Transcription of Neural Ordinary Differential Equations
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Neural Ordinary Differential Equations
Neural Ordinary Differential EquationsRicky T. Q. Chen*, Yulia Rubanova*, Jesse Bettencourt*, David DuvenaudUniversity of TorontoBackground: Ordinary Differential Equations (ODEs)-Model the instantaneous change of a state.(explicit form)-Solving an initial value problem (IVP) corresponds to integration.(solution is a trajectory)-Euler method approximates with small steps:Residual Networks interpreted as an ODE Solver-Hidden units look like:-Final output is the composition:Haber & Ruthotto (2017). E (2017). Residual Networks interpreted as an ODE Solver-Hidden units look like:-Final output is the composition:-This can be interpreted as an Euler discretization of an & Ruthotto (2017). E (2017). -In the limit of smaller steps:Deep Learning as Discretized Differential EquationsMany deep learning networks can be interpreted as ODE Numerical SchemeResNet, RevNet, ResNeXt, EulerPolyNetApproximation to Backward EulerFractalNetRunge-KuttaDenseNetRunge- KuttaLu et al.
Continuous Normalizing Flows Instantaneous Change of variables (iCOV): - For a Lipschitz continuous function - In other words, With an invertible F: Continuous Normalizing Flows 1D: 2D: Data Discrete-NF CNF. Is the ODE being correctly solved? Stochastic Unbiased Log Density.
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