Neural Network

Learning from imperfect data is a central theme in machine learning, connecting practical questions of robustness to fundamental questions of learnability. Here we examine attribute noise: learning from corrupted inputs while keeping the labels intact, a setting that has received considerably less analytical attention than its label-noise counterpart. We consider two types of corruption models: additive noise and replacement noise. Through experiments with multi-layer perceptrons (MLPs) on corrupted classification datasets, we find that neural networks remain robust, maintaining well-above-chance accuracy even when inputs are >90% corrupted -- far beyond human recognition. To understand this robustness, we analyze infinite-width networks in the heavy-corruption regime using a mean-field-inspired approach and derive a leading-order decision rule for the classification outcome: the network implements a prototype rule, the nearest-class-mean, assigning each test point to the class whose training-set average it most closely resembles. This leading-order decision rule is universal across a broad range of MLP architectures, holding for any depth, as well as a wide class of activation functions and noise distributions. The same centroid mechanism closely matches finite-width network behavior in our experiments and provides an interpretable and analytically tractable account of why learning can succeed even when individual training examples carry almost no signal.

We investigate limitations of learning $\tanh$ neural networks from point evaluations under finite-precision computations and $L^p$ accuracy guarantees, building on Berner, Grohs, and Voigtländer (2023). Our approach is based on a novel construction of sharply localized bump functions via iterated $\tanh$ activations. Using this mechanism, we show that, in a finite-precision setting, no adaptive randomized algorithm based on $m$ samples can achieve a convergence rate higher than the Monte Carlo rate $O(m^{-1/p})$ in the $L^p$ norm, unless the sampling budget grows exponentially with the size of the network parameters and architecture. The results reveal fundamental limitations imposed by finite precision on the learnability of classes containing localized bump functions, extending previous results for ReLU networks to the $\tanh$ setting.

We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing \emph{tensorizable networks} to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.

A multi-similarity Siamese neural network unifies zero-day anomaly detection and one-shot classification in optical networks, achieving over 99% accuracy and instant adaptability across lightpaths and unseen anomaly types without any retraining.

While physics-informed neural networks (PINNs) have shown strong potential for process modeling, physical equations are only enforced as soft constraints during training, and thus, they do not guarantee constraint satisfaction at inference. We propose a framework, called piecewise-linear Karush--Kuhn--Tucker hard-constrained PINNs (PL-KKT-hPINNs), that strictly enforces nonlinear equality constraints through piecewise-linear projection. This extends the KKT-hPINN framewor, which exactly enforces linear equalities through the Karush--Kuhn--Tucker (KKT) conditions associated with orthogonally projecting neural network outputs onto the constraint feasible region. The method is demonstrated on a continuous stirred-tank reactor (CSTR) case study for both one and two inputs. Results show that PL-KKT-hPINN preserves predictive accuracy comparable to that of a standard neural network while achieving substantially lower constraint violations. In addition, the proposed model shows improved robustness in low-data regimes, yielding lower RMSE than the unconstrained neural network for limited training sample sizes. These results demonstrate that PL-KKT-hPINN provides a computationally efficient and physically consistent framework for surrogate modeling of nonlinear chemical engineering systems.

Criticality has been proposed as a key organizing principle in biological neural systems, yet its origin and relevance in artificial neural networks remain unclear. We analyze hidden-state dynamics in trained long short-term memory (LSTM) networks and show that small networks near their optimal training epochs (steps) exhibit scale-free avalanche statistics and branching parameters close to unity, indicative of near-critical dynamics, while larger models remain subcritical. To explain the coexistence of subcritical branching with robust $1/f^β$ noise, we introduce a mixture branching process framework that links heterogeneous branching dynamics to long-range temporal correlations. These results identify critical-like behavior in LSTMs as an emergent, capacity-dependent dynamical regime.

Deploying neural networks on unconventional hardware demands architectures that co-optimize task accuracy and platform-specific constraints such as energy cost, physical non-idealities, and numerical precision. Existing neural architecture search (NAS) methods are typically tailored to a single hardware family, limiting cross-platform comparison and generalization. We introduce Unconventional Hardware Neural Architecture Search (UH-NAS), a hardware-agnostic, LLM-guided NAS framework that integrates language models as evolutionary operators to co-optimize accuracy and inference energy. By exposing hardware as a swappable backend with per-platform energy models, physical constraints, and non-ideality simulators, UH-NAS enables fair system-level comparisons across various backends without modifying the search algorithm. Tested on optical MZI hardware, UH-NAS discovers more diverse, robust architectures than conventional baselines while outperforming existing LLM-to-NAS approaches. Additional ablations on architecture robustness under non-idealities and the role of system prompts highlight the importance of architecture-hardware co-design for emerging computing platforms.

In this work, we develop theoretical foundation for flow matching with neural-network-parameterized conditional velocity fields. We establish convergence guarantees for gradient descent in the over-parameterized 2-layered ReLU neural network regime. We derive generalization bounds for the conditional velocity-field matching objective. Building on these results, we provide Wasserstein-distance guarantees for the samples generated by the induced flow. Our analysis is based on generalization bound for multi-task representation learning with unbounded losses, which may be of independent interest beyond flow-based generative modeling. These theoretical results are validated through extensive experiments on both synthetic and real-world image benchmarks.

We introduce Temporal Sheaf Neural Networks (TSNN), a temporal link prediction framework that equips each node with a time-varying orthogonal frame and compares node states only after explicit transport between local coordinate systems. In contrast to existing continuous-time graph models that operate in a shared global embedding space, TSNN models node-specific and evolving interaction semantics through dynamic local frames. The model parameterizes per-node frames via efficient low-rank Householder products, preserves stored hidden states exactly under frame updates, and uses a geometric-residual decoder that anchors predictions on transported distances while learning residual corrections. All computations are strictly causal and use only the pre-event history. We show that the symmetric degree-normalized sheaf Laplacian is orthogonally similar to the symmetric normalized graph Laplacian, with the random-walk normalized form similar in the corresponding degree metric; the full-active, feature-scaled diffusion used by TSNN is exactly a metric-gradient step on the combinatorial sheaf Dirichlet energy, with a degree-free monotone-descent and non-expansiveness guarantee. Frame drift perturbs updates only linearly. Across TGB v2 link-prediction and temporal-heterogeneous leaderboards, together with the DGB benchmark suite, TSNN matches or surpasses the strongest prior methods on most benchmarks, with the largest improvements on graphs exhibiting strong node-role heterogeneity. Ablations confirm the distinct benefit of dynamic frames, orthogonal transport, and geometric-residual decoding.

