}{du} s + \frac{\partial f}{\partial p}$$

It turns out that $\frac{df}{du} s$ does not require computing the full Jacobian. This operation, known as Jacobian-vector products or jvps, are the primitive operation of forward-mode automatic differentation and thus special seeding of a forward-mode AD tool gives a faster and more robust algorithm than a finite difference form. When done correctly, this operation is computed without ever building the full Jacobian. A trick for this does exist in the finite difference sense as well:

$$\frac{df}{du} s \approx \frac{f(u + \epsilon s) - f(u)}{\epsilon}$$

since it is equivalent to the directional derivative. This is explained in more detail in these lectures (or accompanying video).

In the same vein, continuous adjoints of ODE solves boil down to defining a differential equation which is solved backwards and that differential equation which is solved backwards has a term which is $\frac{df}{du}^T s$, i.e. Jacobian transposed times a vector, also known as the vector-Jacobian product because it's equivalent to $s^T \frac{df}{du}$ when transposed. It turns out that this is the primitive operation of reverse-mode AD, which then allows for computing this operation without fully building the Jacobian. There is no analogue for this operation with finite differencing, which means that there's a pretty massive performance gain from doing this properly. Our paper A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions measures this effect on a stiff partial differential equation, getting:

enter image description here

The takeaway from this plot is that using these AD tricks results in a few orders of magnitude performance improvements (by avoiding the Jacobian construction, which are the "seeding" versions on the left, the right shows the difference that different AD techniques make, which itself is another few orders of magnitude). When people note that the Julia differential equation adjoint solvers are much faster than the adjoints from Sundials COVDES and IDAS on large equations, this part right here is one of the major factors because Sundials does not embed a reverse AD engine into its adjoint code to do the vjp definitions, and instead falls back to using a numerical formulation unless the user provides a vjp override, which is seemingly to be uncommon to do but from these plots clearly should be done more often.

Summary

In total, what can we takeaway so far about differentiating solvers?

  • There are some advantages to differentiating solvers, but there are also some advantages to mixing in analytical continuous adjoints. It's context-dependent which is better.
  • Even when mixing in analytical continuous derivative rules, these are best defined with automatic differentiation within their constructed equations, so one cannot avoid AD completely if one wishes to achieve full performance on arbitrary models.
  • For cases which converge to some kind of implicitly defined solution, using special adjoint tricks will be much better than direct differentiation of the solver.

There's still a lot more to mention, especially as stochastic simulation gets involved, but I'll cut this here for now. As you can see, there's still some open questions that are being investigated in the field, so if you find this interesting please feel free to get in touch.

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