When performing predictions that use Machine Learning (ML), we are mainly interested in performance and interpretability. This generates a natural trade-off, where complex models generally have higher skills but are harder to explain and thus trust. Interpretability is particularly important in the climate community, where we aim at gaining a physical understanding of the underlying phenomena. Even more so when the prediction concerns extreme weather events with high impact on society. In this paper, we perform probabilistic forecasts of extreme heatwaves over France, using a hierarchy of increasingly complex ML models, which allows us to find the best compromise between accuracy and interpretability. More precisely, we use models that range from a global Gaussian Approximation (GA) to deep Convolutional Neural Networks (CNNs), with the intermediate steps of a simple Intrinsically Interpretable Neural Network (IINN) and a model using the Scattering Transform (ScatNet). Our findings reveal that CNNs provide higher accuracy, but their black-box nature severely limits interpretability, even when using state-of-the-art Explainable Artificial Intelligence (XAI) tools. In contrast, ScatNet achieves similar performance to CNNs while providing greater transparency, identifying key scales and patterns in the data that drive predictions. This study underscores the potential of interpretability in ML models for climate science, demonstrating that simpler models can rival the performance of their more complex counterparts, all the while being much easier to understand. This gained interpretability is crucial for building trust in model predictions and uncovering new scientific insights, ultimately advancing our understanding and management of extreme weather events.

We provide a new interpretation of Hessian locally linear embedding (HLLE), revealing that it is essentially a variant way to implement the same idea of locally linear embedding (LLE). Based on the new interpretation, a substantial simplification can be made, in which the idea of "Hessian" is replaced by rather arbitrary weights. Moreover, we show by numerical examples that HLLE may produce projection-like results when the dimension of the target space is larger than that of the data manifold, and hence one further modification concerning the manifold dimension is suggested. Combining all the observations, we finally achieve a new LLE-type method, which is called tangential LLE (TLLE). It is simpler and more robust than HLLE.

We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regularization is not supposed to be needed in the original algorithm. The existence of one special type of result, which we call ``projection pattern'', is mathematically proved in the situation that an exact local linear relation is achieved in each neighborhood of the data. These special patterns as well as some other bizarre results that may occur in more general situations are shown by numerical examples on the Swiss roll with a hole embedded in a high dimensional space. It is observed that all these bad results can be effectively prevented by using regularization.

