Neural Network

We present a data-adaptive method for parameter-efficient fine-tuning of large neural networks. Standard low-rank adaptation methods improve efficiency by restricting each layer update to a fixed low-rank form, but this static parameterization can be too rigid when the appropriate correction depends on the input and on the evolving depth-wise computation of the network. Our approach replaces a purely layer-local adapter with a shared queryable memory of low-rank update atoms. For each block of layers, the model forms a query from the current low-rank state and a running summary of previous blocks, uses this query to retrieve a content-dependent combination of shared update components via attention, and applies the resulting routed operator within the low-rank bottleneck. In this way, the method retains the efficiency and scalability of low-rank adaptation while allowing the effective update to vary across inputs and to share reusable structure across layers. The resulting architecture provides a principled middle ground between static LoRA-style updates and fully generated parameter updates: it remains compact and parameter-efficient while supporting dynamic, context-sensitive adaptation. Further, we incorporate instruction-regularization by augmenting routing logits with a language-induced prior over update atoms, thereby biasing the selection of low-rank transformations toward semantically relevant directions without generating unconstrained parameter updates. Experiments on noisy non-linear regression tasks and LLM fine-tuning suggest that this queryable update-memory formulation can improve final test performance and training stability compared to standard low-rank adaptation, while using a comparable number of trainable parameters.

This paper proposes a time-warping transfer learning method, a technique for temporally rescaling the learned dynamics of a recurrent neural network (RNN) with a Long Short-Term Memory (LSTM) layer to enable task transfer across fuel moisture classes. Fuel moisture content (FMC) is divided into idealized classes based on characteristic lag time. Large quantities of real-time data are available for 10h fuels from sensors on weather stations, but observations of other fuel classes are sparse in space and time. We use transfer learning to adapt an RNN pretrained on 10h FMC to predict FMC for 1h, 100h, and 1000h fuels. We validate this method using data from a landmark field study conducted in Oklahoma that was used to calibrate the state-of-the-art Nelson fuel moisture model.

A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a ``Newton neural tangent kernel'' (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data (i.e., a global minimizer with zero loss). We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. The eigenvalues of the NTK for gradient descent accumulate at zero, leading to slow convergence for target data with high-frequency components. In contrast, the NNTK has uniformly lower bounded eigenvalues if the regularization parameter is selected appropriately, allowing Newton's method to converge more quickly for data with high-frequency components. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows. This complicates deriving the training dynamics in the overparameterized limit as well as proving the convergence of the finite-width dynamics thereto. The analysis identifies a scaling formula for selecting the regularization parameter, which we show can vanish at a suitable rate as the number of hidden units becomes larger. We prove that, for sufficiently large numbers of hidden units, the regularized Hessian remains positive definite during training and the Newton updates for individual NN parameters converge to zero, showing that the model behaves as a linearization around the initialization.

In this paper, we propose a modular framework for 6D pose estimation based on keypoint heatmap regression. Our approach combines YOLOv10m for object detection with a ResNet18-based network that predicts 2D heatmaps from RGB images. Keypoints extracted from these heatmaps are used to estimate the 6D object pose via the PnP RANSAC algorithm. We compare different keypoint selection strategies to assess their impact on pose accuracy. Additionally, we extend the baseline by incorporating depth data using a cross-fusion architecture, which enables interaction between RGB and depth features at multiple stages. We further explore general training improvements, such as experimenting with activation functions and learning rate scheduling strategies to improve model performance. Our best RGB-only model achieved a mean ADD-based accuracy of 84.50%, while the RGB-D fusion model reached 92.41% on the LINEMOD dataset. The code is available at https://github.com/ameermasood/HeatNet.

Diabetic Retinopathy (DR) is a common complication of diabetes that can lead to blindness of people. Detecting DR at the earliest stage is essential to prevent irreversible eye damage. Microaneurysm dots are the first signs of DR. As the dots are tiny and of low contrast, detecting mild DR is a very challenging task. Federated learning (FL) preserves data privacy, which is a major concern for medical image processing. FL is a collaborative learning method, which shares only the model parameters with a server, without sharing the patient data to a central server. Inspired by classical FL, we propose a federated learning-based quantum neural network (federated QNN) for this task. We implemented the models with limited samples and few learnable parameters from the E-ophtha and Retina MNIST datasets. The crossevaluation efficiency of the proposed federated quantum neural network system for privacy-preserving early detection of diabetic retinopathy (FQPDR) in Kaggle dataset images indicates the robustness of the light weight learning models. FQPDR performances are inspiring while considering existing non-FL and FL methods.

We study the evolution of hidden-weight spectra in wide neural networks trained by (stochastic) gradient descent. We develop a two-level dynamical mean-field theory (DMFT) that jointly tracks bulk and outlier spectral dynamics for spiked ensembles whose spike directions remain statistically dependent on the random bulk. We apply this framework to two settings: (1) infinite-width nonlinear networks in mean-field/$μ$P scaling and (2) deep linear networks in the proportional high-dimensional limit, where width, input dimension, and sample size diverge with fixed ratios. Our theory predicts how outliers evolve with training time, width, output scale, and initialization variance. In deep linear networks, $μ$P yields width-consistent outlier dynamics and hyperparameter transfer, including width-stable growth of the leading NTK mode toward the edge of stability (EoS). In contrast, NTK parameterization exhibits strongly width-dependent outlier dynamics, despite converging to a stable large-width limit. We show that this bulk+outlier picture is descriptive of simple tasks with small output channels, but that tasks involving large numbers of outputs (ImageNet classification or GPT language modeling) are better described by a restructuring of the spectral bulk. We develop a toy model with extensive output channels that recapitulates this phenomenon and show that edge of the spectrum still converges for sufficiently wide networks.

Neural network verification is an active and rapidly maturing research area, with a growing ecosystem of solvers and tools. The VNN-LIB standard was introduced to support interoperability in this ecosystem, but Version1.0 has several serious short-comings as a formal foundation: it lacks a precise syntax, semantics, and type system, offers limited expressivity, and relies on externally defined ONNX models whose semantics are informal and constantly evolving. The latter distinguishes VNN-LIB from established standards such as SMT-LIB, where queries are self-contained and have fixed semantics. In this paper we address these challenges by developing the theoretical foundations of VNN-LIB2.0. Our key contribution is the introduction of the notion of a \emph{network theory}, which abstractly characterises the minimal semantic interface required from a neural network model format. This abstraction enables VNN-LIB to be defined independently of any specific ONNX version while remaining compatible with evolving model representations. Building on this foundation, we present a formal syntax for a more expressive query language, a type system for it over the numeric domains provided by the network theory, and finally a formal semantics. To ensure internal consistency, the standard is mechanised in the Agda theorem prover. VNN-LIB~2.0 therefore provides robust and rigorous foundations for trustworthy neural network verification.

