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Deep learning has emerged as a promising paradigm to give access to highly accurate predictions of molecular and materials properties. A common short-coming shared by current approaches, however, is that neural networks only give point estimates of their predictions and do not come with predictive uncertainties associated with these estimates. Existing uncertainty quantification efforts have primarily leveraged the standard deviation of predictions across an ensemble of independently trained neural networks. This incurs a large computational overhead in both training and prediction that often results in order-of-magnitude more expensive predictions. Here, we propose a method to estimate the predictive uncertainty based on a single neural network without the need for an ensemble. This allows us to obtain uncertainty estimates with virtually no additional computational overhead over standard training and inference. We demonstrate that the quality of the uncertainty estimates matches those obtained from deep ensembles. We further examine the uncertainty estimates of our methods and deep ensembles across the configuration space of our test system and compare the uncertainties to the potential energy surface. Finally, we study the efficacy of the method in an active learning setting and find the results to match an ensemble-based strategy at order-of-magnitude reduced computational cost.

We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester's bijection, which implies a stronger partition theorem. For m=0, our result reduces to Bessenrodt's refinement of Euler's Theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a connection to a generalization of Euler's theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of n with alternating sum k such that the multiplicity of each even part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is also bounded by 2m+1. We provide a combinatorial proof as well., Comment: 15 pages

Stanley defined a partition function t(n) as the number of partitions $\lambda$ of n such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)-t(n). As a consequence, we see that t(n) has the same parity as the ordinary partition function p(n) for any n. A simple combinatorial explanation of this fact is also provided., Comment: 8 pages