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. In particular, a neural ordinary differential equation (neural ODE) is an ordinary differential equation of the form

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

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. Applying the Euler method with a unit time step to a neural ODE yields the forward propagation equation of a residual neural network:

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

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

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

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

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). For instance, an UDE version of the Lotka-Volterra model can be written as

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

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

See also

• Physics-informed neural networks

External links

• Steve Brunton lecture on neural ODEs