Random-feature neural networks (RFNNs), including architectures with fixed hidden layers and analytically determined output weights, offer fast training but often suffer from issues due to dense representations of the hidden layer activation. Their reliance on dense feature mappings and least squares solvers can limit scalability and numerical stability, particularly for high-dimensional or stiff systems. Specifically, the activation matrix is observed to be low-rank and extremely ill-conditioned. In this work, we propose a sparse framework for RFNNs that integrates structured sparsity into the hidden layer activations that increases the rank and employs Sparse Singular Value Decomposition (sSVD) for solving the resulting linear least squares problem scalably and efficiently while catering to the bad condition number. We explore the theory behind Lanczos-Golub-Kahan Bidiagonalization technique for sparse SVD and conduct some experiments to identify some limitations and justify the requirement for orthogonalization step in our application. Then, we demonstrate that the proposed method maintains or improves solution accuracy for solving the benchmark one-dimensional steady convection-diffusion equations case having stronger advection, while achieving substantial gains in training efficiency and robustness compared to standard dense implementations.

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity $\mathcal{O}(n^2 + ne)$, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to $10.6 \times$ compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

Individual treatment effects are not point-identified from data. The Probability of Necessity and Sufficiency (PNS) circumvents this limitation by characterizing individual-level causality through intersection bounds derived from combined experimental and observational data. In finite samples, however, standard plug-in estimators systematically fail: they violate structural probability constraints and suffer from extremum bias induced by max-min operators, yielding spuriously narrow intervals. We propose a neural framework for finite-sample PNS estimation that resolves both pathologies. We introduce an anchored neural architecture that guarantees structural constraint satisfaction by construction. To correct extremum bias, we employ precision-corrected intersection-bound inference, leveraging Epistemic Neural Networks for scalable, high-dimensional uncertainty quantification. Empirical evaluations confirm that this approach maintains nominal coverage and exact constraint validity in high-dimensional regimes where standard estimators systematically undercover.

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.

Building on the abstract Generalized Singular Value Decomposition (GSVD) theory of Brown et al. [2025], we prove that most modern neural architectures admit a generalized SVD representation in which they are left-invertible before a final linear layer, with no change in input-output behavior. Furthermore, the left-invertible nonlinear portion of the input-output behavior can be made to be \emph{norm preserving}, meaning that perturbations in the left-invertible ``embedding'' (the activations prior to the final linear layer in this representation) correspond proportionally to changes in the input, i.e., distance in feature space can be calibrated directly to distance in input space. We provide a data-driven algorithm for estimating this representation from trained models and propose a model architecture that naturally facilitates the decomposition. We then provide a proof-of-concept that the learned representation can be used to identify adversarial perturbations to model inputs, and develop the theory necessary for future applications to areas such as model bias and invertibility.

It has been known for a long time that initializing weight matrices to be orthogonal instead of having i.i.d. Gaussian components can improve training performance. This phenomenon can be analyzed using finite-width corrections, where the infinite-width statistics are supplemented by a power series in $1/\mathrm{width}$. In particular, recent empirical results by Day et al. show that the tensors appearing in this treatment stabilize for large depth, as opposed to the tensors of i.i.d.-initialized networks. In this article, we derive explicit layer-wise recursion relations for the tensors appearing in the finite-width expansion of the network statistics in the case of orthogonal initializations. We also provide an extension of recently-introduced Feynman diagrams for the corresponding recursions in the i.i.d.-case which are valid to all orders in $1/\mathrm{width}$. Finally, we show explicitly that the recursions we derive reproduce the stability of the finite-width tensors which was observed for activation functions with vanishing fixed point. This work therefore provides a theoretical explanation for the stability of nonlinear networks of finite width initialized with orthogonal weights, closing a long-standing gap in the literature. We validate our theoretical results experimentally by showing that numerical solutions of our recursion relations and their analytical large-depth expansions agree excellently with Monte-Carlo estimates from network ensembles.

These lecture notes introduce some topics of classical statistical physics, particularly those that are relevant for neural networks and deep learning. Statistical physics is treated as a branch of probability theory or statistics, with the goal of making concepts such as phase transitions and the renormalization group accessible to readers without prior knowledge of physics. We introduce the Boltzmann-Gibbs distribution and the thermodynamic potentials on a finite configuration space, notably for Ising spins and spin-glass models on a lattice, and then define phase transitions as discontinuities that arise in the limit that the number of lattice points goes to infinity. We further introduce Hopfield networks and Boltzmann machines, which are governed by the same energy function as spin-glass models, and discuss the learning algorithm for restricted Boltzmann machines. In this algorithm hidden neurons are integrated out as in the renormalization group. Finally, modern deep learning is introduced, whose early developments were in part motivated by restricted Boltzmann machines in that they carry many layers of hidden neurons. A description of large language models is given.

Deep networks with continuous piecewise affine activations induce polyhedral partitions of the input space, making the number of realized affine regions a natural measure of expressive capacity and a key determinant of how well the model can approximate nonlinear target functions. In practice, standard training realizes far fewer region refinements in data-visited neighborhoods than the architecture could in principle support, while existing region-count theory is primarily architectural and offers little guidance on how optimization shapes the realized partition near the data. Our theory provides a sufficient condition under which bringing neuron switching surfaces sufficiently close to data points ensures their intersection with local neighborhoods, which in turn implies a strict increase in the local affine-region count, yielding a principled training-time handle for seeding data-relevant partitions early in optimization. Guided by these results, we propose a plug-and-play region-seeding regularizer that encourages early partitioning while allowing task-driven refinement to dominate later in training. Experiments show that the regularizer increases the number of realized affine regions via exact enumeration and improves overall performance on toy datasets, while also improving early-stage accuracy and achieving comparable (or slightly improved) final accuracy on ImageNet-1k for classical models.

Quantile regression is a fundamental tool for distributional learning but poses significant optimization challenges for deep models due to the non-smoothness of the pinball loss. We propose ConquerNet, a class of \textbf{con}volution-smoothed \textbf{qu}antil\textbf{e} \textbf{R}eLU neural \textbf{net}works, which yield smooth objectives while preserving the underlying quantile structure. We establish general nonasymptotic risk bounds for ConquerNet under mild conditions, providing minimax guarantees over Besov function classes. In numerical studies, we demonstrate that the proposed approach outperforms standard quantile neural networks at multiple quantile levels, showing improved estimation accuracy and training efficiency across the board, with particularly pronounced advantages at high and low quantiles.

