Collocation points
Where you decide the equation matters.
1 min read
Collocation points are where you decide the equation matters. Get their placement wrong and the network will satisfy the residual everywhere you looked and nowhere you didn't.
Sampling strategies
| Strategy | Cost | When it helps |
|---|---|---|
| Uniform grid | Cheap | Smooth solutions, low dimension |
| Latin hypercube | Cheap | Better coverage in 3D and above |
| Adaptive (RAR) | Expensive | Shocks, boundary layers, steep fronts |
Residual-based adaptive refinement adds points where $|r_\theta|$ is largest, which is exactly where the network is currently wrong. In practice, resampling every few thousand iterations is enough.