Spiking Neural Networks (SNNs) provide a naturally temporal machine-learning framework. Their neurons maintain an internal state and propagate information through discrete spikes, enabling low-latency temporal inference. Although SNNs are often associated with asynchronous neuromorphic processors, many scientific real-time inference systems rely on conventional synchronous field-programmable gate arrays (FPGAs) and high-level synthesis (HLS) workflows. In this paper we present an extension of hls4ml that enables clock-driven deployment of SNNs trained in pytorch onto FPGA firmware. We demonstrate the workflow using a dense quantised SNN trained on the Heidelberg Spiking Digits dataset where it achieves inference latencies of approximately $34μ$s. We validate the generated design through software reference comparisons, HLS C simulation, HLS synthesis, export, and Vivado synthesis reports. This work opens up the hls4ml toolkit to neuromorphic computing, allowing streamlined optimisation, synthesis, and deployment of SNN models for real-time inference.

With AI increasingly deployed in safety-critical systems, providing formal robustness guarantees for the underlying models is essential. Existing verification methods either rely on overly conservative approximations or incur prohibitive computational costs. For example, the use of lp-norm perturbations in video settings encodes the belief that the adversary can inject noise in every video frame. In practice, adversarial perturbations exhibit structured spatial and temporal correlations, constrained to lower-dimensional, semantically meaningful subspaces. In this work, we study robustness verification of 3D CNNs processing video and volumetric inputs, targeting applications in action recognition (UCF-101), autonomous driving (Udacity), and medical imaging (MedMNIST) exploiting realistic assumptions on adversarial strength by modelling them as spatio-temporal constraints - where the attacker can modify either a subset of frames or patches within a set of consecutive frames. We demonstrate that modelling realistic constraints enables tighter approximations. We introduce Spatio-Temporal Bound Propagation (STBP), a verification framework that computes an exact closed-form characterization of the first convolutional layer and propagates certified bounds through subsequent layers using scalable approximations. Computing the exact closed form provides the tightest bounds for the first convolutional layer. Thus, we utilise approximation methods in the remainder of the network. To spur further progress in this field, we propose ST-Bench, a verification benchmark for autonomous driving and activity recognition, to systematically evaluate verifiable robustness. Compared to existing verification-based approaches, STBP provides stronger robustness guarantees with significantly improved scalability, achieving 1.7x higher certified robust accuracy under identical perturbation budgets.

This paper presents a unified system designed to support precision agriculture by integrating advanced weather prediction, crop recommendation, and a question-answering tool for farmers. We propose two deep learning models -- a Transformer-based Graph Neural Network and a Spatio-Temporal Graph Convolutional Network (STGCN) -- to forecast weather conditions for the next 30 days using data from 1,359 locations in Nepal. The STGCN outperforms the Transformer-based model in accuracy (MSE ~0.011 vs. 0.013), effectively modeling both spatial and temporal dependencies in climate data. These predictions are combined with static soil properties such as pH, moisture, and organic content to generate localized crop recommendations through a scoring algorithm that matches each crop's optimal growing conditions. Additionally, we develop a Retrieval-Augmented Generation (RAG) chatbot that leverages domain-specific agricultural documents to answer farmers' questions in natural language. The entire system is deployed via a mobile application, offering real-time suggestions and conversational support. User feedback confirms the system's usability and relevance, especially in rural settings where personalized farming guidance is limited. Overall, our approach demonstrates how combining machine learning models with local agricultural data can empower farmers with actionable insights, promoting more informed decisions, better crop yields, and increased resilience to climate variability.

Convolutions have successfully transitioned from image processing to the complex realm of non-Euclidean higher-order domains, particularly in hypergraphs. Despite the success in convolution, the exploration of a popular architecture named U-Net remains largely unexplored for hypergraph data due to the lack of well-defined pooling and unpooling operations. This work pioneers the study of U-Net architectures for hypergraph data, addressing the critical challenge of designing effective pooling and unpooling operations that retain maximal structural information from the input hypergraph. Motivated by hierarchical clustering, we propose to construct the pooling and unpooling operators all at once by cutting the clustering dendrogram at different granularities, named the Parallel Hierarchical Pooling (PHPool) and Unpooling (PHUnpool) operators. Unlike existing pooling methods that risk local structural damage through a sequential learning procedure, our PHPool operators are designed in a global and parallel manner to ensure fidelity to the original hypergraph structure with efficient computation while the PHUnpool operators are tailored to perform inverse operations of the PHPools for hypergraph reconstruction. We validate our model through hypergraph reconstruction simulation, hypergraph classification, and node-level anomaly detection, where it demonstrates superior performance over existing state-of-the-art graph and hypergraph deep learning methods.

Fundamental investigations into how different molecular encoding methods affect molecular property prediction remain relatively limited. In this study, we extensively examined the optimal molecular encoding methods for molecular properties prediction using two prevalent structure designs: a classical neural network model (MLP) and a Transformer encoder-based model (MLP+TL). For molecular encoding methods, we investigated several types of fingerprints, including traditional topological fingerprints, substructure-based fingerprints, and string-based representations. These two models were trained on seven well-known molecular datasets to evaluate different input molecular encoding methods based on evaluation metrics. On several biologically relevant classification tasks, including toxicity, mutagenicity, and side-effect prediction, our models consistently achieved average AUC values above 0.9. Rather than relying on external post-hoc explanation methods such as the local interpretable model-agnostic explanation (LIME) or the Deep SHapley Additive exPlanations (SHAP), we leveraged the model's intrinsic attention weights as an internal interpretability signal for identifying potentially important feature. The MLP+TL model using MACCS and PubChem as input can capture chemically interpretable groups that determined the major blood-brain barrier (BBB) permeability and mutagenicity in Salmonella typhimurium. In particular, a comparison between Morphine and Heroin highlighted the role of hydroxyl-related substructures in BBB permeability prediction, which was consistently reflected in the attention weights. Overall, our findings provide practical guidance for selecting effective molecular encoding methods and contribute to the development of interpretable molecular informatics approaches for drug discovery.

