We describe Bayesian Layers, a module designed for fast experimentation with neural network uncertainty. It extends neural network libraries with drop-in replacementsforcommonlayers.

We introduce the STATION, an open-world multi-agent environment for autonomous scientific discovery. The Station simulates a complete scientific ecosystem, where agents can engage in long scientific journeys that include reading papers from peers, formulating hypotheses, collaborating with peers, submitting experiments, and publishing results. Importantly, there is no centralized system coordinating their activities. Utilizing their long context, agents are free to choose their own actions and develop their own narratives within the Station. Experiments demonstrate that AI agents in the Station achieve new state-of-the-art performance on a wide range of benchmarks, spanning mathematics, computational biology, and machine learning, notably surpassing AlphaEvolve in circle packing. A rich tapestry of unscripted narratives emerges, such as agents collaborating and analyzing other works rather than pursuing myopic optimization. From these emergent narratives, novel methods arise organically, such as a new density-adaptive algorithm for scRNA-seq batch integration that borrows concepts from another domain. The Station marks a first step towards autonomous scientific discovery driven by emergent behavior in an open-world environment, representing a new paradigm that moves beyond rigid pipelines.

Machine Learning (ML) is applicable to scientific problems, i.e. to those which have a well defined answer, only if this answer can be brought to a peculiar form ${\cal G}: X\longrightarrow Z$ with ${\cal G}(\vec x)$ expressed as a combination of iterated Heaviside functions. At present it is far from obvious, if and when such representations exist, what are the obstacles and, if they are absent, what are the ways to convert the known formulas into this form. This gives rise to a program of reformulation of ordinary science in such terms -- which sounds like a strong enhancement of the constructive mathematics approach, only this time it concerns all natural sciences. We describe the first steps on this long way.

Rapid robot motion generation is critical in Human-Robot Collaboration (HRC) systems, as robots need to respond to dynamic environments in real time by continuously observing their surroundings and replanning their motions to ensure both safe interactions and efficient task execution. Current sampling-based motion planners face challenges in scaling to high-dimensional configuration spaces and often require post-processing to interpolate and smooth the generated paths, resulting in time inefficiency in complex environments. Optimization-based planners, on the other hand, can incorporate multiple constraints and generate smooth trajectories directly, making them potentially more time-efficient. However, optimization-based planners are sensitive to initialization and may get stuck in local minima. In this work, we present a novel learning-based method that utilizes a Flow Matching model conditioned on a single-view point cloud to learn near-optimal solutions for optimization initialization. Our method does not require prior knowledge of the environment, such as obstacle locations and geometries, and can generate feasible trajectories directly from single-view depth camera input. Simulation studies on a UR5e robotic manipulator in cluttered workspaces demonstrate that the proposed generative initializer achieves a high success rate on its own, significantly improves the success rate of trajectory optimization compared with traditional and learning-based benchmark initializers, requires fewer optimization iterations, and exhibits strong generalization to unseen environments.

We describe Bayesian Layers, a module designed for fast experimentation with neural network uncertainty. It extends neural network libraries with drop-in replacements for common layers. This enables composition via a unified abstraction over deterministic and stochastic functions and allows for scalability via the underlying system. These layers capture uncertainty over weights (Bayesian neural nets), pre-activation units (dropout), activations ("stochastic output layers"), or the function itself (Gaussian processes). They can also be reversible to propagate uncertainty from input to output. We include code examples for common architectures such as Bayesian LSTMs, deep GPs, and flow-based models. As demonstration, we fit a 5-billion parameter "Bayesian Transformer" on 512 TPUv2 cores for uncertainty in machine translation and a Bayesian dynamics model for model-based planning. Finally, we show how Bayesian Layers can be used within the Edward2 language for probabilistic programming with stochastic processes.

For distributed training, the batch statistics are usually estimated locally on a subset of the training minibatch ("ghost batch normalization" [ We now define the three models in full. These inputs first pass through a single fully connected linear layer of width 1000. We then apply a series of residual blocks. LeCun normal initialization [48] to preserve the variance in the absence of non-linearities. We then apply a series of residual blocks.

We analyze the relation between spectral initializers and theoretical limits of \emph{descending} phase retrieval algorithms (dPR). In companion paper [104], for any sample complexity ratio, $α$, \emph{parametric manifold}, ${\mathcal {PM}}(α)$, is recognized as a critically important structure that generically determines dPRs abilities to solve phase retrieval (PR). Moreover, overlap between the algorithmic solution and the true signal is positioned as a key ${\mathcal {PM}}$'s component. We here consider the so-called \emph{overlap optimal} spectral initializers (OptSpins) as dPR's starting points and develop a generic \emph{Random duality theory} (RDT) based program to statistically characterize them. In particular, we determine the functional structure of OptSpins and evaluate the starting overlaps that they provide for the dPRs. Since ${\mathcal {PM}}$'s so-called \emph{flat regions} are highly susceptible to \emph{local jitteriness} and as such are key obstacles on dPR's path towards PR's global optimum, a precise characterization of the starting overlap allows to determine if such regions can be successfully circumvented. Through the presented theoretical analysis we observe two key points in that regard: \textbf{\emph{(i)}} dPR's theoretical phase transition (critical $α$ above which they solve PR) might be difficult to practically achieve as the ${\mathcal {PM}}$'s flat regions are large causing the associated OptSpins to fall exactly within them; and \textbf{\emph{(ii)}} Opting for so-called ``\emph{safer compression}'' and slightly increasing $α$ (by say $15\%$) shrinks flat regions and allows OptSpins to fall outside them and dPRs to ultimately solve PR. Numerical simulations are conducted as well and shown to be in an excellent agreement with theoretical predictions.