Deep neural networks (DNNs) exhibit crackling-like avalanches whose origin lacks a mechanistic explanation. Here, I derive a stochastic theory of deep information propagation (DIP) by incorporating Central Limit Theorem (CLT)-level fluctuations. Four effective couplings $(r, h, D_1, D_2)$ characterize the dynamics, yielding a Landau description of the static exponents and a Directed Percolation (DP) structure of activity cascades. Tuning the couplings selects between avalanche dynamics generated by a Brownian Motion (BM) in a logarithmic trap and an absorbed free BM, each corresponding to a distinct universality classes. Numerical simulations confirm the theory and demonstrate that activation function design controls the collective dynamics in random DNNs.

Abstract--Anthropomorphic underactuated hands are valued for their structural simplicity and inherent adaptability. However, the uncertainty arising from interdependent joint motions makes it challenging to capture various grasp states during hand-object interaction without increasing structural complexity through multiple embedded sensors. This motivates the need for an approach that can extract rich grasp-state information from a single sensing source while preserving the simplicity of underactuation. This study proposes an anthropomorphic underactuated hand that achieves comprehensive grasp state estimation, using only tendon-based proprioception provided by series elastic actuators (SEAs). Our approach is enabled by the design of a compact SEA with high accuracy and reliability that can be seamlessly integrated into sensorless fingers. By coupling accurate proprioceptive measurements with potential energy-based modeling, the system estimates multiple key grasp state variables, including contact timing, joint angles, relative object stiffness, and external disturbances. Finger-level experimental validations and extensive hand-level grasp functionality demonstrations confirmed the effectiveness of the proposed approach. NTHROPOMORPHIC robotic hands have been widely adopted to replicate the functionality of the human hand. Among various actuation strategies, underactuated hands are extensively employed due to their structural simplicity and adaptability to diverse object geometries [1], [2].

Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems utilize teams of large language models (LLMs) to generate candidate solutions to a problem as human readable code. These candidate solutions are then 'evolved' with the goal of improving them beyond what an LLM can produce in a single shot. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop. We describe some lessons learned for others in the field interested in exploring the use of these systems.

The move-acceptance hyper-heuristic was recently shown to be able to leave local optima with astonishing efficiency (Lissovoi et al., Artificial Intelligence (2023)). In this work, we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic Cliff$_d$ and Jump$_m$ function classes. (i) Instead of randomly choosing between the only-improving and any-move acceptance operator, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on Jump$_m$ functions with gap parameter $m$ from $Ω(n^{2m-1})$ to $O(n^{m+1})$. (ii) We then replace the all-moves acceptance operator with the operator that only accepts worsenings. Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to $O(n^3 \log n)$ independent of the gap size. In general, we prove a remarkably good runtime of $O(n^{k+1} \log n)$ for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT$_k$, which contains a large number of functions having $k$ successive local optima, and which contains the commonly studied Jump$_m$ and Cliff$_d$ functions for $k=2$.

We introduce Local Reinforcement-Based Selection of Auxiliary Objectives (LRSAO), a novel approach that selects auxiliary objectives using reinforcement learning (RL) to support the optimization process of an evolutionary algorithm (EA) as in EA+RL framework and furthermore incorporates the ability to unlearn previously used objectives. By modifying the reward mechanism to penalize moves that do no increase the fitness value and relying on the local auxiliary objectives, LRSAO dynamically adapts its selection strategy to optimize performance according to the landscape and unlearn previous objectives when necessary. We analyze and evaluate LRSAO on the black-box complexity version of the non-monotonic Jump function, with gap parameter $\ell$, where each auxiliary objective is beneficial at specific stages of optimization. The Jump function is hard to optimize for evolutionary-based algorithms and the best-known complexity for reinforcement-based selection on Jump was $O(n^2 \log(n) / \ell)$. Our approach improves over this result to achieve a complexity of $\Theta(n^2 / \ell^2 + n \log(n))$ resulting in a significant improvement, which demonstrates the efficiency and adaptability of LRSAO, highlighting its potential to outperform traditional methods in complex optimization scenarios.

We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and $\tanh$), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations.

This paper presents a computational method for generating virtual 3D morphologies of functional materials using low-parametric stochastic geometry models, i.e., digital twins, calibrated with 2D microscopy images. These digital twins allow systematic parameter variations to simulate various morphologies, that can be deployed for virtual materials testing by means of spatially resolved numerical simulations of macroscopic properties. Generative adversarial networks (GANs) have gained popularity for calibrating models to generate realistic 3D morphologies. However, GANs often comprise of numerous uninterpretable parameters make systematic variation of morphologies for virtual materials testing challenging. In contrast, low-parametric stochastic geometry models (e.g., based on Gaussian random fields) enable targeted variation but may struggle to mimic complex morphologies. Combining GANs with advanced stochastic geometry models (e.g., excursion sets of more general random fields) addresses these limitations, allowing model calibration solely from 2D image data. This approach is demonstrated by generating a digital twin of all-solid-state battery (ASSB) cathodes. Since the digital twins are parametric, they support systematic exploration of structural scenarios and their macroscopic properties. The proposed method facilitates simulation studies for optimizing 3D morphologies, benefiting not only ASSB cathodes but also other materials with similar structures.

Sequential design of real and computer experiments via Gaussian Process (GP) models has proven useful for parsimonious, goal-oriented data acquisition purposes. In this work, we focus on acquisition strategies for a GP model that needs to be accurate within a predefined range of the response of interest. Such an approach is useful in various fields including synthetic chemistry, where finding molecules with particular properties is essential for developing useful materials and effective medications. GP modeling and sequential design of experiments have been successfully applied to a plethora of domains, including molecule research. Our main contribution here is to use the threshold-weighted Continuous Ranked Probability Score (CRPS) as a basic building block for acquisition functions employed within sequential design. We study pointwise and integral criteria relying on two different weighting measures and benchmark them against competitors, demonstrating improved performance with respect to considered goals. The resulting acquisition strategies are applicable to a wide range of fields and pave the way to further developing sequential design relying on scoring rules.

To help with my holiday shopping this year, I recently turned to a new personal assistant online. "I'm looking for a Christmas present for my mother, who spends long hours working," I typed. "Is there something she can use in her office every day?" "Of course!" came the instant reply. "Does your mother have any specific preferences or needs for her office? For example, does she need organization tools, desk accessories, or something to help her relax during breaks?"

In this study, we explore the application of Topological Data Analysis (TDA) and Lipschitz-Killing Curvatures (LKCs) as powerful tools for feature extraction and classification in the context of biomedical multiomics problems. TDA allows us to capture topological features and patterns within complex datasets, while LKCs provide essential geometric insights. We investigate the potential of combining both methods to improve classification accuracy. Using a dataset of biomedical images, we demonstrate that TDA and LKCs can effectively extract topological and geometrical features, respectively. The combination of these features results in enhanced classification performance compared to using each method individually. This approach offers promising results and has the potential to advance our understanding of complex biological processes in various biomedical applications. Our findings highlight the value of integrating topological and geometrical information in biomedical data analysis. As we continue to delve into the intricacies of multiomics problems, the fusion of these insights holds great promise for unraveling the underlying biological complexities.