The Simulator Invented an Oscillation

Approximate a Changing State by Following Its Rate

BOTIE

A physical model may tell us how quickly a state is changing without giving us a convenient formula for its value later. Before a computer could draw a simulated trajectory, numerical methods gave people a way to approximate one from local changes. Euler's 1768 integral-calculus volume is part of that development.

The step rule named for Euler starts with the current state and its current rate of change. It uses that rate to estimate the change over a chosen interval, then repeats from the new estimate. This makes a changing process calculable in separate steps, but it also assumes the initial rate is adequate for each whole step.

That assumption is what Botie will test. An oversized step can make a cooling process oscillate even when the continuous model settles smoothly. Modern libraries offer several solvers because accuracy and stability depend on the problem and the step rule. We start with explicit Euler so the source of the error is visible.

Sources for this history

A Cooling Pump Became Hotter Than It Started

Botie simulates a pump cooling toward room temperature to generate training examples. With a large timestep, the temperature jumps below room temperature and then above it again. Is the pump oscillating, or did our numerical procedure invent that behaviour?

A simulator approximates continuous change with finite steps. Before learning from its output, we need to inspect the approximation.

Constant-rate scalar cooling model; the result is an approximation, not a measured physical response.

Step 1 of 1

Cooling uses the temperature above ambient

The rate is proportional to the remaining difference.

The initial rate is -5 degrees per second, giving an Euler next value of 25.
Quantity	Value
Temperature	30
Ambient	20
Cooling rate k	0.5 per second
Timestep	1 second

Temperature
Value
30
Ambient
Value
20
Cooling rate k
Value
0.5 per second
Timestep
Value
1 second

The initial rate is -5 degrees per second, giving an Euler next value of 25.

The state and its derivative are different quantities

The derivative is held at its starting value for the duration of one Euler step.

Inspect the simulator’s time step

Botie’s generated cooling records contain an unexpected oscillation.

BOTIE

Botie’s generated cooling records contain an unexpected oscillation.

Choose an example

Current temperature

Ambient temperature

Cooling rate k

Timestep

Temperature after one Euler step

Follow the operations

1. Current temperature
```
30
```
2. Cooling rate k
```
0.5
```
3. Ambient temperature
```
20
```
4. Subtract
```
10
```
5. Multiply
```
5
```
6. Timestep
```
1
```
7. Multiply
```
5
```
8. Subtract
```
25
```
9. Result
```
25
```

Constant-rate scalar cooling model; the result is an approximation, not a measured physical response.

1. The state and its derivative are different quantities

Temperature is the state. Its derivative measures how quickly temperature changes. The cooling rule says a larger difference from ambient produces a larger rate toward ambient.

A simulator stores the current state and advances it repeatedly. A learned surrogate may later approximate that update, so numerical errors in the generated targets can become training errors.

This simplified cooling law assumes a constant ambient temperature and constant rate parameter.

2. Use the current rate for one finite interval

Euler multiplies the current derivative by timestep and adds the resulting change to the current state.

rate = -k × (temperature - ambient)
next = temperature + dt × rate
next = 30 + 1 × (-0.5 × 10) = 25

The true cooling rate changes during the interval. Keeping the starting rate for the whole step introduces discretisation error.

A reference for this particular cooling law

The analytic solution gives us an independent numerical target for a known equation.

Python

import math
exact = 20 + (30-20)*math.exp(-.5*1)
print(round(exact,3))

Output

26.065

One Euler step gave 25. The discrepancy is numerical error under this model, before asking whether the model describes a real pump.

Check yourself

What did the numerical step assume?

Question 1 of 2

What is one Euler step?

Use the information below.

T=30, ambient=20, k=.5, dt=1.

3. A large step can change the apparent physics

The excess temperature is multiplied by one minus k times timestep. Above a product of one, Euler crosses ambient; above two, the excess grows in magnitude with alternating sign.

The exact solution of this particular positive-k cooling model approaches ambient monotonically from above. Its oscillation is therefore a numerical artifact, not evidence of a physical mode.

Euler excess multiplier = 1 - k × dt
exact excess after one step = initial excess × exp(-k × dt)

Compare smaller steps at the same final time. A smaller step changes the numerical approximation; a different final time changes the physical question.

4. Validate the simulator before training on it

For this model, compare against the analytic solution. For harder models, use timestep refinement, independent methods and physical invariants. Separate numerical agreement from agreement with observed equipment.

Do not make the learned model reproduce a coarse simulator and then use agreement with that same simulator as proof of physical accuracy.

The Python rollout exposes the full trajectory. No physical device is controlled and the teaching model omits changing ambient conditions.