TL;DR: We measure variable $x$ every $10$ minutes, solve a differential equation $\frac{\mathrm{d}y}{\mathrm{d}t}$ where $y=f(x)$. We are interested in the time it takes for the cumulative value of $y$ to reach a certain value. Let's call that the output $z$. We can repeat the experiment multiple times to get the $z$ for various sequences of measured $x$. Which type of ML model might be best suited to predict $z$ from an ongoing measurement of $x$ in real-time.
We have a differential equation,
$\frac{\mathrm{d}y}{\mathrm{d}t}$ where $y=f(x)$
(We know the value of $y$ at time $t=0$, $y_{t=0}$.)
Tricky part: $x$ is measured in real-time. We get a measurement of its value every $10$ minutes.
x data:
| t (min) | x (units) |
|---|---|
| $0$ | $x_{t=0} = 0.1$ |
| $10$ | 0.3 |
| $20$ | 0.25 |
| ' | ' |
| ' | ' |
Because of this we solve the differential equation at time steps of $10$ minutes and use the value of $y$ at the end of each step as the starting value for the next step.
| t (min) | Starting value of y | Solution of diff. equn |
|---|---|---|
| $0-10$ | $y_{t=0}$ | $y_{t=10}$ |
| $10-20$ | $y_{t=10}$ | $y_{t=20}$ |
| $20-30$ | $y_{t=20}$ | $y_{t=30}$ |
| ' | ' | ' |
| ' | ' | ' |
As a result, we have $y$ vs $t$ corresponding to real-time measurements of $x$.
Next, we calculate values of cumulative $y$ vs $t$, and define $z$ as the value of $t$ at which the value of cumulative $y$ reaches a pre-decided threshold value.
Based on all of that, we have our final data,
| t (min) | x (units) | z (min) |
|---|---|---|
| $0$ | $0.1$ | $180$ |
| $10$ | $0.3$ | |
| $20$ | $0.25$ | |
| ' | ' | |
| ' | ' | |
| $180$ | ' |
We can repeat this experiment multiple times and generate values of $z$ for various sequences of $x$.
Question: What ML model do you think would be better suited for such data? Our objective is to be able to predict $z$ for ongoing real-life measurements of $x$.