Recurrent neural networks maintain a hidden state $h_t$, but its probabilistic meaning is often unclear. We study hidden-state stability through \emph{backward coherence}: the extent to which $h_t$ can be reconstructed from $h_{t+1}$ by a learned backward projector $g_φ$. Under contraction and summable backward drift, the hidden-state sequence forms a quasi-reverse-martingale. This yields almost-sure convergence, rates under mixing, an interpretable limiting representation, finite pathwise stopping times, and a theoretical framework for time-uniform confidence sequences. Simulations support the theory. Backward-coherence regularisation reduces the empirical quasi-martingale total $\hat Q$ by $43$--$58$, reaches stability $28$--$44$ earlier than an unregularised RNN, and gives tracking-error recovery consistent with geometric bounds. Additional tests confirm echo-state forgetting rates bounded by $ρ$ and verify the increment-sum tube $R_t$ with $100$ simultaneous coverage, although $R_t$ is conservative; in practice, the defect-tail proxy $\hat Q_t$ is the more useful monitor. The backward-coherence loss is also equivalent to minimising a Kullback--Leibler divergence in a Gaussian backward model, linking the method to variational inference. Extensions cover $φ$-mixing inputs, change-point tracking, and finite-sample concentration. Three real-data studies further validate the approach. On PhysioNet 2012 ICU data, the Reverse Martingale RNN (RMRNN) matches RNN mortality-prediction AUC while reaching stable representations 13 hours earlier. On FRED-MD, it reduces one-month-ahead forecast error by about fourfold under concept drift. On UCI Human Activity Recognition, it maintains lower post-transition tracking error with geometric decay. The guarantees apply under the stated assumptions; universality is not claimed.

The Forward-Forward (FF) algorithm offers a biologically inspired alternative to backpropagation by replacing gradient-based credit assignment with local, forward-only objectives. While recent extensions have adapted FF to convolutional neural networks (CNNs), existing formulations rely on static channel-class partitions and struggle to perform effectively in complex tasks. In this work, we introduce a learnable channel-class assignment mechanism that enables adaptive, data-driven specialization of convolutional channels, supported by entropy and orthogonality regularization to promote learning performance. We further propose a loss-aware layer contribution strategy that adaptively weights intermediate-layer predictions based on their validation performance, enhancing the effectiveness of forward-only inference. Integrated into residual CNNs, the proposed method achieves consistently superior performance across CIFAR-10, CIFAR-100, and Tiny-ImageNet compared to existing similar forward-only methods. Notably, it establishes new state-of-the-art performance among FF-based models, substantially narrowing the gap with backpropagation. These findings demonstrate that introducing learnable channel specialization and layer contribution weighting significantly enhances the representational capacity of forward-only learning in deep CNNs.

Tensor networks provide efficient representations for compressing large neural networks. By carefully designing shapes and topologies, they can significantly reduce memory and computational costs. However, identifying implicit low-rank structures in large foundation models remains challenging due to their enormous scale and un-structured weight distributions. We propose an adaptive tensorization method that discovers inherent low-rank structure in a target tensor by index ordering. Experiments on weight and KV-cache compression demonstrate improved reconstruction quality compared to baselines.

While confidence calibration is essential for trustworthy decision-making in safety-critical applications, the robustness of calibrated GNNs to adversarial structural perturbations remains largely unexplored. However, studying calibration attacks on graphs presents unique technical challenges: (1) the discrete nature of graph structures complicates gradient-based optimization, (2) existing underconfidence objectives fail to drive predictions toward uniform distributions, and (3) GNNs are highly sensitive to edge perturbations, often causing unintended label changes that violate attack constraints. To address these challenges, we propose a \textbf{Unified Graph Calibration Attack (UGCA)} framework designed for \textbf{worst-case (white-box) analysis} of GNN calibration robustness. UGCA introduces a KL-divergence loss to encourage uniform predictive distributions, a reranking mechanism to reduce label flipping, a hybrid loss to recover labels when violations occur, and beam search to explore a broader adversarial search space. We further provide theoretical insights linking model generalization, dataset complexity, and calibration vulnerability, showing that models with higher accuracy or trained on datasets with more classes are more susceptible under this threat model. Extensive experiments demonstrate that UGCA substantially increases Expected Calibration Error while preserving classification accuracy. Our code is publicly available at https://github.com/CaptainCuong/Graph-Calibration-Attack.git.

Advances in novel materials science enable structures to function as intelligent machines by embedding memory and learning capabilities directly into materials. Our work introduces a physical adaptive material motor unit neural network,leveraging a new generation of controllable actuators composed of wood- and carbon black-based composites, sensitive to temperature and relative humidity. These material actuators are assembled into a motor unit-like structure inspired by muscle contraction trigger, forming an intelligent machine capable of dynamic shading control that can be used, for example, in buildings. The machine is governed by a neural network trained on over 350 experimental data points collected under diverse environmental conditions. By establishing a new data-aware backpropagation training, we show that the machine predicts shading responses and learns to predict appropriate behaviour incrementally as the database expands. We also demonstrate the ability of the machine to optimise configurations to achieve similar shading outputs under two distinct conditions.

