Solving differential equations with RNNs

To gain some intuition on how we can use RNNs for time series, let us tailor the representation of the solution of a differential equation as a time series.

Consider the famous differential equation (Newton's equation of motion for damped harmonic oscillations, scaled in terms of dimensionless time)

$$\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t),$$

where $\eta$ is a constant used in scaling time into a dimensionless variable and $F(t)$ is an external force acting on the system. The constant $\eta$ is a so-called damping.

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