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1. Birth and Death of a Rose

   Geng, Chen; Zhang, Yunzhi; Wu, Shangzhe; Wu, Jiajun (June 2025, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR))
2. Global convergence of neuron birth-death dynamics

   Rotskoff, Grant; Jelassi, S; Bruna, Joan; Vanden-Eijnden, Eric. (June 2019, International Conference on Machine Learning)

   Neural networks with a large number of units ad- mit a mean-field description, which has recently served as a theoretical explanation for the favor- able training properties of “overparameterized” models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appro- priate assumptions. In this work, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We im- plement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean- field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of units. We illustrate our algorithms with two models to provide intuition for the mech- anism through which convergence is accelerated 

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3. Identifiability and inference of phylogenetic birth–death models

   https://doi.org/10.1016/j.jtbi.2023.111520  

   Legried, Brandon; Terhorst, Jonathan (July 2023, Journal of Theoretical Biology)
4. A class of identifiable phylogenetic birth–death models

   https://doi.org/10.1073/pnas.2119513119  

   Legried, Brandon; Terhorst, Jonathan (August 2022, Proceedings of the National Academy of Sciences)

   In a striking result, Louca and Pennell [S. Louca, M. W. Pennell, Nature 580, 502–505 (2020)] recently proved that a large class of phylogenetic birth–death models is statistically unidentifiable from lineage-through-time (LTT) data: Any pair of sufficiently smooth birth and death rate functions is “congruent” to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. As Louca and Pennell argue, this fact has distressing implications for the thousands of studies that have utilized birth–death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth–death models is indeed identifiable. Specifically, we show that piecewise constant birth–death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant timetree and some knowledge of the present-day population. Subject to mild regularity conditions, we further show that any unidentifiable birth–death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic. 

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5. Unifying Phylogenetic Birth–Death Models in Epidemiology and Macroevolution

   https://doi.org/10.1093/sysbio/syab049  

   MacPherson, Ailene; Louca, Stilianos; McLaughlin, Angela; Joy, Jeffrey B; Pennell, Matthew W (June 2021, Systematic Biology)

   Albert, James (Ed.)

   Abstract Birth–death stochastic processes are the foundations of many phylogenetic models and are widely used to make inferences about epidemiological and macroevolutionary dynamics. There are a large number of birth–death model variants that have been developed; these impose different assumptions about the temporal dynamics of the parameters and about the sampling process. As each of these variants was individually derived, it has been difficult to understand the relationships between them as well as their precise biological and mathematical assumptions. Without a common mathematical foundation, deriving new models is nontrivial. Here, we unify these models into a single framework, prove that many previously developed epidemiological and macroevolutionary models are all special cases of a more general model, and illustrate the connections between these variants. This unification includes both models where the process is the same for all lineages and those in which it varies across types. We also outline a straightforward procedure for deriving likelihood functions for arbitrarily complex birth–death(-sampling) models that will hopefully allow researchers to explore a wider array of scenarios than was previously possible. By rederiving existing single-type birth–death sampling models, we clarify and synthesize the range of explicit and implicit assumptions made by these models. [Birth–death processes; epidemiology; macroevolution; phylogenetics; statistical inference.] 

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