In machine learning, a neural differential equation is a differential equation whose right-hand side is parametrized by the weights θ of a neural network.
[1] In particular, a neural ordinary differential equation (neural ODE) is an ordinary differential equation of the form
In classical neural networks, layers are arranged in a sequence indexed by natural numbers.
In neural ODEs, however, layers form a continuous family indexed by positive real numbers.
Specifically, the function
maps each positive index t to a real value, representing the state of the neural network at that layer.
Neural ODEs can be understood as continuous-time control systems, where their ability to interpolate data can be interpreted in terms of controllability.
[2] Neural ODEs can be interpreted as a residual neural network with a continuum of layers rather than a discrete number of layers.
[1] Applying the Euler method with a unit time step to a neural ODE yields the forward propagation equation of a residual neural network:
with ℓ being the ℓ-th layer of this residual neural network.
While the forward propagation of a residual neural network is done by applying a sequence of transformations starting at the input layer, the forward propagation computation of a neural ODE is done by solving a differential equation.
More precisely, the output
of the neural ODE is obtained by solving the initial value problem
In physics-informed contexts where additional information is known, neural ODEs can be combined with an existing first-principles model to build a physics-informed neural network model called universal differential equations (UDE).
[3][4][5][6] For instance, an UDE version of the Lotka-Volterra model can be written as[7]
= α x − β x y +
= − γ y + δ x y +
{\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy+f_{\theta }(x(t),y(t)),\\{\frac {dy}{dt}}&=-\gamma y+\delta xy+g_{\theta }(x(t),y(t)),\end{aligned}}}
are correction terms parametrized by neural networks.
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