Physics-Informed Neural Network (PINN) is a way of including knowledge in the form of equations in Machine Learning methods. Beyond equations, knowledge exists in other forms, such as text and network structure. While existing PINN-based approaches discover equation parameters from data, they rely solely on experimental measurements. We propose a new PINN framework that enriches parameter discovery by incorporating auxiliary knowledge sources. We instantiate our framework for microbiology, where generalised Lotka-Volterra (gLV) serves as a biological foundation for modelling microbial communities. We demonstrate that incorporating knowledge improves microbial community modelling. Our framework enriches the gLV parameters using peer-reviewed metagenomics literature, as text provides biological context on external influences that gLV alone cannot capture. We combine this knowledge with experimental measurements of microbial abundance using a data-driven integration approach. We integrate network-based structural knowledge by explicitly modelling microbial interactions. Our knowledge-inclusive framework infers microbial networks, revealing ecological insights. We validate these findings against ecological roles documented in the literature. We evaluate on real and simulated datasets spanning human- and plant-associated microbial communities. Our framework improves over the state-of-the-art by up to 53%, even without knowledge. Knowledge addition yields gains of up to 23% in Bray-Curtis Dissimilarity-based accuracy and 47% in $\mathrm{R}^2$.

Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.

Accurate vessel trajectory prediction is essential for safe and efficient maritime operations, enabling collision avoidance and supporting route optimization. Although memory-augmented neural networks have recently shown strong performance in pedestrian and road-vehicle trajectory prediction by selectively retrieving relevant information from an external memory, their potential for vessel trajectory prediction remains underexplored. This paper presents an empirical investigation of memory-based trajectory prediction using Automatic Identification System (AIS) data. Experiments on data from the Gulf of Mexico and the New York Bight demonstrate consistent and substantial performance gains over a range of deep learning baselines that do not incorporate an external memory.

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.

Deformable 3D Gaussian Splatting (D-3DGS) re-constructs dynamic scenes from monocular video by deforming a canonical set of 3D Gaussians through a positional-encoded MLP of frame time t. Although fitted to a continuous variable, the MLP couples no two values of t in its architecture and effectively predicts discrete per-frame offsets, leaving temporal smoothness to emerge only as a byproduct of optimisation. We redesign the deformation field as a stack of Closed-form Continuous-time (CfC) cells, a Liquid Neural Network (LNN), that is the closed-form solution of the Liquid Time-constant ODE while preserving every other part of the D-3DGS pipeline. Each cell exposes a sigmoidal time gate that interpolates between two candidate hidden states, baking a learned smooth response to t into the loss landscape without invoking any numerical solver. On the eight D-NeRF and seven NeRF-DS scenes the liquid field matches or exceeds the MLP baseline in aggregate, with its largest gains concentrated on the scenes with the most high-frequency articulated motion. The result is a near-zero-friction architectural design that turns the discrete MLP deformation field into an explicit continuous-time function of t.

Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PINN optimisation is sensitive to numerical precision. Existing implementations commonly use either single precision (FP32), which is computationally efficient but prone to failure modes, or double precision (FP64), which is robust but substantially expensive. This creates a trade-off between computational efficiency and numerical accuracy. To reduce the computational cost of double-precision training while retaining prediction accuracy, we propose a curvature-aware precision controller that adapts numerical precision during training rather than treating it as a fixed implementation choice. The proposed method reuses curvature information derived from the limited-memory BFGS (L-BFGS) optimiser to construct a precision controller, retaining FP32 when lower precision is sufficient and promoting computation to FP64 when the training dynamics indicate numerical sensitivity or precision-limited stagnation. We evaluate the proposed approach on four canonical PINN failure-mode benchmarks and an irradiance-driven ordinary differential equation example. We further test the proposed approach across different neural network architectures. The method consistently matches or even slightly exceeds full FP64 solution accuracy while reducing training time relative to full double-precision training on all benchmark equations. The obtained results indicate that precision sensitivity in PINN optimisation is phase-dependent, and that selectively applying higher precision only during numerically critical stages can lower computational cost without sacrificing predictive accuracy.

Many researchers investigated neural networks with some of their weights fixed to values randomly drawn from a given distribution, e.g., $N(0, I)$. Our proposed HalfNet draws random weights from $N(0, Σ)$, where $Σ$, which defines the geometry of the distribution, has a low-rank factorization that we learn from data. Experiments on MNIST and CIFAR-10 demonstrate that HalfNet can match the performance of fully trained multilayer perceptrons while using substantially fewer parameters. Spectral analysis indicates that much of the predictive power of neural networks lies in the geometry of their weight space rather than in the precise values of individual parameters, and we observe that accuracy scales smoothly with rank. HalfNet is not a neural architecture trick for low-rank structure; it implements a data-dependent random embedding that can also be interpreted through supervised metric learning, or random-feature and kernel perspectives.

A common heuristic used to explain the generalization of first-order gradient methods on non-convex neural networks is that "flat interpolators generalize well" (Hochreiter and Schmidhuber, 1994; Keskar et al., 2017), where flatness can be measured by the trace of the Hessian of the empirical loss. However, Dinh et al. 2017) showed that, using symmetry of the network that can change flatness while keeping the population and empirical losses unchanged, any interpolator can be made sharper or flatter. This result makes the earlier heuristic statement vacuous. In this paper, we show that for learning an unknown multi-index model with $2$-layer non-convex homogeneous neural networks, there is a connection between flatness and generalization, despite the existence of symmetries. This connection pertains to the "flattest" interpolators, i.e., the interpolators that have orderwise minimum flatness among all interpolators. First, we show that there exists a natural class of non-generalizing interpolators whose flatness cannot be made closer to the flattest possible, even using symmetries. Second, we show that for data generated by a sum of single-index models, if the approximation error and label noise are low, any flattest interpolator achieves small population loss, i.e., the flattest interpolators always generalize. This establishes a direct link between flatness and generalization which applies to a large class of activations and realistic data distributions.

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

We investigate the geometric structure of stationary plateaus that arise in the loss landscape of two-layer neural networks with smooth activation functions. We focus on the phenomenon of "neuron splitting" where duplicating a hidden neuron yields an affine set of stationary points in a wider network. We provide a comprehensive classification of all stationary points on these plateaus, determining under what conditions they constitute local minima or saddle points. Our characterization hinges on a per-neuron curvature object we term the "inner Hessian" matrix. Our analysis reveals that the definiteness of the inner Hessian and the choice of splitting coefficients jointly dictate the local geometry of the plateau. We show that "splitting" a local minimum can yield either a mixture of local minima and saddles or an all-saddle plateau, with a concrete sure-saddle region identified under mild assumptions. In contrast, splitting a saddle point always produces a plateau of saddle points. Our results unify and extend prior landscape analyses, elucidating when and how model expansion preserves or alters the nature of stationary points. These findings offer new geometric insights into the effects of width expansion and reparameterization in neural networks.