The massive computational costs of scaling modern deep learning architectures have driven the widespread use of parameter-efficient low-rank structures, such as LoRA and low-rank factorization. However, theoretical guarantees for their expressive power are less explored, often relying on restrictive priors like a pretrained base matrix, ReLU activations or non-verifiable singularity conditions. We first investigate the limits of neural networks constrained strictly to low-rank manifolds without pretrained dense priors. We demonstrate a theoretical paradox: while purely rank-1 layers can exactly interpolate arbitrary scalar datasets, they collapse for function approximations. To overcome this bottleneck without surrendering parameter efficiency, we introduce a unified \textit{Structural Correspondence} framework. We prove that augmenting low-rank layers with only a minimal sparse diagonal component, say a Diagonal plus Low-Rank (DLoR) structure, is sufficient to reach Universal Approximation. We show that any full-rank transformation can be exactly reconstructed using these DLoR components by trading off network width (additive decomposition) or depth (multiplicative decomposition). By tracking asymptotic Taylor remainders, we prove that DLoR neural networks fully restore the Universal Approximation Theorem for general activation functions. Finally, we establish that multiplicative depth provides superior parameter-to-expressivity scaling compared to additive width. Our results show that dense matrices and specific activation functions are not topological prerequisites for universal expressivity.

Uncertainty quantification (UQ) in graph neural networks (GNNs) is crucial in high-stakes domains but remains a significant challenge. In graph settings, message passing often relies on strong assumptions such as exchangeability, which are rarely satisfied in practice, and achieving reliable UQ typically requires costly resampling or post-hoc calibration. To address these issues, we introduce Quantile-free Prediction Interval GNN (QpiGNN), a framework that builds on quantile regression (QR) to enable GNN-based UQ by directly optimizing coverage and interval width without requiring quantile inputs or post-processing. QpiGNN employs a dual-head architecture that decouples prediction and uncertainty, and is trained with label-only supervision through a quantile-free joint loss. This design allows efficient training and yields robust prediction intervals, with theoretical guarantees of asymptotic coverage and near-optimal width under mild assumptions. Experiments on 19 synthetic and real-world benchmarks show QpiGNN achieves average 22% higher coverage and 50% narrower intervals than baselines, while ensuring efficiency and robustness to noise and structural shifts.

Continuous-time (CT) Transformers improve irregular and long-range modeling over CT-RNNs by exploiting inputs or outputs embeddings with continuous dynamics. However, the core scaled-dot-product-attention (SDPA) mechanism remains inherently discrete. We propose FLUID (Flexible Unified Information Dynamics), a CT Transformer that incorporates continuous dynamics directly into the attention computation by replacing it with Liquid Attention Network (LAN). LAN reinterprets attention logits as continuous dynamical system and reformulates them as the solution to a linear ODE modulated by input-dependent nonlinear recurrent gates. Theoretically, we establish stability guarantees for LAN dynamics and show that it serves as an interpolating middle ground between SDPA and CT-RNNs, recovering each as special case under well-defined parameterization of its gating functions. LAN also introduces an explicit attention-sink gate to eliminate disproportionate attention mass on uninformative nodes. FLUID replaces standard residual connections with input-dependent Liquid Hyper-Connections to adaptively regulate interlayer information flow. Empirically, we evaluate FLUID on a broad set of learning tasks, including (i) irregular time-series, (ii) long-range modeling, (iii) lane-keeping control of autonomous vehicles, and (iv) learning physical dynamics under a scarce data regime. Across all the tasks, FLUID consistently matches or outperforms CT baselines, achieving improvements of up to 47% in certain scenarios and enhancing generalization under distributional shifts. Additionally, FLUID demonstrates superior noise robustness and a self-correcting inductive bias in autonomous vehicle control. We also provide a detailed analysis of key hyperparameters to guide tuning and show that FLUID occupies an intermediate position among competing approaches in terms of runtime and memory efficiency.

We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.

Climate classification plays a vital role in agricultural planning, hydrological studies, and climate science. One of the most widely used systems for classifying global climate zones is the Köppen-Trewartha (KT) classification. However, the KT classification is fundamentally deterministic, offering discrete labels to spatial locations without accounting for uncertainties in classification. In this paper, we provide a framework for probabilistic modeling of climatic zones. We implement a feedforward artificial neural network (ANN) for classification, allowing for efficient, uncertainty-aware categorization of climatic regions, thereby offering a more nuanced understanding of transitional climate zones compared to traditional deterministic methods. We apply this method to the Sahara Desert region over the 30-year period of 1960 - 1989, using data at more than 400,000 space-time locations from the first 11 years to train our model. We assess the model's short- and long-term classification capabilities to evaluate its stability and accuracy over time. We also compare the probabilistic classification from our model with the traditional KT classification. In addition, we use fluctuation analysis methods to highlight the temporal evolution of climatic zones across the Sahara region and identify areas undergoing significant flux of probabilities of their climate classes, providing insights into broader trends in desertification.

Physical neural networks (PNNs) exploit diverse physical substrates---spanning molecular, wetware, and solid-state devices---to perform embodied neural computation close to physical processes. However, their operational heterogeneity in signal modalities, timing regimes, reset requirements, calibration lifecycles, and observability prevents their integration as interchangeable execution endpoints in edge, fog, and cloud workflows. We present CP$^2$N$^2$, a substrate-aware control-plane architecture that bridges software-level orchestration and physical neural computing. CP$^2$N$^2$ introduces a three-plane model that decouples high-level orchestration, twin-state synchronisation, and substrate-specific execution under versioned Physical Neural Resource Contracts. It extends Model Context Protocol (MCP)-style discovery and invocation with explicit lifecycle management, atomic leases, telemetry freshness verification, and fail-closed safety boundaries. We evaluate a reference prototype across contrasting behavioural twins, a same-host multi-process deployment under fault injection, and the Cortical Labs SDK Simulator. In a 945-request concurrency campaign, CP$^2$N$^2$ enforces typed contention and fail-closed recovery with sub-millisecond local orchestration overhead. Furthermore, in a 160-decision Agent-to-PNN campaign with a hosted LLM planner, independent enforcement ensured that 100% of executed actions remained strictly safe, verified, and reconciled despite planner-level inconsistencies. CP$^2$N$^2$ thus provides an enforceable systems abstraction for integrating scarce, stateful physical AI substrates into agentic computing environments.

Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both $\mathbb{Z}_2$ and $\mathrm{U}(1)$ lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in $\mathrm{U}(1)$ quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.

Information plane (IP) analysis has been suggested to study the training dynamics of deep neural networks through mutual information (MI) between inputs, representations, and targets. However, its statistical validity is often compromised by the difficulty of estimating MI from samples of high-dimensional, deterministic representations. In this work, we perform IP analyses on binary neural networks (BNNs) where activations are discrete and MI is finite. We characterise the finite-sample behaviour of the plug-in entropy estimator and identify regimes for sample size $N$ and representation dimensionality $D$ under which MI estimates are reliable. Outside these regimes, we show that empirical MI estimates saturate to $\log_2 N$, rendering IP trajectories uninformative. Restricting attention to the reliable regime, we train 375 BNNs to investigate the existence of late-stage compression phases and the relationship between compressed representations and generalisation performance. Our results show that while late-stage compression is frequently observed, compressed latent representations do not consistently correlate with improved generalization performance. Instead, the relationship between compression and generalisation is highly dependent on task, architecture, and regularisation.

