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Forward invariance is a long-studied property in control theory that is used to certify that a dynamical system stays within some pre-specified set of states for all time, and also admits robustness guarantees (e.g., the certificate holds under perturbations). We propose a general framework for training and provably certifying robust forward invariance in Neural ODEs. We apply this framework in two settings: certified adversarial robustness for image classification, and certified safety in continuous control. Notably, our method empirically produces superior adversarial robustness guarantees compared to prior work on certifiably robust Neural ODEs (including implicit-depth models). 

Rapidly growing interest in the nanoparticle-mediated delivery of DNA and RNA to plants requires a better understanding of how nanoparticles and their cargoes translocate in plant tissues and into plant cells. However, little is known about how the size and shape of nanoparticles influence transport in plants and the delivery efficiency of their cargoes, limiting the development of nanotechnology in plant systems. In this study we employed non-biolistically delivered DNA-modified gold nanoparticles (AuNPs) of various sizes (5–20 nm) and shapes (spheres and rods) to systematically investigate their transport following infiltration into Nicotiana benthamiana leaves. Generally, smaller AuNPs demonstrated more rapid, higher and longer-lasting levels of association with plant cell walls compared with larger AuNPs. We observed internalization of rod-shaped but not spherical AuNPs into plant cells, yet, surprisingly, 10 nm spherical AuNPs functionalized with small-interfering RNA (siRNA) were the most efficient at siRNA delivery and inducing gene silencing in mature plant leaves. These results indicate the importance of nanoparticle size in efficient biomolecule delivery and, counterintuitively, demonstrate that efficient cargo delivery is possible and potentially optimal in the absence of nanoparticle cellular internalization. Overall, our results highlight nanoparticle features of importance for transport within plant tissues, providing a mechanistic overview of how nanoparticles can be designed to achieve efficacious biocargo delivery for future developments in plant nanobiotechnology.