As neural networks are increasingly deployed in safety-critical domains, testing is essential to evaluate and improve their reliability. Existing testing methods, whether black-box or white-box, primarily use global mutation or coverage-guided strategies, both of which struggle to efficiently uncover diverse model failures while remaining proximate to the original data distribution and semantics. We propose BayesWarp, a testing framework that addresses this limitation by mutating decision-critical input regions identified via interpretable saliency techniques and adaptively guiding the testing process using an uncertainty-aware Bayesian Optimization strategy, enabling the discovery of diverse failures while preserving distributional and semantic proximity to the original data. Evaluation on MNIST, CIFAR-10, and ImageNet across six neural network models shows that BayesWarp improves failure discovery, failure diversity, test case quality, and critical neuron coverage under a fixed mutation budget. These results demonstrate that BayesWarp improves testing effectiveness. Moreover, fine-tuning with the generated failure cases leads to improvements in model performance.

Deep Neural Networks (DNNs) are increasingly being deployed in security-critical and safety-sensitive applications, which makes rigorous testing essential to identify and mitigate model weaknesses. Existing DNN testing approaches explore either the input space or a learned latent space. While latent-space generation can better maintain plausibility than direct input-space mutation, current methods still face a trade-off among exploration controllability, failure diversity, and seed-relative semantic drift. To overcome these limitations, we propose Latte, a black-box testing framework that generates semantically proximate, diverse, and fault-revealing test cases by leveraging the latent space. Specifically, Latte encodes each input seed with a pre-trained VQ-VAE and performs a seed-centered, one-step latent mutation along directions defined by anchors sampled from alternative classes, followed by quantization and decoding back to the input space. This explores local neighborhoods around each seed within the learned latent manifold, resulting in a larger number and broader diversity of oracle-triggering prediction discrepancies under the same budget. We evaluated Latte on 5 datasets and 10 DNN models in single-model and multi-model testing scenarios. Across the evaluated datasets and models, Latte improves fault exposure and behavioral diversity under matched testing budgets. Under the single-model setting, it also maintains low seed-relative semantic drift with respect to the source seeds.

Determining the neutrino mass ordering remains a central open problem in particle physics. While next-generation long-baseline experiments are expected to resolve this question, current data provide limited sensitivity because the spectral differences between normal and inverted ordering are subtle and entangled with parameter degeneracies. We investigate a machine-learning strategy for mass-ordering determination using a feed-forward neural-network classifier trained on synthetic long-baseline datasets generated with three-flavour oscillation probabilities, matter effects, and statistical fluctuations. We evaluate the classifier against standard $χ^2$ and $\log\mathcal{L}$ approaches using common discrimination metrics, including receiver-operating-characteristic curves, to quantify sensitivity and to illustrate how operating points can be selected to prioritise purity or efficiency. We find that the neural network achieves performance comparable to conventional fits for the scenarios studied, providing a flexible, independent cross-check of established analyses. The framework can be extended to incorporate systematic uncertainties and to explore joint inference of oscillation parameters, and it may also serve as a pedagogical tool for introducing machine-learning methods in neutrino physics.

Training quantum neural networks (QNNs) on quantum hardware is currently bottlenecked by the cost of gradient estimation: standard parameter-shift methods require a number of circuit evaluations that grows quadratically with the number of trainable parameters, making hardware-based optimisation impractical beyond small system sizes. In this work, we introduce a training framework that reduces this cost to logarithmic in the number of qubits, making gradient-based QNN optimisation feasible on near-term hardware at increasing scales. Our framework combines three co-designed ingredients: (i) a structured, subspace-preserving Butterfly circuit architecture with $O(n \log n)$ parameters and logarithmic depth; (ii) a layer-wise training strategy that confines on-hardware optimisation to one small, well-structured layer at a time; and (iii) a parallelised parameter-shift rule that exploits the commuting structure within each Butterfly layer to extract all gradients in a constant number of circuit executions. Together these reduce the number of distinct circuit evaluations per optimisation step from $O(n^2)$ to $O(\log n)$. We validate the framework on clinical data imputation using the MIMIC-III electronic health record dataset, a demanding benchmark sensitive to optimisation instability and model variance. Hybrid classical-quantum models are trained directly on IonQ Forte Enterprise trapped-ion hardware at 16 qubits without performance degradation relative to ideal or noisy simulation and via tensor-network simulation at 32 qubits, with 32-qubit inference executed on hardware. The resulting models match or exceed strong classical neural baselines in downstream patient survival prediction while exhibiting reduced variance across runs, demonstrating that the proposed framework enables practical, scalable QNN training under realistic hardware constraints.

Neural networks suffer from shortcut learning, where learned features generalize well to the training set but not to in-distribution (ID) or out-of-distribution (OOD) test sets. Existing studies are all based on a few standard benchmarks, which are shape-driven. Numerous application domains, however, are texture-driven. In this work, we present shortcut learning analysis for texture-driven domains, and compare it with that of a standard benchmark. We show that texture-driven domains suffer from low-frequency shortcuts. They make the majority of their decisions based on a few low-frequency components (LFCs) with a skewed spectral behavior, despite that their classification information is in higher-frequency, fine-grained details. Pruning LFCs from training and test sets eliminates the shortcut and provides a more balanced spectral behavior, improving the ID accuracy by up to 8%. We show that low-frequency shortcuts make the models highly vulnerable to OOD corruptions, leading up to 70% accuracy drop compared to the ID accuracy. Pruning LFCs significantly improves robustness to low-frequency corruptions, by up to 40%, and introduces a trade-off for high-frequency corruptions; the balanced spectral behavior provides a better generalization performance, whereas the increased dependence on high-frequency features reduces it. OOD accuracy depends on the interaction between these two factors.