Understanding how biological and artificial neural networks implement computation from connectivity is a central problem in neuroscience and machine learning. In neural systems, structural and functional connectivity are known to diverge, motivating approaches that move beyond direct connections alone. Here, we show that the spatial and temporal function of recurrent neural networks (RNNs) trained on hierarchically modular tasks can be recovered by modelling the network as a graph and analysing the multi-hop pathways between input and output units. In particular, decomposing these pathways by hop length reveals how the network temporally routes information. This perspective reframes regularisation: if function is implemented through multi-hop communication, then standard penalties such as L1 regularisation, which act only on individual weights, constrain single-hop structure rather than the multi-hop pathways that support computation. Motivated by this view, we introduce resolvent-RNNs (R-RNNs), which constrain multi-hop pathways and thereby induce temporal sparsity beyond that achieved by standard L1 regularisation. Compared with L1 regularisation, R-RNNs achieve improved performance by inducing temporal sparsity that matches the task structure, even when the task signal is sparse. Moreover, R-RNNs exhibit stronger sparsity-function alignment, reflected in their increased robustness under strong regularisation. Together, our results identify multi-hop communication as a key principle linking structure to function in recurrent networks, and suggest that sparsity should be defined over functional pathways rather than individual parameters.

The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose probability distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \geq 2$, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator, and no supremum evaluation is necessary. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics-informed neural networks (SV-PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks, the underlying principle is independent of the trial space and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher-order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second-order linear elliptic problems spanning high-frequency and multi-scale solutions, indefinite operators, variable coefficients, and non-standard domains, in which SV-PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L-BFGS steps.

We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.

Solving inverse problems in dynamical systems governed by high-dimensional coupled ordinary differential equations (ODEs) is a ubiquitous challenge in scientific machine learning. In many real-world applications, researchers seek to uncover unknown parameters or model unknown dynamics even as the underlying physics is only partially characterized, and observations are sparse and limited to specific measurable channels. While physics-informed neural networks (PINNs) are ideal for inverse inference under partial observability, existing PINNs typically rely on task-specific joint optimization, which suffers from optimization difficulties and poor generalization. In this paper, we propose a meta-inverse physics-informed neural network (MI-PINN) that reformulates inverse modeling as a two-stage meta-learning problem. MI-PINN first learns a physics-aware representation across multiple tasks, and then performs inverse modeling by optimizing task-specific unknowns while keeping the learned representation fixed. This two-stage formulation significantly reduces the parameter search dimension, thereby improving sample efficiency and enabling accurate inference. To handle multi-scale dynamics common in these high-dimensional ODE systems, we further introduce an adaptive clustering-based multi-branch learning scheme. We demonstrate the effectiveness of MI-PINN on whole-body physiologically based pharmacokinetic (PBPK) models with up to 33 coupled ODEs, using paracetamol and theophylline under intravenous and oral dosing scenarios. Experimental results show that MI-PINN enables accurate recovery of masked kinetic parameters and reconstruction of missing mechanistic terms despite limited clinical observations.

We introduce \emph{Dynamical Physics-Modeled Neural Networks} (DynPMNNs), a continuous-time deep learning architecture in which each hidden layer is defined as the solution of an ordinary differential equation. Unlike classical feed-forward networks, this approach replaces static activation functions with time-evolving dynamical systems, providing a biologically inspired interpretation of hidden-layer behavior and enabling the integration of physically meaningful models. The framework is rigorously grounded in Reproducing Kernel Banach Spaces (RKBSs), allowing DynPMNNs to be characterized as finite-dimensional solutions of an abstract training problem and revealing structural connections with standard neural networks. We present a concrete implementation based on the FitzHugh--Nagumo model for neuronal activation, where numerical ODE solvers are embedded into the computational graph via Euler-type schemes. Both network weights and dynamical parameters are trained jointly. Through experiments on the California Housing dataset, we compare DynPMNNs with Neural ODEs (NODEs) and Closed-form Continuous-Time Networks (CfCs). Despite using fewer trainable parameters, DynPMNNs achieve competitive performance. These results position DynPMNNs as a principled bridge between dynamical systems and deep learning, with promising directions for further research in expressivity, stability, and physics-based modeling.

Graph Neural Networks (GNNs) achieve strong performance on node classification tasks but remain difficult to interpret, particularly with respect to which input features drive their predictions. Existing global GNN explainers operate at the structural level identifying recurring subgraph motifs, but none explain model behaviour globally at the level of input node attributes. We propose GRAFT, a posthoc global explanation framework that identifies class-level feature importance profiles for GNNs. The method combines diversity-guided exemplar selection, Integrated Gradients-based attribution, and aggregation to construct a global view of feature influence for each class, which can be further expressed as concise natural language rules using a large language model with self-refinement. We evaluate GRAFT across multiple datasets, architectures, and experimental settings, demonstrating its effectiveness in capturing model-relevant features, supporting bias analysis, and enabling feature-efficient transfer learning. In addition, we introduce a structured human evaluation protocol to assess the interpretability of generated rules along dimensions such as accuracy and usefulness. Our results suggest that GRAFT provides a practical and interpretable approach for analysing feature-level behaviour in GNNs, bridging quantitative attribution with human-understandable explanations.

The proliferation of large-scale and structurally complex data has spurred the integration of machine learning methods into statistical modeling. Recurrent neural networks (RNNs), a foundational class of models for time-dependent data, can be viewed as nonlinear extensions of classical autoregressive moving average models. Despite their flexibility and empirical success in machine learning, RNNs often suffer from limited interpretability and slow training, which hinders their use in statistics. This paper proposes the Parallelized RNN (ParaRNN), a novel model composed of multiple small recurrent units. ParaRNN admits an additive representation that decouples recurrent dynamics into interpretable components, whose behavior can be characterized through recurrence features. This interpretability enables its applications in nonparametric regression for time-dependent data, while the design also allows efficient parallelization. The approximation capacity and non-asymptotic prediction error bounds in a nonparametric regression setting are established for ParaRNN. Empirical results on three sequential modeling tasks further demonstrate that ParaRNN achieves performance comparable to vanilla RNNs while offering improved interpretability and efficiency.

