Package: paddlesde
PyTorch Implementation of Differentiable SDE SolversThis library providesstochastic differential equation (SDE)solvers with GPU support and efficient backpropagation.InstallationpipinstalltorchsdeRequirements:Python >=3.6 and PyTorch >=1.6.0.DocumentationAvailablehere.ExamplesQuick exampleimportpaddleimportpaddlesdebatch_size,state_size,brownian_size=32,3,2t_size=20classSDE(paddle.nn.Layer):noise_type='general'sde_type='ito'def__init__(self):super().__init__()self.mu=paddle.nn.Linear(state_size,state_size)self.sigma=paddle.nn.Linear(state_size,state_size*brownian_size)# Driftdeff(self,t,y):returnself.mu(y)# shape (batch_size, state_size)# Diffusiondefg(self,t,y):returnself.sigma(y).reshape([batch_size,state_size,brownian_size])sde=SDE()y0=paddle.full((batch_size,state_size),0.1)ts=paddle.linspace(0,1,t_size)# Initial state y0, the SDE is solved over the interval [ts[0], ts[-1]].# ys will have shape (t_size, batch_size, state_size)ys=paddlesde.sdeint(sde,y0,ts)Notebookexamples/demo.ipynbgives a short guide on how to solve SDEs, including subtle points such as fixing the randomness in the solver and the choice ofnoise types.Latent SDEexamples/latent_sde.pylearns alatent stochastic differential equation, as in Section 5 of[1].
The example fits an SDE to data, whilst regularizing it to be like anOrnstein-Uhlenbeckprior process.
The model can be loosely viewed as avariational autoencoderwith its prior and approximate posterior being SDEs. This example can be run viapython-mexamples.latent_sde--train-dir<TRAIN_DIR>The program outputs figures to the path specified by<TRAIN_DIR>.
Training should stabilize after 500 iterations with the default hyperparameters.Neural SDEs as GANsexamples/sde_gan.pylearns an SDE as a GAN, as in[2],[3]. The example trains an SDE as the generator of a GAN, whilst using aneural CDE[4]as the discriminator. This example can be run viapython-mexamples.sde_ganCitationIf you found this codebase useful in your research, please consider citing either or both of:@article{li2020scalable,
  title={Scalable gradients for stochastic differential equations},
  author={Li, Xuechen and Wong, Ting-Kam Leonard and Chen, Ricky T. Q. and Duvenaud, David},
  journal={International Conference on Artificial Intelligence and Statistics},
  year={2020}
}@article{kidger2021neuralsde,
  title={Neural {SDE}s as {I}nfinite-{D}imensional {GAN}s},
  author={Kidger, Patrick and Foster, James and Li, Xuechen and Oberhauser, Harald and Lyons, Terry},
  journal={International Conference on Machine Learning},
  year={2021}
}References[1] Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, David Duvenaud. "Scalable Gradients for Stochastic Differential Equations".International Conference on Artificial Intelligence and Statistics.2020.[arXiv][2] Patrick Kidger, James Foster, Xuechen Li, Harald Oberhauser, Terry Lyons. "Neural SDEs as Infinite-Dimensional GANs".International Conference on Machine Learning2021.[arXiv][3] Patrick Kidger, James Foster, Xuechen Li, Terry Lyons. "Efficient and Accurate Gradients for Neural SDEs". 2021.[arXiv][4] Patrick Kidger, James Morrill, James Foster, Terry Lyons, "Neural Controlled Differential Equations for Irregular Time Series".Neural Information Processing Systems2020.[arXiv]This is a research project, not an official Google product.