Meta-Solver for Neural Ordinary Differential Equations

Towards robust neural ODEs using parametrized solvers.

Main idea

Each Runge-Kutta (RK) solver with s stages and of the p-th order is defined by a table of coefficients (Butcher tableau). For s=p=2, s=p=3 and s=p=4 all coefficient in the table can be parametrized with no more than two variables [1].

Usually, during neural ODE training RK solver with fixed Butcher tableau is used, and only the right-hand side (RHS) function is trained. We propose to use the whole parametric family of RK solvers to improve robustness of neural ODEs.

Requirements

• pytorch==1.7
• apex==0.1 (for training)

Examples

For CIFAR-10 and MNIST demo, please, check examples folder.

Meta Solver Regimes

In the notebook examples/cifar10/Evaluate model.ipynb we show how to perform the forward pass through the Neural ODE using different types of Meta Solver regimes, namely

• Standalone
• Solver switching/smoothing
• Solver ensembling
• Model ensembling

In more details, usage of different regimes means

• Standalone

  • Use one solver during inference.
  • This regime is applied in the training and testing stages.

• Solver switching / smoothing

  • For each batch one solver is chosen from a group of solvers with finite (in switching regime) or infinite (in smoothing regime) number of candidates.
  • This regime is applied in the training stage

• Solver ensembling

  • Use several solvers durung inference.
  • Outputs of ODE Block (obtained with different solvers) are averaged before propagating through the next layer.
  • This regime is applied in the training and testing stages.

• Model ensembling

  • Use several solvers durung inference.
  • Model probabilites obtained via propagation with different solvers are averaged to get the final result.
  • This regime is applied in the training and testing stages.

Selected results

Different solver parameterizations yield different robustness

We have trained a neural ODE model several times, using different u values in parametrization of the 2-nd order Runge-Kutta solver. The image below depicts robust accuracies for the MNIST classification task. We use PGD attack (eps=0.3, lr=2/255 and iters=7). The mean values of robust accuracy (bold lines) and +- standard error mean (shaded region) computed across 9 random seeds are shown in this image.

Solver smoothing improves robustness

We compare results of neural ODE adversarial training on CIFAR-10 dataset with and without solver smoothing (using normal distribution with mean = 0 and sigma=0.0125). We choose 8-steps RK2 solver with u=0.5 for this experiment.

• We perform training using FGSM random technique described in https://arxiv.org/abs/2001.03994 (with eps=8/255, alpha=10/255).
• We use cyclic learning rate schedule with one cycle (36 epochs, max_lr=0.1, base_lr=1e-7).
• We measure robust accuracy of resulting models after FGSM (eps=8/255) and PGD (eps=8/255, lr=2/255, iters=7) attacks.
• We use premetanode10 architecture from sopa/src/models/odenet_cifar10/layers.py that has the following form Conv -> PreResNet block -> ODE block -> PreResNet block -> ODE block -> GeLU -> Average Pooling -> Fully Connected
• We compute mean and standard error across 3 random seeds.

References

[1] Wanner, G., & Hairer, E. (1993). Solving ordinary differential equations I. Springer Berlin Heidelberg