Road traffic accidents remain a significant global concern, with the majority attributed to human factors such as driver distraction and fatigue. This study proposes a camera-based approach to derive useful indicators to assess driver attentiveness and alertness. The proposed pipeline jointly satisfies the stringent real-time requirements imposed by the critical application and minimizes the computational requirements to allow for deployment on a tight computational budget. To this end, we develop a lightweight multi-task neural network that predicts multiple indicators for the face region in a single forward pass. The developed model is integrated into a complete execution workflow to produce a real-time estimate of attentiveness, fatigue, and engagement in distracting activities.

We present a PyTorch package that compiles neural networks and their weights from Turing machine descriptions, producing models that exactly simulate the specified machine without any training. Given a transition function and a set of terminal states, the package constructs a model whose forward pass corresponds to one step of the Turing machine. Two architectures are implemented, each realizing a different theoretical result: (1) a transformer with self-attention, cross-attention, and feedforward layers based on Wei, Chen, and Ma (2021), and (2) a recurrent network based on Siegelmann and Sontag (1995) that encodes the stack in a Cantor set. We develop the constructions from first principles, showing how ReLU networks implement Boolean circuits (AND, OR, NOT, XOR gates and their composition into DNF formulas and binary adders) and how hard attention implements positional lookup on the tape. The package serves as a concrete, runnable reference for the symbolic-neural bridge, and as a foundation for future work on the stability of constructed solutions under gradient-based optimization. Code is available at https://github.com/jonrbates/turing.

Training neural networks is difficult to interpret, particularly for newcomers. We introduce NeuroViz, an interactive visualization tool that supports real-time exploration of fully connected neural network training. Users can configure network architecture, activation functions, learning rates, and datasets, then observe activations, weight updates, and loss progression. NeuroViz visualizes weight changes in direct correspondence with activation signals in both forward and backward passes, enabling users to distinguish pre- and post-update states within individual epochs and view dynamically updating per-neuron equations. We conduct a comparative user study with 31 participants against six established visualization tools and we achieved the highest usability score (SUS 80.97, in the 'excellent' range), with mean rankings of 2.47 for clarity and 2.23 for usefulness (lower is better). Over 70% of participants reported that the visualizations substantially increased their perception of neural network training transparency. The implemented instance is accessible at https://neuroviz.org.

Gestational Diabetes Mellitus (GDM) is a high-prevalence pregnancy complication that requires accurate early risk stratification to reduce maternal and fetal morbidity. However, real-world clinical deployment of machine learning is hindered by two coupled constraints: (i) label scarcity, where a large fraction of electronic health records (EHR) lack confirmed diagnostic labels, and (ii) data privacy, which prevents sharing patient-level data across hospitals. This paper proposes FedTGNN-SS, a privacy-preserving federated semi-supervised framework for clinical tabular EHR. Each hospital builds a local k-nearest-neighbor patient similarity graph and trains a topology-adaptive GNN encoder. To robustly exploit unlabeled records, FedTGNN-SS combines (1) prototype-guided pseudo-labeling with neighborhood agreement, (2) adaptive graph refinement that periodically updates the k-NN graph using learned embeddings, (3) clinical-aware consistency augmentation applied only to continuous variables, and (4) privacy-safe prototype sharing that exchanges only class-level centroids. Across three diabetes-related datasets (GDM: N = 3,525; Pima: N = 768; Early Stage: N = 520) under 10%-80% missing labels per silo, FedTGNN-SS achieves 56 significant wins ($p < 0.05$) against 11 federated baselines and attains strong AUROC under extreme scarcity (Pima: 0.8037 at 80% missing, Early Stage: 0.9634 at 80% missing).

Integration of Answer Set Programming (ASP) with neural networks has emerged as a promising tool in Neuro-symbolic AI. While existing approaches extend the capabilities of ASP to real world domains, their reasoning pipelines depend on classical solvers, which is a bottleneck for scalability. To tackle this problem, we propose a new method to compute stable models, called decision-propagation (DProp), which alternates falsity decisions and truth propagations. Successful DProp computations are shown to capture the stable model semantics. We then develop Neural DProp (NDProp), a differentiable extension of DProp with neural computation for decisions and fuzzy evaluation for propagations. We evaluate the capabilities of NDProp for learning decision heuristics as well as neuro-symbolic integration, and compare it with existing neuro-symbolic approaches. The results show that NDProp can learn to efficiently compute stable models, and it improves accuracy and scalability on neuro-symbolic benchmarks.

Human cognition emerges from coordinated spiking dynamics in distributed neural circuits, where information is encoded via both firing rates and precise spike timing determined by brain rhythms. Inspired by this notion, we propose a brain-inspired learning primitive in which cognition-level neural synchrony emerges through iterative bottom-up and top-down interactions between micro-scale dynamics of spiking neurons and a macro-scale mechanism of oscillatory synchronization. Specifically, we model each parcel (e.g., a cortical region or an image pixel) in the target system as a spiking neuron embedded in a predefined connectivity scaffold. Low-level information is encoded in a spatiotemporal domain, where neurons are selectively grouped and fire spontaneously over time through self-organized dynamics. In the bottom-up route, oscillatory synchronization is formed from past spiking activity accumulated over a finite memory window. Since brain dynamics operate in a regime of partial and transient synchronization rather than global phase locking, we model oscillatory coordination using a time-delayed synchronization formulation, which enables a top-down modulation of heterogeneous neural spiking for a large-scale distributed system. Together, we devise a spiking-by-synchronization neural network (S2-Net) that uses rhythmic timing as a control mechanism for efficient information processing. Promising results have been achieved across a broad range of tasks, including neural activity decoding, energy-efficient signal processing, temporal binding and semantic reasoning.

Spiking Neural Networks (SNNs) are widely regarded as an energy-efficient paradigm for modeling and processing temporal and event-driven information. Incorporating delays in SNNs has been proven to be an effective mechanism for improving spike alignment in event-driven tasks. However, existing delay learning approaches predominantly assign static delays to individual synapses, resulting in a large number of delay parameters and limited adaptability to input-dependent activity dynamics. To this end, we propose a Congestion-Aware Dynamic Axonal Delay (CADAD) mechanism, which decomposes the delay into a channel-wise static base delay for temporal structuring and a global, activity-conditioned shift that dynamically regulates the state update rate under varying spike intensities. The delay parameters are learned using differentiable linear interpolation and discretized at inference time, preserving the benefits of dynamic delay modulation while incurring only minimal additional cost. Experiments on speech benchmarks, including the Spiking Heidelberg Dataset, Spiking Speech Commands, and Google Speech Commands, demonstrate that introducing congestion-aware delays into synaptic signal transmission effectively improves accuracy on temporal tasks, notably achieving 93.75% accuracy on SHD, 80.69% accuracy on SSC, and 95.58% on GSC-35, while reducing the parameter count by approximately 50% compared to state-of-the-art delay-based methods with the same architecture.

