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We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent control problems with obstacle and collision avoidance. These problems immediately become high-dimensional, even for moderate phase-space dimensions per agent. Our approach fuses the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system. We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generation phase nor solutions from another algorithm. Our model uses empirically effective HJB penalizers for efficient training. By training on a distribution of initial states, we ensure the controls' optimality is achieved on a large portion of the state-space. Our approach is grid-free and scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate our approach's effectiveness on a 150-dimensional multi-agent problem with obstacles., Comment: 6 pages, 3 figures, 2021 European Control Conference

We compare the discretize-optimize (Disc-Opt) and optimize-discretize (Opt-Disc) approaches for time-series regression and continuous normalizing flows (CNFs) using neural ODEs. Neural ODEs are ordinary differential equations (ODEs) with neural network components. Training a neural ODE is an optimal control problem where the weights are the controls and the hidden features are the states. Every training iteration involves solving an ODE forward and another backward in time, which can require large amounts of computation, time, and memory. Comparing the Opt-Disc and Disc-Opt approaches in image classification tasks, Gholami et al. (2019) suggest that Disc-Opt is preferable due to the guaranteed accuracy of gradients. In this paper, we extend the comparison to neural ODEs for time-series regression and CNFs. Unlike in classification, meaningful models in these tasks must also satisfy additional requirements beyond accurate final-time output, e.g., the invertibility of the CNF. Through our numerical experiments, we demonstrate that with careful numerical treatment, Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs. Disc-Opt reduced costs in six out of seven separate problems with training time reduction ranging from 39% to 97%, and in one case, Disc-Opt reduced training from nine days to less than one day., Comment: 19 pages, 8 figures

The lunar cycle has long been suspected to influence biological phenomena. Folklore alludes to such a relationship, but previous scientific analyses have failed to find significant associations. It has been shown that lunar cycles indeed have effects on animals; significant associations between human circadian rhythms and lunar cycles have also been reported. We set out to determine whether a significant statistical correlation exists between the lunar phase and sex determination during conception. We found that significant associations (\textit{p}-value $< 5 \times 10^{-5}$) exist between the average sex ratio (male:female) and the lunar month. The likelihood of conception of a male is at its highest point five days after the full moon, whereas the highest likelihood of female conception occurs nineteen days after the full moon. Furthermore, we found that the strength of this influence is correlated with the amount of solar radiation (which is proportional to moonlight). Our results suggest that sex determination may be influenced by the moon cycle, which suggests the possibility of lunar influence on other biological phenomena. We suggest for future research the exploration of similar effects in other phenomena involving humans and other species., 12 pages, 12 figures