13 Ordinary Differential Equations

Stan provides a number of different methods for solving systems of ordinary differential equations (ODEs). All of these methods adaptively refine their solutions in order to satisfy given tolerances, but internally they handle calculations quite a bit differently.

Because Stan’s algorithms requires gradients of the log density, the ODE solvers must not only provide the solution to the ODE itself, but also the gradient of the ODE solution with respect to parameters (the sensitivities). Two fundamentally different approaches are available in Stan to solve this problem, each having very different computational cost depending on the number of ODE states \(N\) and the number of parameters \(M\) being used:

  • A forward sensitivity solver expands the base ODE system with additional ODE equations for the gradients of the solution. For each parameter, an additional full set of \(N\) sensitivity states are added meaning that the full ODE solved has \[N \, + N \cdot M\] states.

  • An adjoint sensitivity solver starts by solving the base ODE system forward in time to get the ODE solution and then solves another ODE system (the adjoint) backward in time to get the gradients. The forward and reverse solves both have \(N\) states each. There is additionally one quadrature problem solved for every parameter.

The adjoint sensitivity approach scales much better than the forward sensitivity approach. Whereas the computational cost of the forward approach scales multiplicatively in the number of ODE states \(N\) and parameters \(M\), the adjoint sensitivity approach scales linear in states \(N\) and parameters \(M\). However, the adjoint problem is harder to configure and the overhead for small problems actually makes it slower than solving the full forward sensitivity system. With that in mind, the rest of this introduction focuses on the forward sensitivity interfaces. For information on the adjoint sensitivity interface see the Adjoint ODE solver

Two interfaces are provided for each forward sensitivity solver: one with default tolerances and default max number of steps, and one that allows these controls to be modified. Choosing tolerances is important for making any of the solvers work well – the defaults will not work everywhere. The tolerances should be chosen primarily with consideration to the scales of the solutions, the accuracy needed for the solutions, and how the solutions are used in the model. For instance, if a solution component slowly varies between 3.0 and 5.0 and measurements of the ODE state are noisy, then perhaps the tolerances do not need to be as tight as for a situation where the solutions vary between 3.0 and 3.1 and very high precision measurements of the ODE state are available. It is also often useful to reduce the absolute tolerance when a component of the solution is expected to approach zero. For information on choosing tolerances, see the control parameters section.

The advantage of adaptive solvers is that as long as reasonable tolerances are provided and an ODE solver well-suited to the problem is chosen the technical details of solving the ODE can be abstracted away. The catch is that it is not always clear from the outset what reasonable tolerances are or which ODE solver is best suited to a problem. In addition, as changes are made to an ODE model, the optimal solver and tolerances may change.

With this in mind, the four forward solvers are rk45, bdf, adams, and ckrk. If no other information about the ODE is available, start with the rk45 solver. The list below has information on when each solver is useful.

If there is any uncertainty about which solver is the best, it can be useful to measure the performance of all the interesting solvers using profile statements. It is difficult to always know exactly what solver is the best in all situations, but a profile can provide a quick check.

  • rk45: a fourth and fifth order Runge-Kutta method for non-stiff systems (Dormand and Prince 1980; Ahnert and Mulansky 2011). rk45 is the most generic solver and should be tried first.

  • bdf: a variable-step, variable-order, backward-differentiation formula implementation for stiff systems (Cohen and Hindmarsh 1996; Serban and Hindmarsh 2005). bdf is often useful for ODEs modeling chemical reactions.

  • adams: a variable-step, variable-order, Adams-Moulton formula implementation for non-stiff systems (Cohen and Hindmarsh 1996; Serban and Hindmarsh 2005). The method has order up to 12, hence is commonly used when high-accuracy is desired for a very smooth solution, such as in modeling celestial mechanics and orbital dynamics (Montenbruck and Gill 2000).

  • ckrk: a fourth and fifth order explicit Runge-Kutta method for non-stiff and semi-stiff systems (Cash and Karp 1990; Mazzia, Cash, and Soetaert 2012). The difference between ckrk and rk45 is that ckrk should perform better for systems that exhibit rapidly varying solutions. Often in those situations the derivatives become large or even nearly discontinuous, and ckrk is designed to address such problems.

For a discussion of stiff ODE systems, see the stiff ODE section. For information on the adjoint sensitivity interface see the Adjoint ODE solver section. The function signatures for Stan’s ODE solvers can be found in the function reference manual section on ODE solvers.

References

Ahnert, Karsten, and Mario Mulansky. 2011. “Odeint—Solving Ordinary Differential Equations in C++.” arXiv 1110.3397.
Cash, J. R., and Alan H. Karp. 1990. “A Variable Order Runge-Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides.” ACM Transactions on Mathematical Software 16 (3): 201–22. https://doi.org/10.1145/79505.79507.
Cohen, Scott D, and Alan C Hindmarsh. 1996. CVODE, a Stiff/Nonstiff ODE Solver in C.” Computers in Physics 10 (2): 138–43.
Dormand, John R, and Peter J Prince. 1980. “A Family of Embedded Runge-Kutta Formulae.” Journal of Computational and Applied Mathematics 6 (1): 19–26.
Mazzia, F., J. R. Cash, and K. Soetaert. 2012. “A Test Set for Stiff Initial Value Problem Solvers in the Open Source Software R: Package deTestSet.” Journal of Computational and Applied Mathematics 236 (16): 4119–31.
Montenbruck, Oliver, and Eberhard Gill. 2000. Satellite Orbits: Models, Methods and Applications. Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-58351-3.
Serban, Radu, and Alan C Hindmarsh. 2005. CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS.” In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 257–69. American Society of Mechanical Engineers.