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Trained neural networks (NN) are attractive as surrogate models to replace costly calculations in physical simulations, but are often unknowingly applied to states not adequately represented in the training dataset. We present the novel technique of soft checksums for scientific machine learning, a general-purpose method to differentiate between trustworthy predictions with small errors on in-distribution (ID) data points, and untrustworthy predictions with large errors on out-of-distribution (OOD) data points. By adding a check node to the existing output layer, we train the model to learn the chosen checksum function encoded within the NN predictions and show that violations of this function correlate with high prediction errors. As the checksum function depends only on the NN predictions, we can calculate the checksum error for any prediction with a single forward pass, incurring negligible time and memory costs. Additionally, we find that incorporating the checksum function into the loss function and exposing the NN to OOD data points during the training process improves separation between ID and OOD predictions. By applying soft checksums to a physically complex and high-dimensional non-local thermodynamic equilibrium atomic physics dataset, we show that a well-chosen threshold checksum error can effectively separate ID and OOD predictions.

PHOTO: 2 color, 2 b&w BY KENT PORTER/PRESS DEMOCRAT MAP: b&w no credit Wildland fire; Caption: 1. California Department of Forestry firefighters from Middletown guide a backfire away from a home on Butts Canyon Road. 2. A structure burns along Butts Canyon Road in Middletown on Saturday evening, as a 2,500-acre wildfire raged through Butts Canyon in Lake County. 3. Two firefighters from the U.S. Forest Service light a backfire Saturday night to ward off an approaching blaze in an effort to protect a structure on Butts Canyon Road near Middletown. 4. A ranch worker drives off in an ATV in an attempt to round up cattle Saturday on Butts Canyon Road.

In this note we prove that the level-set flow of the topologist’s sine curve is a smooth closed curve. In Lauer (Geom Funct Anal 23(6): 1934–1961, 2013 ) it was shown by the second author that under the level-set flow, a locally connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we give the first example of a domain whose boundary is not locally connected for which the level-set flow is instantaneously smooth. Our methods also produce an example of a nonpath-connected set that instantly evolves into a smooth closed curve.

In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we give the first example of a domain whose boundary is not locally-connected for which the level-set flow is instantaneously smooth. Our methods also produce an example of a non path-connected set that instantly evolves into a smooth closed curve.

This article demonstrates that state collective bargaining laws are important determinants of union and nonunion public employee compensation. State laws that provide stronger bargaining rights and ensure closure to the bargaining process increase the direct effect of police unions on compensation. Moreover, indirect threat effects on the pay of nonunion police also increase with stronger bargaining laws. In each law category investigated, nonunion police receive most of the compensation premium enjoyed by unionized police. Previous studies that have not adequately controlled for these effects of bargaining laws have therefore underestimated the full effect of public-sector unions on compensation.

This article demonstrates that state collective bargaining laws are important determinants of union and nonunion public employee compensation. State laws that provide stronger bargaining rights and ensure closure to the bargaining process increase the direct effect of police unions on compensation. Moreover, indirect threat effects on the pay of nonunion police also increase with stronger bargaining laws. In each law category investigated, nonunion police receive most of the compensation premium enjoyed by unionized police. Previous studies that have not adequately controlled for these effects of bargaining laws have therefore underestimated the full effect of public-sector unions on compensation.