Finding missing optimal states for completed neural PDE models

Determine how an optimizer can find the missing optimal states introduced by derivative completion when the loss is minimized over a nonconvex neural PDE model.

Background

The paper proves that fixed-width translated-kernel models may have nonattained infima because minimizing sequences converge to derivative combinations that are absent from the original translate-only representation. Derivative completion incorporates these limiting states and restores variational attainment under the stated assumptions.

However, attainment of an optimum in the completed class does not guarantee that parameter optimization will locate it. The paper explicitly identifies the optimizer’s ability to find such missing optimal states as a separate unresolved task, because the loss remains nonconvex in the model parameters.