Leia: Learning with EvolutIon equAtions

     

A PyTorch toolbox for solving learning tasks with neural ODEs. An important element of this toolbox is that it allows for time-dependent weights (controls), and costs involving integrals of the state. Generally speaking, there is flexibility in using different functionals and weight penalties (beyond simply $L^2$, for instance).

A sample experiment may be found in generate_fig.py, with the main modules being a simple instantiation of the neural ODE

model = NeuralODE(device, 
                  data_dim=2, 
                  hidden_dim=5, 
                  augment_dim=1, 
                  non_linearity='relu',
                  architecture='bottleneck', 
                  T=10, 
                  time_steps=20, 
                  fixed_projector=False, 
                  cross_entropy=False)

and then of the optimization algorithm

trainer = Trainer(model, 
                        optimizer_anode, 
                        device, 
                        cross_entropy=False, 
                        turnpike=True,
                        bound=0., 
                        fixed_projector=False)

Citing

If you are using this toolbox for your scientific publication, we would be very appreciative if you were to cite one of our following articles on this topic.

1. Turnpike in optimal control of PDEs, ResNets, and beyond

@article{geshkovski2022turnpike,
  title={Turnpike in optimal control of PDEs, ResNets, and beyond},
  author={Geshkovski, Borjan and Zuazua, Enrique},
  journal={Acta Numerica},
  volume={31},
  pages={135--263},
  year={2022},
  publisher={Cambridge University Press}
}

2. Large-time asymptotics in deep learning

@article{esteve2021large,
  title={Large-time asymptotics in deep learning},
  author={Esteve-Yag{\"u}e, Carlos and Geshkovski, Borjan and Pighin, Dario and Zuazua, Enrique},
  year={2021}
}

3. Sparse approximation in learning via neural ODEs

@article{esteve2023sparsity,
  title={Sparsity in long-time control of neural ODEs},
  author={Esteve-Yag{\"u}e, Carlos and Geshkovski, Borjan},
  journal={Systems \& Control Letters},
  volume={172},
  pages={105452},
  year={2023},
  publisher={Elsevier}
}