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Finding Nash equilibrial policies for two-player differential games requires solving Hamilton-Jacobi-Isaacs (HJI) PDEs. Self-supervised learning has been used to approximate solutions of such PDEs while circumventing the curse of dimensionality. However, this method fails to learn discontinuous PDE solutions due to its sampling nature, leading to poor safety performance of the resulting controllers in robotics applications when player rewards are discontinuous. This paper investigates two potential solutions to this problem: a hybrid method that leverages both supervised Nash equilibria and the HJI PDE, and a value-hardening method where a sequence of HJIs are solved with a gradually hardening reward. We compare these solutions using the resulting generalization and safety performance in two vehicle interaction simulation studies with 5D and 9D state spaces, respectively. Results show that with informative supervision (e.g., collision and near-collision demonstrations) and the low cost of self-supervised learning, the hybrid method achieves better safety performance than the supervised, self-supervised, and value hardening approaches on equal computational budget. Value hardening fails to generalize in the higher-dimensional case without informative supervision. Lastly, we show that the neural activation function needs to be continuously differentiable for learning PDEs and its choice can be case dependent. 

The Steptoean Positive Carbon Isotope Excursion (SPICE) event at ca. 497−494 Ma was a major carbon-cycle perturbation of the late Cambrian that coincided with rapid diversity changes among trilobites. Several scenarios (e.g., climatic/oceanic cooling and seawater anoxia) have been proposed to account for an extinction of trilobites at the onset of SPICE, but the exact mechanism remains unclear. Here, we present a chemostratigraphic study of carbonate carbon and carbonate-associated sulfate sulfur isotopes (δ13Ccarb and δ34SCAS) and elemental redox proxies (UEF, MoEF, and Corg/P), augmented by secular trilobite diversity data, from both upper slope (Wangcun) and lower slope (Duibian) successions from the Jiangnan Slope, South China, spanning the Drumian to lower Jiangshanian. Redox data indicate locally/regionally well-oxygenated conditions throughout the SPICE event in both study sections except for low-oxygen (hypoxic) conditions within the rising limb of the SPICE (early-middle Paibian) at Duibian. As in coeval sections globally, the reported δ13Ccarb and δ34SCAS profiles exhibit first-order coupling throughout the SPICE event, reflecting co-burial of organic matter and pyrite controlled by globally integrated marine productivity, organic preservation rates, and shelf hypoxia. Increasing δ34SCAS in the “Early SPICE” interval (late Guzhangian) suggests that significant environmental change (e.g., global-oceanic hypoxia) was under way before the global carbon cycle was markedly affected. Assessment of trilobite range data within a high-resolution biostratigraphic framework for the middle-late Cambrian facilitated re-evaluation of the relationship of the SPICE to contemporaneous biodiversity changes. Trilobite diversity in South China declined during the Early SPICE (corresponding to the End-Marjuman Biomere Extinction, or EMBE, of Laurentia) and at the termination of the SPICE (corresponding to the End-Steptoean Biomere Extinction, or ESBE, of Laurentia), consistent with biotic patterns from other cratons. We infer that oxygen minimum zone and/or shelf hypoxia expanded as a result of locally enhanced productivity due to intensified upwelling following climatic cooling, and that expanded hypoxia played a major role in the EMBE at the onset of SPICE. During the SPICE event, global-ocean ventilation promoted marine biotic recovery, but termination of SPICE-related cooling in the late Paibian may have reduced global-ocean circulation, triggering further redox changes that precipitated the ESBE. Major changes in both marine environmental conditions and trilobite diversity during the late Guzhangian demonstrate that the SPICE event began earlier than the Guzhangian-Paibian boundary, as previously proposed.