If the memory state can be addressed by its content, itisfurthermore called content-addressable.

Neural networks are widely used, yet their analysis and verification remain challenging. In this work, we present a Lean 4 formalization of neural networks, covering both deterministic and stochastic models. We first formalize Hopfield networks, recurrent networks that store patterns as stable states. We prove convergence and the correctness of Hebbian learning, a training rule that updates network parameters to encode patterns, here limited to the case of pairwise-orthogonal patterns. We then consider stochastic networks, where updates are probabilistic and convergence is to a stationary distribution. As a canonical example, we formalize the dynamics of Boltzmann machines and prove their ergodicity, showing convergence to a unique stationary distribution using a new formalization of the Perron-Frobenius theorem.

Here, we are motivated by both the Hopfield network and by strong error-correcting codes (ECCs).

Visual planning, by offering a sequence of intermediate visual subgoals to a goal-conditioned low-level policy, achieves promising performance on long-horizon manipulation tasks. To obtain the subgoals, existing methods typically resort to video generation models but suffer from model hallucination and computational cost. We present Vis2Plan, an efficient, explainable and white-box visual planning framework powered by symbolic guidance. From raw, unlabeled play data, Vis2Plan harnesses vision foundation models to automatically extract a compact set of task symbols, which allows building a high-level symbolic transition graph for multi-goal, multi-stage planning. At test time, given a desired task goal, our planner conducts planning at the symbolic level and assembles a sequence of physically consistent intermediate sub-goal images grounded by the underlying symbolic representation. Our Vis2Plan outperforms strong diffusion video generation-based visual planners by delivering 53\% higher aggregate success rate in real robot settings while generating visual plans 35$\times$ faster. The results indicate that Vis2Plan is able to generate physically consistent image goals while offering fully inspectable reasoning steps.

Project and task scheduling under uncertainty remains a fundamental challenge in program and project management, where accurate estimation of task durations and dependencies is critical for delivering complex, multi project systems. The Program Evaluation and Review Technique provides a probabilistic framework to model task variability and critical paths. In this paper, the author presents a novel formulation of PERT scheduling as an energy minimization problem within a Hopfield neural network architecture. By mapping task start times and precedence constraints into a neural computation framework, the networks inherent optimization dynamics is exploited to approximate globally consistent schedules. The author addresses key theoretical issues related to energy function differentiability, constraint encoding, and convergence, and extends the Hopfield model for structured precedence graphs. Numerical simulations on synthetic project networks comprising up to 1000 tasks demonstrate the viability of this approach, achieving near optimal makespans with minimal constraint violations. The findings suggest that neural optimization models offer a promising direction for scalable and adaptive project tasks scheduling under uncertainty in areas such as the agentic AI workflows, microservice based applications that the modern AI systems are being built upon.

The scramjet engine is a key propulsion system for hypersonic vehicles, leveraging supersonic airflow to achieve high specific impulse, making it a promising technology for aerospace applications. Understanding and controlling the complex interactions between fuel injection, turbulent combustion, and aerodynamic effects of compressible flows are crucial for ensuring stable combustion in scramjet engines. However, identifying stable modes in scramjet combustors is often challenging due to limited experimental measurement means and extremely complex spatiotemporal evolution of supersonic turbulent combustion. This work introduces an innovative deep learning framework that combines dimensionality reduction via the Residual Convolutional Neural Network-beta-Variational Autoencoder (Res-CNN-beta-VAE) model with unsupervised clustering (K-means) to identify and analyze dynamical combustion modes in a supersonic combustor. By mapping high-dimensional data of combustion snapshots to a reduced three-dimensional latent space, the Res-CNN-beta-VAE model captures the essential temporal and spatial features of flame behaviors and enables the observation of transitions between combustion states. By analyzing the standard deviation of latent variable trajectories, we introduce a novel method for objectively distinguishing between dynamic transitions, which provides a scalable and expert-independent alternative to traditional classification methods. Besides, the unsupervised K-means clustering approach effectively identifies the complex interplay between the cavity and the jet-wake stabilization mechanisms, offering new insights into the system's behavior across different gas-to-liquid mass flow ratios (GLRs).

Soft robots are distinguished by their flexible and adaptable, allowing them to perform tasks that are nearly impossible for rigid robots. However, controlling their configuration is challenging due to their nonlinear material response and infinite deflection degrees of freedom. A potential solution is to discretize the infinite-dimensional configuration space of soft robots into a finite but sufficiently large number of functional shapes. This study explores a co-design strategy for pneumatically actuated soft grippers with multiple encoded stable states, enabling desired functional shape and stiffness reconfiguration. An energy based analytical model for soft multistable grippers is presented, mapping the robots' infinite-dimensional configuration space into discrete stable states, allowing for prediction of the systems final state and dynamic behavior. Our approach introduces a general method to capture the soft robots' response with the lattice lumped parameters using automatic relevance determination regression, facilitating inverse co-design. The resulting computationally efficient model enables us to explore the configuration space in a tractable manner, allowing the inverse co-design of our robots by setting desired targeted positions with optimized stiffness of the set targets. This strategy offers a framework for controlling soft robots by exploiting the nonlinear mechanics of multistable structures, thus embodying mechanical intelligence into soft structures.

Dialectical frameworks are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. Their generality allow them to cover a number of different approaches with varying forms of representing the argumentation structure. Boolean regulatory networks are used to model the dynamics of complex biological processes, taking into account the interactions of biological compounds, such as proteins or genes. These models have proven highly useful for comprehending such biological processes, allowing to reproduce known behaviour and testing new hypotheses and predictions in silico, for example in the context of new medical treatments. While both these approaches stem from entirely different communities, it turns out that there are striking similarities in their appearence. In this paper, we study the relation between these two formalisms revealing their communalities as well as their differences, and introducing a correspondence that allows to establish novel results for the individual formalisms.

The learning sparse gating network selects a set of experts for each token to be processed; however, this may lead to differences in the number of tokens processed by each expert over several successive iterations, i.e., the expert load fluctuations, which reduces computational parallelization and resource utilization. To this end, we traced and analyzed loads of each expert in the training iterations for several large language models in this work, and defined the transient state with "obvious load fluctuation" and the stable state with "temporal locality". Moreover, given the characteristics of these two states and the computational overhead, we deployed three classical prediction algorithms that achieve accurate expert load prediction results. For the GPT3 350M model, the average error rates for predicting the expert load proportion over the next 1,000 and 2,000 steps are approximately 1.3% and 1.8%, respectively. This work can provide valuable guidance for expert placement or resource allocation for MoE model training. Based on this work, we will propose an expert placement scheme for transient and stable states in our coming work.

One of the main goals of developmental biology is to reveal the gene regulatory networks (GRNs) underlying the robust differentiation of multipotent progenitors into precisely specified cell types. Most existing methods to infer GRNs from experimental data have limited predictive power as the inferred GRNs merely reflect gene expression similarity or correlation. Here, we demonstrate, how physics-informed neural networks (PINNs) can be used to infer the parameters of predictive, dynamical GRNs that provide mechanistic understanding of biological processes. Specifically we study GRNs that exhibit bifurcation behavior and can therefore model cell differentiation. We show that PINNs outperform regular feed-forward neural networks on the parameter inference task and analyze two relevant experimental scenarios: 1. a system with cell communication for which gene expression trajectories are available and 2. snapshot measurements of a cell population in which cell communication is absent. Our analysis will inform the design of future experiments to be analyzed with PINNs and provides a starting point to explore this powerful class of neural network models further.