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The PINN regularization thresholds need not be comparable under the stated count condition

Statement refuted

For $d \geq 2$ , $0 < α \leq 1$ , and positive constants in equation (3.1), the count condition $m_{r} = O (m_{b}^{d / (d - 1)})$ alone does not imply that the two threshold coordinates have bounded max-to-min ratio. In particular, choosing $λ_{m}^{R} = {\hat{λ}}_{m}^{R}$ satisfies the stated coordinatewise lower bound and norm upper bound, while that ratio can diverge.

Statement references

Shin et al., equations (3.1)–(3.2) and Appendix C: threshold and minimizer comparison

Shin et al., Appendix C, equation (21): claimed regularizer-coordinate comparability

Facts & Assumptions

Given: The threshold definitions for $d \geq 2$ , positive fixed $λ_{r}, λ_{b}, c_{r}, c_{b}$ , and the sample-count condition in the Statement.

[F1]

The exact thresholds are ${\hat{λ}}_{r, m}^{R} = \frac{3 λ_{r} (\sqrt{d})^{2 α} c_{r}^{- 2 α / d} m_{r}^{- α / d}}{C_{m}}, {\hat{λ}}_{b, m}^{R} = \frac{3 λ_{b} (\sqrt{d})^{2 α} c_{b}^{- 2 α / (d - 1)} m_{b}^{- α / (d - 1)}}{C_{m}},$ where the same positive $C_{m}$ divides both coordinates (Empirical Hölder-regularized PINN objective).

Proof

Proof strategy: explicit sample-count sequence.

1.1givenF1algebrachoose

Set $m_{r} = m_{b} = n$ for integers $n \geq 1$ . Since $d / (d - 1) > 1$ , we have $n \leq n^{d / (d - 1)}$ , so the required count condition holds. Choose $λ_{m}^{R} = {\hat{λ}}_{m}^{R}$ ; then the coordinatewise lower-bound condition is equality and the max norm is exactly the prescribed comparison scale.

2.1step 1.1F1algebra∎

Cancelling the common factor $C_{m}$ in [F1] gives $\frac{λ_{r, m}^{R}}{λ_{b, m}^{R}} = \frac{{\hat{λ}}_{r, m}^{R}}{{\hat{λ}}_{b, m}^{R}} = \frac{λ_{r} c_{r}^{- 2 α / d}}{λ_{b} c_{b}^{- 2 α / (d - 1)}} n^{α / (d - 1) - α / d} = C n^{α / (d (d - 1))},$ where $C > 0$ is independent of $n$ . The exponent is positive, so the ratio tends to infinity. Thus the two coordinates need not be comparable under the stated count condition.

Depends on

Empirical Hölder-regularized PINN objective

Used by

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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)