Equilibrium propagation (EP) is a biologically plausible alternative to backpropagation (BP), but its effectiveness can degrade in deeper and more challenging learning settings. In parallel, dendritic neural networks have demonstrated improved performance and generalization when trained with BP, suggesting that structured, biologically inspired architectures may enhance learning. In this work, we investigate the integration of dendritic neural networks with equilibrium propagation using an advanced EP framework. We evaluate the proposed dendritic EP model on MNIST, Kuzushiji-MNIST (KMNIST), and Fashion-MNIST (FMNIST), considering both shallow and deeper architectures. Our results show that dendritic EP achieves performance comparable to standard EP on simple tasks, while providing consistent improvements on more challenging datasets and deeper models. In particular, dendritic EP significantly outperforms standard EP on KMNIST and FMNIST, and approaches the performance of dendritic networks trained with backpropagation through time.To further understand these improvements, we analyze the evolution of hidden states during the free phase. We observe that dendritic EP exhibits higher activation magnitudes and more distributed hidden-state activity compared to standard EP, indicating that dendritic structure alters the internal network dynamics. These findings suggest that incorporating dendritic structure can enhance the effectiveness of biologically plausible learning algorithms, especially in regimes where standard EP struggles. Our work highlights the importance of architectural design for improving biologically inspired training methods.

Riemannian neural networks have proven effective in solving a variety of machine learning tasks. The key to their success lies in the development of principled Riemannian analogs of fundamental building blocks in deep neural networks (DNNs). Among those, Riemannian batch normalization (BN) layers have shown to enhance training stability and improve accuracy. In this paper, we propose BN layers for neural networks on complex domains. The proposed layers have close connections with existing Riemannian BN layers. We derive essential components for practical implementations of BN layers on some complex domains which are less studied in previous works, e.g., the Siegel disk domain. We conduct experiments on radar clutter classification, node classification, and action recognition demonstrating the efficacy of our method.

Spiking Neural Networks (SNNs) provide a promising framework for energy-efficient and biologically grounded computation; however, scalable learning in deep recurrent architectures with sparse connectivity remains a major challenge. In this work, we propose a structured multi-layer recurrent SNN architecture composed of locally dense recurrent layers augmented with sparse small-world long-range projections to a readout population. The long-range connectivity is largely fixed, preserving routing efficiency and hardware scalability, while synaptic adaptation is performed using strictly local plasticity mechanisms. To enable supervised learning without backpropagation or surrogate gradients, we introduce a biologically motivated learning framework that combines: (i) population-based winner-take-all (WTA) teaching signals at the output layer, (ii) fixed random broadcast alignment feedback pathways, and (iii) low-dimensional modulatory neuron populations that gate synaptic updates through three-factor learning rules with eligibility traces. This design supports deep recurrent computation with sparse global communication and purely local synaptic updates. We analyze the algorithmic properties, computational complexity, and hardware feasibility of the proposed approach, and demonstrate stable learning and competitive performance on benchmark classification tasks. The results highlight the potential of structured recurrence and neuromodulatory learning to enable scalable, hardware-compatible SNN training beyond gradient-based methods.

In this paper, we study norm-based regularization methods for neural networks. We compare existing penalization approaches and introduce two regularization strategies that extend classical ridge- and lasso-type penalties to neural network models. The first strategy modifies weight decay by incorporating the covariance structure of the input features into a ridge-type $\ell_2$ penalty, allowing regularization to account for feature dependence. The second combines an $\ell_1$ sparsity penalty with covariance-aware $\ell_2$ regularization, producing neural network weights that are both sparse and structurally informed. Monte Carlo simulations are used to evaluate these methods under different data-generating settings, followed by two real-data applications on building cooling-load prediction and leukemia cell-type classification from high-dimensional gene expression data. Across simulated and real-data examples, the proposed regularizers improve predictive performance on unseen data and provide more effective complexity control than standard norm-based penalties, particularly when features are correlated or high-dimensional.

Spiking neural networks (SNNs) are a promising paradigm for energy-efficient event-driven computation, but large-scale SNN execution remains challenging because sparse spike communication and synchronization can dominate runtime. Existing solutions across CPU, GPU, ASIC, and FPGA platforms offer different trade-offs between programmability, efficiency, and scalability. To address this gap, we present NeuroRing, a modular and scalable SNN accelerator based on a stream-dataflow architecture and a bidirectional ring topology, implemented in High-Level Synthesis (HLS) on FPGAs. NeuroRing supports modular single- and multi-FPGA deployment and is compatible with existing SNN workflows through integration with the NEST simulator. We evaluate NeuroRing on the cortical microcircuit benchmark and a Sudoku constraint-satisfaction workload. Results show that NeuroRing preserves the key activity statistics of the NEST reference model, achieves faster-than-real-time execution of the full-scale cortical microcircuit with a real-time factor (RTF) of 0.83, exhibits meaningful strong and weak scaling, and provides competitive energy efficiency on two programmable FPGAs. These results position NeuroRing as a flexible and scalable platform for both neuroscience simulation and broader event-driven applications.

Foundation models are deep neural networks (such as GPT-5, Gemini3, and Opus4) trained on large datasets that can perform diverse downstream tasks -- text and code generation, question answering, summarization, image classification, and so on. The philosophy of foundation models is to put effort into a single, large (${\sim}10^{12}$-parameter) general-purpose model that can be adapted to many downstream tasks with no or minimal additional training. We argue that the rise of foundation models presents an opportunity for hardware engineers: in contrast to when different models were used for different tasks, it now makes sense to build special-purpose, fixed hardware implementations of neural networks, manufactured and released at the roughly 1-year cadence of major new foundation-model versions. Beyond conventional digital-electronic inference hardware with read-only weight memory, we advocate a more radical re-thinking: hardware in which the neural network is realized directly at the level of the physical design and operates via the hardware's natural physical dynamics -- \textit{Physical Foundation Models} (PFMs). PFMs could enable orders-of-magnitude advantages in energy efficiency, speed, and parameter density. For ${\sim}10^{12}$-parameter models, this would both reduce the high energy burden of AI in datacenters and enable AI in edge devices that today are power-constrained to far smaller models. PFMs could also enable inference hardware for models much larger than current ones: $10^{15}$- or even $10^{18}$-parameter PFMs seem plausible by some measures. We present back-of-the-envelope calculations illustrating PFM scaling using an optical example -- a 3D nanostructured glass medium -- and discuss prospects in nanoelectronics and other physical platforms. We conclude with the major research challenges that must be resolved for trillion-parameter PFMs and beyond to become reality.

