Neural differential equation

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

$${\displaystyle {\frac {\mathrm {d} \mathbf {h} (t)}{\mathrm {d} t}}=f_{\theta }(\mathbf {h} (t),t).}$$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 ${\displaystyle h:\mathbb {R} _{\geq 0}\to \mathbb {R} }$ 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]

Connection with residual neural networks

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:

$${\displaystyle \mathbf {h} _{\ell +1}=f_{\theta }(\mathbf {h} _{\ell },\ell )+\mathbf {h} _{\ell },}$$

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 ${\displaystyle \mathbf {h} _{\text{out}}}$ associated to the input ${\displaystyle \mathbf {h} _{\text{in}}}$ of the neural ODE is obtained by solving the initial value problem

$${\displaystyle {\frac {\mathrm {d} \mathbf {h} (t)}{\mathrm {d} t}}=f_{\theta }(\mathbf {h} (t),t),\quad \mathbf {h} (0)=\mathbf {h} _{\text{in}},}$$

and assigning the value ${\displaystyle \mathbf {h} (T)}$ to ${\displaystyle \mathbf {h} _{\text{out}}}$.

Universal differential equations

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]

$${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy+f_{\theta }(x(t),y(t)),\\{\frac {dy}{dt}}&=-\gamma y+\delta xy+g_{\theta }(x(t),y(t)),\end{aligned}}}$$

where the terms ${\displaystyle f_{\theta }}$ and ${\displaystyle g_{\theta }}$ are correction terms parametrized by neural networks.