The graphical representation of the brain offers critical insights into diagnosing and prognosing neurodegenerative disease via relationships between regions of interest (ROIs). Despite recent emergence of various Graph Neural Networks (GNNs) to effectively capture the relational information, there remain inherent limitations in interpreting the brain networks. Specifically, convolutional approaches ineffectively aggregate information from distant neighborhoods, while attention-based methods exhibit deficiencies in capturing node-centric information, particularly in retaining critical characteristics from pivotal nodes. These shortcomings reveal challenges for identifying disease-specific variation from diverse features from different modalities. In this regard, we propose an integrated framework guiding diffusion process at each node by a downstream transformer where both short- and long-range properties of graphs are aggregated via diffusion-kernel and multi-head attention respectively. We demonstrate the superiority of our model by improving performance of pre-clinical Alzheimer's disease (AD) classification with various modalities. Also, our model adeptly identifies key ROIs that are closely associated with the preclinical stages of AD, marking a significant potential for early diagnosis and prevision of the disease.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

Alternatives to soil-based horticulture, such as hydroponics, have been developed to respond to food distribution concerns for dense urban centers. A new system was developed to track an individual lettuce plant's growth in a hydroponic environment, utilizing streams of measured information and available models to continuously update the growth trajectory estimates for a plant. These "digital twin" models were integrated into an operating hydroponic greenhouse, with custom horticultural and sensor hardware to grow and measure relevant information. To aid in updating model parameters, plant yield was continuously measured with a custom neural network, using RGB-D images of the plants as an input. The network, trained on a collected dataset of 1300 images, was able to estimate mass within 1.5 g of the ground-truth value. After integration into the custom system, digital twin growth projections could approximate future yield between one and four days in the future, maintaining around a 2 g forecasting error.

The success of the transformer architecture as the backbone of modern LLMs is in large part due to its use of attention layers. An attention layer follows the standard neural network paradigm: it takes the residual stream as input and thereby produces context-dependent query, key, and value vectors. However, we find that model performance meaningfully improves when deeper layers learn only a context-free value vector to preserve the original token information, without drawing on any context from the residual stream. When the model has access to this context-free value vector, adding back the context-dependent component provides little additional benefit for aggregate benchmark performance. Such context-free value vectors can be stored as sparse model parameters, eliminating the need to recompute or persistently cache these values. Through systematic ablations on the key design choices for such context-free value vectors, we propose Bank of Values (BoV), a new way of computing value vectors in attention by learning a lookup table of token-specific value vectors for each of the last third of layers. Across 135M and 780M models, BoV improves validation loss over standard attention and, at 780M, the average score across 21 benchmarks, matching the previous best method that adds token information to the value vector with less compute and memory.

We introduce Lie groupoid equivariant neural networks as a specialization of recently proposed topological category-equivariant neural networks to the differentiable setting. Lie groupoid equivariant neural networks are composed from Lie groupoid lifting convolutions and Lie groupoid convolution layers, and we show how for suitable Lie groupoids they are equivalent to certain Lie algebroid-equivariant neural networks. We additionally describe groupoid invariant global pooling as a generalization of group invariant global pooling. Furthermore, we show that each of the aforementioned layers is a special case of recently introduced admissible category-equivariant layers by demonstrating that they define continuous natural transformations between continuous feature functors.

State-space models are traditionally based on physical knowledge, but multi-step predictions from these physical models can be poor due to model inaccuracy. Black-box deep learning has shown promise as an alternative. However, these methods rely on the availability of large datasets and potentially available physical knowledge is neglected. We propose the PG-RSSNN, a physics-guided recurrent state-space neural network that incorporates recurrent structures to enable the use of non-saturating activation functions in multi-step prediction. It mitigates the vanishing gradients and eliminates the risk of numerical divergence in training seen in existing structures that feed back state estimates. Results across multiple systems with various physical model imperfections, from linear state-space models with Gaussian noise to a robotic arm and a cascaded water tank system, show that the proposed PG-RSSNN maintains stable training behavior, and improves multi-step predictions, as compared with black-box neural networks and physics-only models, even with limited training data and when physical models are only partially known.

Modern neural networks are highly susceptible to adversarial perturbations. In this work, we identify that part of this vulnerability stems from the sensitivity of the widely used fully connected (FC) classifiers to such perturbations. In contrast, simple $\ell_2$ distance-based classifiers exhibit significantly greater robustness. We provide thorough theoretical and empirical analysis showing that while FC classifiers' high sensitivity makes them discriminative, it also makes them vulnerable. Conversely, $\ell_2$-classifiers' insensitivity grants robustness but limits performance. Motivated by this trade-off, we propose a novel $\ell_2$-reclassifier based on a Hybrid Prototype Mixing (HPM) framework. This method retains the discriminative power of FC classifiers while leveraging the robustness of $\ell_2$ distance. It yields $\ell_2$-distance-based predictions by fusing two prototype types: (1) stable, dataset-level prototypes updated via EMA, and (2) dynamic, batch-level prototypes generated from the FC classifier's predictions using a Straight-Through Estimator (STE). However, this dynamic, STE-based architecture introduces significant challenges for evaluation, such as gradient obfuscation and forward discontinuity. To address this, we propose a new, rigorous evaluation protocol, the Mixed Surrogate Attack (MSA), which uses multiple surrogates along with powerful AutoAttack to ensure a fair and robust assessment. Extensive experiments demonstrate that our lightweight, plug-and-play module, with minimal fine-tuning, effectively enhances the adversarial robustness of various existing SOTA adversarially trained models.

