Machine Learning and PINNs

1 ML Roles in Thermofluids

Machine learning can be used for:

surrogate modeling,
parameter identification,
reduced-order modeling,
closure modeling,
inverse heat-transfer problems.

2 PINN Concept

A physics-informed neural network approximates the solution:

[ T(x,t)T_(x,t) ]

The loss includes the PDE residual:

[ _{PDE} = _i ^2 ]

plus initial and boundary condition losses.

3 Example: Coffee ODE PINN

Newton cooling:

[ + (T-T_)=0 ]

Residual:

[ r(t)= + (T_-T_) ]

4 Project Mapping

Project	ML Task	PINN Task
Steel cube	surrogate for final temperature	1D heat equation PINN
Coffee	fit cooling parameters	ODE-PINN
L-pipe	surrogate for outlet temperature	plug-flow ODE-PINN
Cavity	(Nu(Ra)) model	advection-diffusion PINN
Flat plate	learn Nusselt correlation	boundary-layer PINN

5 Practical Advice

Non-dimensionalize variables.
Start with simple ODEs.
Compare with analytical or classical numerical solutions.
Avoid using PINNs as a black box.
Be careful with loss weighting.

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title: "Machine Learning and PINNs"
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## ML Roles in Thermofluids

Machine learning can be used for:

1. surrogate modeling,
2. parameter identification,
3. reduced-order modeling,
4. closure modeling,
5. inverse heat-transfer problems.

## PINN Concept

A physics-informed neural network approximates the solution:

\[
T(x,t)\approx T_\theta(x,t)
\]

The loss includes the PDE residual:

\[
\mathcal{L}_{PDE}
=
\frac{1}{N}
\sum_i
\left[
\frac{\partial T_\theta}{\partial t}
-
\alpha\nabla^2T_\theta
\right]^2
\]

plus initial and boundary condition losses.

## Example: Coffee ODE PINN

Newton cooling:

\[
\frac{dT}{dt}
+
\beta(T-T_\infty)=0
\]

Residual:

\[
r(t)=
\frac{dT_\theta}{dt}
+
\beta(T_\theta-T_\infty)
\]

## Project Mapping

| Project | ML Task | PINN Task |
|---|---|---|
| Steel cube | surrogate for final temperature | 1D heat equation PINN |
| Coffee | fit cooling parameters | ODE-PINN |
| L-pipe | surrogate for outlet temperature | plug-flow ODE-PINN |
| Cavity | \(Nu(Ra)\) model | advection-diffusion PINN |
| Flat plate | learn Nusselt correlation | boundary-layer PINN |

## Practical Advice

1. Non-dimensionalize variables.
2. Start with simple ODEs.
3. Compare with analytical or classical numerical solutions.
4. Avoid using PINNs as a black box.
5. Be careful with loss weighting.