Judging by the Van der Pol benchmark, your methods are faster than other neural ODE solvers, but are still several magnitude slower than traditional solvers such as Runge-Kutta. Why are many people excited about this?

In these benchmarks we compare the same Runge-Kutta solver (5th order Dormand-Prince) implemented in all of these libraries. None of these libraries actually propose any new stepping methods. The point is to make ODE solvers available in popular deep learning methods to enable deep continuous models such as neural ODEs and continuous normalizing flows. The particular appeal of torchode is its optimized implementation and that it runs multiple independent instances of an ODE solver in parallel when you train on batches, i.e. each instance is solved with its own step size and step accept/reject decisions. This avoids a performance pitfall where the usual batching approach can lead to many unnecessary solver steps in batched training of models with varying stiffness, as we show in the Van der Pol experiment.

Sounds nice. Thanks for the explanation.

This is cool but couldn't they have picked a better name 😂.

Cannot unsee: "torChode" sounds like some wordplay on the Tor project.

One more question: for the Van der Pol benchmark

Using the old faithful ode45 in MATLAB (Runge-Kutta) to run test problem you listed in the poster

vdp1 = @(t, y) [y(2); 25*(1-y(1)^2)*y(2)-y(1)];
tic;
[t,y] = ode45(vdp1,[0 5],[1.2, 0.1]);
toc;

It takes only 0.007329 seconds for marching 389 steps using FP64 on CPU. What is the loop time's unit? What is the bottleneck of the implemented algorithm?

It's been years since I've implemented ODE solvers in my engineering courses, but I recall that ODEs are inherently not parallelizable since the subsequent timesteps require the current time steps to be solved. I've only worked with Euler's and Newton's method though.

Also, people have used neural networks to solve the equation sof radiative transfer and used them in scientific papers

Nice work! Though diffrax is mentioned, it would be interesting to see a direct comparison between diffrax and torchode. Can you give some more details in how they differ in features and performance, apart from the library in which they are implemented?

You are correct except for parallel-in-time integration methods that we also mention in the paper. But the "parallel" in the title refers to solving multiple ODEs in parallel independently which is contrary to what is currently done in ML. At the moment, training on a batch of ODEs means that you treat the batch as one large ODE that is solved jointly. torchode solves them independently from each other but still in parallel by tracking a separate current state, step size, etc. for each sample.

Diffrax is an excellent project and a superset of torchode. torchode solves only ODEs, while diffrax combines ODEs, CDEs and SDEs (maybe more?) in the same framework. The reason why we created torchode is that we wanted to bring its structure and flexibility into the PyTorch ecosystem that is still more popular than JAX. In addition, we were also looking to create an optimized implementation in which we succeeded as far as I am concerned. Even though tracking multiple ODE solvers at once is inherently more complex than solving a batch of ODEs jointly, torchode is as fast as or faster than the other PyTorch ODE solvers in our experiments.

First, the difference in steps is probably due to different tolerances in the step size controller.

The loop times is measured in milliseconds. Of course, that is much slower than what you got in matlab. The difference is that we did all benchmarks on GPU, because that is the usual mode for deep learning even though it is certainly inappropriate for the VdP equation if you were interested in it for anything else but benchmarking the inner loop of an ODE solver on a GPU. I think, you can get similar numbers to your matlab code in diffrax with JIT compilation on a CPU. However, you won't get it with torchode because PyTorch's JIT is not as good as JAX's and specifically this line is really slow on CPUs. Nonetheless, after comparing several alternatives we chose this because, as I said, in practice in most of deep learning only GPU performance matters.

ಠ_ಠ

Is this feature of torchode of solving multiple ODEs at once over some batch dimension comparable to jax.vmapping over that dimension in diffrax?

Yes, it is the same thing. Unfortunately, functorch is not advanced enough yet to just translate diffrax to PyTorch directly. Instead, we had to take care of batching everywhere explicitly to decide how long to loop etc.

Cool. Thanks for the explanation.

Aight! Thanks for the nice answers! I wish a good conference :)

+1 for the paper on parallel in time! Are there any implementation of those algorithms for torchode or diffrax out there?

haha sorry i just couldn't unsee it. Ode2Torch might be a better name xD

I am not aware of any but would be very interested if you find one

I guess you have no choice but to add support for PDEs and rebrand to TorchNDE.