Shallow nanoindentation enables mechanical characterization of thin films, individual phases, and other volume-constrained materials, but the measured hardness is inflated by the indentation size effect (ISE). Classical corrections such as Nix-Gao require a deep linear regime and fail when only shallow measurements are accessible. We present a data-efficient workflow that recovers a high-load reference hardness directly from shallow, size-affected indentation data. Over 700 indentations on three certified steel reference blocks (2-6.5 GPa) were expanded by physics-guided augmentation and used to train conventional regressors (Ridge, Random Forest, XGBoost, neural networks) and a physics-constrained neural network (PCNN) that reconstructs hardness through the bounded form H_ref = H_app/sqrt(1+q), with the signed correction q learned from dimensionless contact-mechanics descriptors (P_max/S^2, W_p/W_tot, H/E_r). On a quarantined fourth steel tested at loads offset from the training schedule, only the constrained formulation generalized: the dimensionless-input PCNN achieved RMSE = 0.372 GPa and MAPE = 5.2%, and remained stable beyond the training load range, where tree-based models failed structurally. Ablation attributed this robustness to the bounded reconstruction and a load-independence regularizer. Deliberate application to fused silica produced systematic failure, delimiting the method to dislocation-mediated crystalline plasticity. A few hundred laboratory indentations thus suffice to train a physically constrained ISE correction operating on single shallow indentations, offering a template for materials lacking analytical size-effect models.

The Hypencoder, proposed by Killingback et al., is a retrieval framework that replaces the fixed inner-product scoring function used in standard bi-encoders with a query-specific neural network (the $q$-net), whose weights are generated by a hypernetwork from the contextualized query embeddings. This design enables more expressive relevance estimation while preserving independent query and document encoding. In this work, we conduct a reproducibility study of the Hypencoder and extend the original analysis in three directions. Our reproduction confirms that the Hypencoder outperforms a similarly trained bi-encoder baseline on in-domain and out-of-domain benchmarks, and that the proposed efficient search algorithm substantially reduces query latency with minimal performance loss. On hard retrieval tasks, we find partial support: the Hypencoder outperforms the baseline on DL-Hard and FollowIR, but not on TREC TOT, where checkpoint incompatibility and fine-tuning sensitivity complicate full verification. Beyond reproduction, we investigate three extensions: (i)~integrating alternative pre-trained encoders into the Hypencoder framework, where we find that performance gains depend on the encoder and fine-tuning strategy; (ii)~comparing query latency against a Faiss-based bi-encoder pipeline, revealing that standard bi-encoder retrieval remains faster under both exhaustive and efficient search settings; and (iii)~evaluating adversarial robustness, where we find that the $q$-net's non-linear scoring does not provide a consistent robustness disadvantage over inner-product scoring. Our code is publicly available at https://github.com/arneeichholtz/Hypencoder-reprod.

This paper extends and explains the Multiple Additive Neural Networks (MANN) methodology, an enhancement to the traditional Gradient Boosting framework, utilizing nearly shallow neural networks instead of decision trees as base learners. This innovative approach leverages neural network architectures, notably Convolutional Neural Networks (CNNs) and Capsule Neural Networks, to extend its application to both structured data and unstructured data such as images and audio. For structured data the advantages of capsule neural networks as feature extractors are used and combined with MANN as a classifier. MANN's unique architecture promotes continuous learning and integrates advanced heuristics to combat overfitting, ensuring robustness and reducing sensitivity to hyperparameter settings like learning rate and iterations. Our empirical studies reveal that MANN surpasses traditional methods such as Extreme Gradient Boosting (XGB) in accuracy across well-known datasets. This research demonstrates MANN's superior precision and generalizability, making it a versatile tool for diverse data types and complex learning environments.

Predictive safety filters (PSFs) leverage model predictive control to enforce constraint satisfaction during deep reinforcement learning (RL) exploration, yet their reliance on first-principles models or Gaussian processes limits scalability and broader applicability. Meanwhile, model-based RL (MBRL) methods routinely employ probabilistic ensemble (PE) neural networks to capture complex, high-dimensional dynamics from data with minimal prior knowledge. However, existing attempts to integrate PEs into PSFs lack rigorous uncertainty quantification. We introduce the Uncertainty-Aware Predictive Safety Filter (UPSi), a PSF that provides rigorous safety predictions using PE dynamics models by formulating future outcomes as reachable sets. UPSi introduces an explicit certainty constraint that prevents model exploitation and integrates seamlessly into common MBRL frameworks. We evaluate UPSi within Dyna-style MBRL on standard safe RL benchmarks and report substantial improvements in exploration safety over prior neural network PSFs while maintaining performance on par with standard MBRL. UPSi bridges the gap between the scalability and generality of modern MBRL and the safety guarantees of predictive safety filters.

This paper presents an application of the biologically realistic JASTAP neural network model to classification tasks. The JASTAP neural network model is presented as an alternative to the basic multi-layer perceptron model. An evolutionary procedure previously applied to the simultaneous solution of feature selection and neural network training on standard multi-layer perceptrons is extended with JASTAP model. Preliminary results on IRIS standard data set give evidence that this extension allows the use of smaller neural networks that can handle noisier data without any degradation in classification accuracy.

Graph-based text classification models typically rely on local neighborhood aggregation and overlook global community structure, despite semantic document graphs exhibiting strong class-consistent clustering. Ignoring this can blur class boundaries and lead to over-smoothing. We propose ModTGCN, a modularity-aware graph neural network for text classification that jointly optimizes cross-entropy and a modularity-based auxiliary objective to promote class-coherent document communities while preserving discriminative representations. The modularity term is computed on a document-document similarity graph derived from transformer embeddings (pretrained or fine-tuned). To improve scalability, we decouple the original heterogeneous TextGCN graph into separate document-word and word-word components, achieving 2x-10x faster training. We further study graph construction strategies, label-aware edge reweighting, and supervision choices for modularity optimization. Experiments on five benchmarks show consistent gains, with larger improvements on complex, low homophily datasets such as Ohsumed and 20NG.

We provide a causal analysis of Binary Spiking Neural Networks (BSNNs) to explain their behavior. We formally define a BSNN and represent its spiking activity as a binary causal model. Thanks to this causal representation, we are able to explain the output of the network by leveraging logic-based methods. In particular, we show that we can successfully use a SAT as well as a SMT solver to compute abductive explanations from this binary causal model. To illustrate our approach, we trained the BSNN on the standard MNIST dataset and applied our SAT-based and SMT-based methods to finding abductive explanations of the network's classifications based on pixel-level features. We also compared the found explanations against SHAP, a popular method used in the area of explainable AI. We show that, unlike SHAP, our approach guarantees that a found explanation does not contain completely irrelevant features.

