CNF Layer Functions

Constructs a continuous-time recurrent neural network, also known as a neural ordinary differential equation (neural ODE), with fast gradient calculation via adjoints [1] and specialized for density estimation based on continuous normalizing flows (CNF) [2] with a direct computation of the trace of the dynamics' jacobian. At a high level this corresponds to the following steps:

1. Parameterize the variable of interest x(t) as a function f(z,θ,t) of a base variable z(t) with known density p_z;
2. Use the transformation of variables formula to predict the density px as a function of the density pz and the trace of the Jacobian of f;
3. Choose the parameter θ to minimize a loss function of p_x (usually the negative likelihood of the data);

!!!note This layer has been deprecated in favor of FFJORD. Use FFJORD with monte_carlo = false instead.

After these steps one may use the NN model and the learned θ to predict the density p_x for new values of x.

DeterministicCNF(model,tspan,basedist=nothing,monte_carlo=false,args...;kwargs...)

Arguments:

• model: A Chain neural network that defines the dynamics of the model.
• basedist: Distribution of the base variable. Set to the unit normal by default.
• tspan: The timespan to be solved on.
• kwargs: Additional arguments splatted to the ODE solver. See the Common Solver Arguments documentation for more details.

Ref [1]L. S. Pontryagin, Mathematical Theory of Optimal Processes. CRC Press, 1987. [2]R. T. Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud. Neural Ordinary Differential Equations. arXiv preprint at arXiv1806.07366, 2019. [3]W. Grathwohl, R. T. Q. Chen, J. Bettencourt, I. Sutskever, D. Duvenaud. FFJORD: Free-Form Continuous Dynamic For Scalable Reversible Generative Models. arXiv preprint at arXiv1810.01367, 2018.

Constructs a continuous-time recurrent neural network, also known as a neural ordinary differential equation (neural ODE), with fast gradient calculation via adjoints [1] and specialized for density estimation based on continuous normalizing flows (CNF) [2] with a stochastic approach [2] for the computation of the trace of the dynamics' jacobian. At a high level this corresponds to the following steps:

1. Parameterize the variable of interest x(t) as a function f(z,θ,t) of a base variable z(t) with known density p_z;
2. Use the transformation of variables formula to predict the density px as a function of the density pz and the trace of the Jacobian of f;
3. Choose the parameter θ to minimize a loss function of p_x (usually the negative likelihood of the data);

After these steps one may use the NN model and the learned θ to predict the density p_x for new values of x.

FFJORD(model,basedist=nothing,monte_carlo=false,tspan,args...;kwargs...)

Arguments:

• model: A Chain neural network that defines the dynamics of the model.
• basedist: Distribution of the base variable. Set to the unit normal by default.
• tspan: The timespan to be solved on.
• kwargs: Additional arguments splatted to the ODE solver. See the Common Solver Arguments documentation for more details.

Ref [1]L. S. Pontryagin, Mathematical Theory of Optimal Processes. CRC Press, 1987. [2]R. T. Q. Chen, Y. Rubanova, J. Bettencourt, D. Duvenaud. Neural Ordinary Differential Equations. arXiv preprint at arXiv1806.07366, 2019. [3]W. Grathwohl, R. T. Q. Chen, J. Bettencourt, I. Sutskever, D. Duvenaud. FFJORD: Free-Form Continuous Dynamic For Scalable Reversible Generative Models. arXiv preprint at arXiv1810.01367, 2018.