ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generated

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A smooth residual can vanish at every collocation point and remain nonzero in population loss

Example

On $U = (0, 1)$ take the residual operator $L [h] = h^{'}$ , target $f = 0$ , boundary operator given by endpoint restriction, and boundary data $g = 0$ . For each integer $m \geq 2$ , set $h_{m} (x) = \frac{1 - \cos (2 π m x)}{2 π m} .$ At the residual collocation points $x_{j} = j / m$ for $1 \leq j < m$ , the residual $L [h_{m}] (x_{j})$ is zero; the exact endpoint values are also zero. Nevertheless, for uniform population measure, the population residual loss is $\int_{0}^{1} ∣ L [h_{m}] (x) ∣^{2} d x = 1 / 2$ . This example concerns unregularized sparse collocation and does not claim that $h_{m}$ minimizes the regularized objective.

Example references

Shin et al., equations (2.4)–(2.6): empirical and population PINN losses

Shin et al., equation (2.5): the population residual loss integrates squared residuals

Facts & Assumptions

Given: The operator, data, collocation set, and trial functions in the Example.

[F1]

The derivative of $h_{m}$ is $h_{m}^{'} (x) = \sin (2 π m x)$ .

[F2]

For every positive integer $m$ , $\int_{0}^{1} \sin^{2} (2 π m x) d x = 1 / 2$ , by $\sin^{2} z = (1 - \cos (2 z)) / 2$ and direct integration.

Verification

Proof strategy: evaluate the derivative at the grid points and integrate its square.

1.1givenF1algebra

Differentiating the displayed formula gives [F1]. At $x_{j} = j / m$ , $h_{m}^{'} (x_{j}) = \sin (2 π j) = 0$ . Also $h_{m} (0) = h_{m} (1) = 0$ since $\cos (0) = \cos (2 π m) = 1$ . Thus the empirical residual and exact boundary errors vanish.

2.1step 1.1F2algebra∎

The population residual loss is $\int_{0}^{1} ∣ h_{m}^{'} (x) ∣^{2} d x = 1 / 2$ by [F2]. Therefore exact collocation fit at these sample points does not by itself force the population residual loss to be small.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Shin, Darbon, and Karniadakis, On the Convergence of Physics-Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs (standard reference, not scraped)