# Differential equations, real time measurements of variables, and ML

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 $$x$$.

• This is quite confusing, have a look at meta.stackexchange.com/questions/66377/what-is-the-xy-problem. Mar 15 at 7:15
• @user2974951 thank you for pointing that out, but I can't figure how else to ask the question. I think it only sounds like an XY problem but isn't one. I've added a tl;dr section which is the only way I think I can make it sound less like an XY problem. Mar 15 at 7:30