We propose EdgeSpike, a co-designed spiking neural network (SNN) framework for autonomous low-power sensing in edge Internet of Things (IoT) architectures. EdgeSpike unifies (i) a hybrid surrogate-gradient and direct-encoding training pipeline, (ii) a hardware-aware neural architecture search (NAS) bounded by per-inference energy and memory budgets, (iii) an event-driven runtime targeting Intel Loihi 2, SpiNNaker 2, and commodity ARM Cortex-M microcontrollers with custom spike-sparse SIMD kernels, and (iv) a lightweight local plasticity rule enabling continual on-device adaptation without backpropagation. The framework is evaluated across five sensing tasks (keyword spotting, vibration-based machine fault detection, surface electromyography gesture recognition, 77 GHz radar human-activity classification, and structural-health acoustic-emission monitoring) on three hardware targets. EdgeSpike achieves a mean classification accuracy of 91.4%, within 1.2 percentage points (pp) of strong INT8 convolutional neural network (CNN) baselines (mean 92.6%), while reducing energy per inference by 18x to 47x on neuromorphic hardware (mean 31x) and by 4.6x to 7.9x on Cortex-M (mean 6.1x). End-to-end latency remains at or below 9.4 ms across all 15 task-hardware configurations. A seven-month, 64-node wireless field deployment confirms a 6.3x extension in projected battery lifetime (from 312 to 1978 days at 2 Wh per node) and bounded accuracy degradation under seasonal drift (0.7 pp with on-device adaptation versus 2.1 pp without). Hardware-aware NAS evaluates 8400 candidates and yields a 12-point Pareto front. EdgeSpike will be released as open source with reproducible training pipelines, hardware-portable runtimes, and benchmark suites.

Physics-informed neural networks (PINNs) approximate solutions of partial differential equations (PDEs) by embedding physical laws into the loss function. In parameterized PDE families, variations in coefficients or boundary/initial conditions define distinct tasks. This makes training individual PINNs for each task computationally prohibitive, while cross-task transfer can be sensitive to task heterogeneity. While meta-learning can reduce retraining cost, existing methods often rely on a single global initialization and may suffer from negative transfer, particularly under feature-scarce coordinate inputs and limited training-task availability. We propose the Learning-Affinity Adaptive Modular Physics-Informed Neural Network (LAM-PINN), a compositional framework that leverages task-specific learning dynamics. LAM-PINN combines PDE parameters with learning-affinity metrics from brief transfer sessions to construct a task representation and cluster tasks even with coordinate-only inputs. It decomposes the model into cluster-specialized subnetworks and a shared meta network, and learns routing weights to selectively reuse modules instead of relying on a single global initialization. Across three PDE benchmarks, LAM-PINN achieves an average 19.7-fold reduction in mean squared error (MSE) on unseen tasks using only 10% of the training iterations required by conventional PINNs. These results indicate its effectiveness for generalization to unseen configurations within bounded design spaces of parameterized PDE families in resource-constrained engineering settings.

Agriculture is a key sector of the economies of developing countries. It serves as a primary source of income and employment for rural populations. However, each year, a large portion of crops is wasted because of pests and diseases. Well-timed prediction of plant diseases is crucial to sustainable, high-quality agricultural production. Detection of plant diseases through conventional methods is both labour-intensive and time-consuming. Researchers have developed image classification based automated techniques for this purpose. Most accurate methods are based on deep convolutional neural networks, which are computationally intensive, with many layers and millions of trainable parameters. In resource-constrained settings, especially in rural areas, it is difficult to deploy deep convolutional neural network models for efficient plant disease identification. To address these issues, an efficient and light-weight Multi-View Convolutional Neural Network is proposed. These additional features aid the proposed model to identify the plant diseases accurately and efficiently with less number of parameters. The proposed model is tested on a benchmark Plantvillage dataset and achieves an improvement of $2.9\%$ in classification accuracy compared to the baseline convolutional neural network model, which was trained only on Red, Green, and Blue (RGB) plant images. Compared with state-of-the-art deep convolutional neural network models, the proposed model is less computationally expensive and achieves comparable accuracy for plant disease identification on the PlantVillage dataset.

Graph neural networks (GNNs) have emerged as a versatile and efficient option for modeling the dynamic behavior of deformable materials. While GNNs generalize readily to arbitrary shapes, mesh topologies, and material parameters, existing architectures struggle to correctly predict the temporal evolution of key physical quantities such as linear and angular momentum. In this work, we propose MomentumGNN -- a novel architecture designed to accurately track momentum by construction. Unlike existing GNNs that output unconstrained nodal accelerations, our model predicts per-edge stretching and bending impulses which guarantee the preservation of linear and angular momentum. We train our network in an unsupervised fashion using a physics-based loss, and we show that our method outperforms baselines in a number of common scenarios where momentum plays a pivotal role.

These lecture notes provide an introduction to the verification of neural networks from a theoretical perspective. We discuss feed-forward neural networks, recurrent neural networks, attention mechanisms, and transformers, together with specification languages and algorithmic verification techniques.

Recurrent Graph Neural Networks (RGNNs) extend standard GNNs by iterating message-passing until some stopping condition is met. Various RGNN models have been proposed in the literature. In this paper, we study three such models: converging RGNNs, where all vertex representations must stabilise; output-converging RGNNs, where only the output classifications must stabilise; and halting RGNNs, where a per-vertex halting classifier determines when to stop. We establish expressiveness relationships between these models: over undirected graphs, converging RGNNs are equally expressive as graded-bisimulation-invariant halting RGNNs, while output-converging RGNNs are at least as expressive. Combined with prior results on halting RGNNs, this shows that, relative to the classifiers expressible in monadic second-order logic (MSO), converging RGNNs express exactly the graded modal $μ$-calculus ($μ$GML), and output-converging RGNNs express at least $μ$GML. These results hold even when restricting to ReLU networks with sum aggregation. The main technical challenge is simulating halting RGNNs by converging ones: without a global halting classifier, vertices may locally decide to halt at different times, causing desynchronisation. We develop a "traffic-light" protocol that enables vertices to coordinate despite this asynchrony. Our results answer an open question from Bollen et al. (2025) and show that the RGNN model of Pflueger et al. (2024) retains full $μ$GML expressiveness even when convergence is guaranteed.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

Stephen Wolfram proclaimed in his 2003 seminal work "A New Kind Of Science" that simple recursive programs in the form of Cellular Automata (CA) are a promising approach to replace currently used mathematical formalizations, e.g. differential equations, to improve the modeling of complex systems. Over two decades later, while Cellular Automata have still been waiting for a substantial breakthrough in scientific applications, recent research showed new and promising approaches which combine Wolfram's ideas with learnable Artificial Neural Networks: So-called Neural Cellular Automata (NCA) are able to learn the complex update rules of CA from data samples, allowing them to model complex, self-organizing generative systems. The aim of this paper is to review the existing work on NCA and provide a unified modular framework and notation, as well as a reference implementation in the open-source library NCAtorch. Supplementary materials, videos, and code are available at the project website: https://www.neural-cellular-automata.org/