Predicting patient-specific facial soft-tissue deformation is critical for iterative orthognathic surgery planning. However, current computational methods face a strict accuracy-efficiency trade-off: high-fidelity Finite Element Methods (FEM) are computationally prohibitive, whereas pure deep learning models often produce biomechanically inconsistent results. While Physics-Informed Neural Networks (PINNs) offer a promising avenue, learning the complex heterogeneous mechanics of bone--soft-tissue interactions with only partial clinical supervision (i.e., outer facial surfaces) remains highly unstable. To overcome these challenges, we present PINNOCHIO, a novel physics-informed framework for facial soft-tissue simulation. PINNOCHIO introduces a hybrid sequential decomposition that explicitly decouples discontinuous bone--soft-tissue interface movements from continuous volumetric hyperelastic deformation. This structural separation enables stable training and facilitates a physics-enabled sim-to-real adaptation strategy, ensuring internal biomechanical consistency without requiring volumetric ground truth. Evaluated on a 40-patient clinical cohort, PINNOCHIO outperforms existing baselines in both surface accuracy and physical validity. Furthermore, it achieves a substantial speedup over FEM, successfully resolving the accuracy-efficiency trade-off to provide a highly reliable and practical tool for interactive surgical planning.

Spiking neural networks (SNNs) are increasingly studied for network intrusion detection, but comparative evidence on how neuron models and spike encodings affect performance remains limited. Evaluation choices can influence results when preprocessing, capture structure, or scenario information crosses the train--test boundary. We evaluate nine snnTorch neuron families with three spike encodings, yielding 27 SNN configurations across four intrusion-detection benchmarks. We screen the design space and then repeat the evaluation using train-only transforms, seed-independent partitions, and capture- or scenario-aware separation. In our study, LeakyParallel/latency ranked first in both 27-configuration evaluations. The leading configurations identified during screening also remained largely consistent under confirmation, indicating that screening preserved the ordering useful for design selection even when the measured performance changed under the stricter protocol. We then examine what latency coding contributes by holding the active spike set fixed and changing only its temporal organization. At the main T=25 operating point, mapping feature magnitude to spike time improved macro-F1 on all five confirmation protocols, while spreading the same spikes through time without that mapping could either improve or reduce performance. This shows that the effect of latency coding comes from both the information carried by spike time and how those spikes interact with the temporal dynamics of the network. Finally, sparse input encoding does not directly translate into sparse internal activity, although latency coding requires fewer fanout-weighted operations than rate coding in the evaluated models. Overall, the results show that SNN design choice, evaluation protocol, spike-timing representation, and computational activity should be examined separately when applying SNNs to static network-flow data.

Accurate modeling of leaf spectral reflectance from physiological and biochemical traits is essential for advancing remote sensing applications in plant science and precision agriculture. Widely used radiative transfer models, such as PROSPECT-PRO, rely on generalized trait-reflectance relationships developed from a wide range of species, which may not fully capture the spectral behavior of specific crops like grapevines. In this study, we developed a trait-to-spectra prediction model using a multi-head attention neural network trained on a grapevine-specific dataset that includes 16 leaf traits measured across multiple varieties, growth stages, and years. The model was evaluated using stratified 5-fold cross-validation and achieved an average coefficient of determination (R^2) of 0.84 and normalized root mean squared error (NRMSE) of 1.52 percent, demonstrating high accuracy and generalizability. When compared to PROSPECT-PRO in forward mode, the neural network exhibited lower mean absolute error (MAE), especially in the near-infrared (NIR) and shortwave-infrared (SWIR) regions. These results emphasize the importance of species-specific modeling approaches and show that integrating biochemical and structural traits into data-driven architectures can significantly improve spectral prediction. The proposed model provides a robust framework for generating accurate leaf-level reflectance data, with potential applications in canopy trait retrieval, vineyard monitoring, and remote sensing-driven crop management.

Neural network compression is commonly achieved by pruning parameters based on local importance scores, e.g., magnitude-based pruning. We propose a complementary approach that compresses models by aggregating neurons with similar functional behavior rather than removing weights independently. Our method encodes a trained network as a polynomial ODE system and applies a lumping method called Approximate Forward Differential Equivalence to identify neurons with approximately matching induced dynamics. A single tolerance parameter, $\varepsilon$, controls the compression level and induces a smooth trade-off between model size and predictive accuracy. We evaluate the method on synthetic datasets derived from nonlinear dynamical systems with known ground-truth behavior and on public regression benchmarks. Across both settings, the proposed approach achieves substantial parameter reduction while preserving accuracy, and consistently compares favorably with magnitude-based pruning and Wanda at similar compression levels. These results suggest that differential equivalence-based aggregation is a principled and effective alternative to conventional weight-centric pruning.

Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics-informed neural networks (PINNs) and their domain-decomposed variants (FBPINNs) offer a mesh-free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum-classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical-to-quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end-to-end automatic differentiation through the classical PINN, the quantum circuit, and the physics-informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave-based inverse problems beyond geophysics, including medical ultrasound tomography and non-destructive evaluation.

Mineral prospectivity modelling (MPM) underpins exploration economics, yet most operational pipelines reduce to data-driven classifiers trained on shallow surface proxies. Such models are blind to the subsurface physics that actually localises ore: heat advection, fluid flow, and lithology-dependent precipitation. We present Korzhinskii-Net, a 2-D radial physics-informed neural network (PINN) that couples Darcy flow, advective-diffusive heat transport, and a softplus-saturated reaction rate into a single differentiable forward model, weakly supervised by surface and remote-sensing proxies. The network is named after Dmitri S. Korzhinskii (1899-1985), whose theory of infiltration metasomatism provides the physical scaffold. We evaluate Korzhinskii-Net on six ore provinces spanning three commodity classes - Udokan (sandstone-hosted Cu), Sukhoi Log, Olimpiada, and Berezovskoye (orogenic Au), Vorontsovskoye (Carlin-type Au), and Dalnegorsk (skarn polymetallic) - under a fair, leakage-controlled 5-fold cross-validation protocol with hard ring-shaped negatives and baseline proxy features disabled. Korzhinskii-Net attains a mean PR-AUC of 0.708 versus 0.235 for the strongest classical baseline (support vector machine), and a mean fractional rank of 0.036 versus 0.475. The improvement is consistent across all six provinces and three commodity systems, suggesting that physics-informed differentiable simulators, even when constrained only by global open-data proxies, can recover localisation patterns that pure feature-based learners systematically miss. We release the full pipeline and evaluation harness as open source.