Cerebellar nuclei, substructures of the cerebellum, transfer information from the cerebellum to other parts of the brain. Using single-cell transcriptomics, Kebschull et al. have now identified a conserved pattern of cerebellar nuclei structure that has been repeated through evolution (see the Perspective by Hatten). Ranging from mice to chickens to humans, cerebellar nuclei are made up of region-specific excitatory neurons and region-invariant inhibitory neurons. In humans, a facet connecting the cerebellum to the frontal cortex is enhanced. Science , this issue p. [eabd5059][1]; see also p. [1411][2] ### INTRODUCTION The brains of extant animals have evolved over hundreds of millions of years from simple circuits. Cell types diversified, connections elaborated, and new brain regions emerged. Models for brain region evolution range from duplication of existing regions to splitting of previously multifunctional regions and de novo assembly from existing cell types. These models, however, have not been demonstrated in vertebrate brains at cell-type resolution. ### RATIONALE We investigated brain region evolution using the cerebellar nuclei as a model system. The cerebellum is a major hindbrain structure in jawed vertebrates, comprising the cerebellar cortex and cerebellar nuclei. It is thought to act as a feedforward model for motor control and cognitive processes. The cerebellar cortex receives and processes inputs and sends outputs to the cerebellar nuclei, which route the results of cerebellar computations to the rest of the brain. Whereas the cerebellar cortex is well conserved across vertebrates, the cerebellar nuclei vary in number, with none in jawless vertebrates, one pair in cartilaginous fishes and amphibians, two pairs in reptiles and birds, and three pairs in mammals. This pattern suggests that extant cerebellar nuclei evolved from a single ancestral nucleus. Cerebellar nuclei thus provide a good model to interrogate brain region evolution. ### RESULTS We characterized the cerebellar nuclei in mice, chickens, and humans using whole-brain and spinal cord projection mapping in cleared samples, single-nucleus RNA sequencing (snRNAseq), and spatially resolved transcript amplicon readout mapping (STARmap) analysis. We first compared the projection patterns of the three cerebellar nuclei of mice. Our data reveal broad projections of all nuclei, which in common target regions are shifted relative to each other. To understand the transcriptomic differences that underlie these shifting projections, we produced a cell-type atlas of the mouse cerebellar nuclei using snRNAseq. We discovered three region-invariant inhibitory cell classes and 15 region-specific excitatory cell types. Excitatory cell types fall into two classes with distinct gene expression and electrophysiological properties. Members of each class are present in every nucleus and are putative sister cell types. STARmap analysis in mice revealed that the organizational unit of the cerebellar nuclei is cytoarchitectonically distinguishable subnuclei, each of which contains the three inhibitory and two excitatory classes. To test whether this archetypal subnucleus is also the evolutionary unit of the cerebellar nuclei, we performed snRNAseq and STARmap on the chicken cerebellar nuclei. We identified four subnuclei, three of which had direct orthologs in mice. Each chicken subnucleus contained the same cell-type set of three inhibitory and two excitatory classes already identified in mice, confirming our hypothesis. Cerebellar nuclei vary in size across vertebrates. In particular, the human lateral nucleus is markedly expanded. To understand this expansion, we performed snRNAseq in humans. We found that the medial and interposed nuclei maintained the archetypal cerebellar nuclei composition. However, the lateral nucleus expanded one excitatory cell class at the expense of the other. Conditional tracing in the mouse lateral nucleus revealed that the cell class expanded in humans preferentially accesses lateral frontal cortices via specific intermediate thalamic nuclei. ### CONCLUSION We identified a conserved cell-type set that forms an archetypal cerebellar nucleus as the unit of cerebellar nuclei organization and evolution. We propose that this archetypal nucleus was repeatedly duplicated during evolution, accompanied primarily by transcriptomic divergence of excitatory neurons and shifts in their projection patterns. Our data support a model of duplication-and-divergence of entire cell-type sets for brain region evolution. ![Figure][3] Evolution of the cerebellar nuclei. Comparative single-cell transcriptomics in mice, chickens, and humans (top left; neurons are color-coded by type), spatial transcriptomic analyses in mice and chickens (top right; neurons are color-coded by type in raw and processed data), and central nervous system (CNS)–wide projection mapping in mice (bottom left; axons in red in a three-dimensional mouse brain) revealed the unit of cerebellar nuclei organization and evolution. This unit (red box) comprises three inhibitory and two excitatory neuron classes (each colored circle indicates a neuron class). Extant cerebellar nuclei likely derived from the duplication and divergence of this unit, with more dynamic gene expression in excitatory neurons (changed color hues), along with projection target shifts. How have complex brains evolved from simple circuits? Here we investigated brain region evolution at cell-type resolution in the cerebellar nuclei, the output structures of the cerebellum. Using single-nucleus RNA sequencing in mice, chickens, and humans, as well as STARmap spatial transcriptomic analysis and whole–central nervous system projection tracing, we identified a conserved cell-type set containing two region-specific excitatory neuron classes and three region-invariant inhibitory neuron classes. This set constitutes an archetypal cerebellar nucleus that was repeatedly duplicated to form new regions. The excitatory cell class that preferentially funnels information to lateral frontal cortices in mice becomes predominant in the massively expanded human lateral nucleus. Our data suggest a model of brain region evolution by duplication and divergence of entire cell-type sets. [1]: /lookup/doi/10.1126/science.abd5059 [2]: /lookup/doi/10.1126/science.abf4483 [3]: pending:yes