A simple way to improve the performance of almost any machine learning model is not to train a single but several models with diverse algorithms which will make slightly distinct kinds of predictions and errors on the same data, and thus improve the average predictions and robustness. However, making predictions using a whole ensemble of models is cumbersome and computationally too expensive to allow deployment to a large number of users, especially if the models are large neural nets. In response to this, we introduce a layer and point wise projection mapping, which maps student and teacher representations into an aligned high-dimensional embedding space during training process. The proposed approach combined with LoRA injection reduces the student model trainable parameters to less than 1% of the teacher model, while significantly improving word error rate (WER) compared to other distillation methods, as demonstrated in ablation studies. Unlike a mixture of experts, our method can be trained rapidly and in parallel.

We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability. Using graph representations associated with finite groups, including Cayley graphs (CG), the proposed model is trained to distinguish solvable and non-solvable groups using structural graph information alone. The framework is evaluated on groups outside the training dataset in order to investigate the extent to which GNNs can learn algebraic properties arising in group theory. More broadly, the present work explores the relationship between algebraic structure and graph-based geometric representations of finite groups. The present study is intended as a proof-of-concept investigation of whether GNNs can learn algebraic properties of finite groups from graph-based representations

To understand how a neural network (NN) functions and makes predictions, it has become increasingly clear that analyzing only the input domain is insufficient -- one must also examine its internal inference mechanisms to capture the complete picture. To explain the internal inference mechanisms of such models, it is essential to analyze the importance of latent representations for a given task. In this paper, we propose the \emph{normalized relevance measure} (NRM) framework -- a novel general explanation procedure that attributes relevance to \emph{arbitrary sets of neurons across layers of arbitrary architectures}. In the NRM framework, relevance of selected neurons is explicitly defined as a normalized signed measure, constructed using simple operations -- marginalization and conditioning based on additive and multiplicative laws -- in analogy to the probability measures. The normalization property further guarantees comparability across layers. The NRM framework subsumes existing propagation-based explanation algorithms by explicitly identifying the underlying quantity being computed. We demonstrate the utility of the framework in computer vision applications, where joint relevance analysis across multiple layers reveals key information flows in VGG16 networks. Overall, the NRM framework provides a general, mathematically grounded approach to understanding how modern NNs propagate information, offering a versatile and broadly applicable foundation for explainable artificial intelligence.

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.

In this paper, we show the possibility of a direct injection of algorithms into neural network architecture. We focus on a complex algorithm, that is, Cocke-Youger-Kasami (CYK) for parsing context-free grammars in Chomsky Normal Form and we propose CYKNN, a simple recurrent neural network architecture for encoding the CYK algorithm in trainable matrix-vector multiplications.We experimented with a very simple grammar with 4 variations showing that our approach outperforms existing LLMs with more than 20B parameters with an in-context learning setting and smaller LLMs of the Qwen family fine-tuned with LoRA. Our attempt paves the way to a different approach to neuro-symbolic methodologies.

We show that contrary to conventional wisdom in the community, graph neural networks (GNNs) are not continuous with respect to all natural modes of graph convergence. As a result, GNNs may generate substantially different latent representations for graphs that are very similar. In particular they assign vastly different latent embeddings to graphs that represent the same underlying object at different resolution scales. We trace this failure of continuity back to a structural obstruction arising from commonly used information-propagation schemes. Building on this insight we then derive a principled modification to standard GNN architectures which equips models with continuity across scales. The proposed modification enables consistent integration of distinct resolutions and reliable generalization between them. We systematically validate our theoretical findings in a wide range of numerical experiments.

We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including $S^2$, $SO(3)$, $\mathrm{SPD}(n)$, and permutation-quotiented point clouds.

Solving high-frequency partial differential equations (PDEs) with neural networks is notoriously difficult due to the spectral bias of conventional architectures. We propose the Multi-Scale Separable Fourier Neural Network (MS-SFNN), a framework designed to overcome this limitation by explicitly encoding multi-scale Fourier features within a separable representation. The network factorizes the solution into $d$ single-coordinate subnetworks with fixed, randomly initialized weights; these subnetworks are combined via element-wise products to form a rich set of basis functions. This separable construction scales linearly with the problem dimension, thus inherently alleviating the curse of dimensionality. Crucially, each subnetwork is equipped with trainable scaling factors coupled with cosine activations, providing an adaptive mechanism for multi-scale frequency selection that endows the model with strong spectral approximation capability. The PDE solution is expressed as a linear combination of these learned basis functions, and the combination coefficients are determined by solving a large-scale least-squares system. To resolve the memory bottleneck in high-frequency or three-dimensional settings, we replace automatic differentiation (AD) with analytical derivatives of the basis functions and use a memory-efficient batched QR decomposition for solving the least-squares systems efficiently. Extensive numerical experiments demonstrate that MS-SFNN achieves superior accuracy and substantially outperforms state-of-the-art methods, including Physics-Informed Neural Network (PINN) and the Separated-Variable Spectral Neural Network (SV-SNN).

Graph Machine Learning as a Service (GMLaaS) platforms increasingly implement explainability interfaces to meet regulatory transparency requirements. However, this transparency creates exploitable vulnerabilities for model extraction attacks. We present the first model extraction attack specifically designed for graph classification under strict black-box constraints where the attacker observes only discrete class labels and binary explanation masks (no probability scores, gradients, or confidence values). Our method (1) uses model explanation outputs to guide Monte Carlo edge sensitivity estimation toward decision boundaries, with Hoeffding concentration guarantees on estimation accuracy and (2) exploits explanation subgraphs to efficiently narrow the boundary search space. Extensive experiments on benchmark graph datasets across multiple domains demonstrate our method's superiority over comparable baselines. These findings demonstrate that such explainability interfaces create exploitable attack surfaces, informing both defensive mechanisms and policy frameworks for explainable AI mandates. The implementation code is provided in https://github.com/LabRAI/XSTEAL/.

Always-on edge systems must keep learning as conditions change under tight compute budgets and must detect unreliable predictions. Bayesian binary neural networks are attractive in this setting, but mean-field Bernoulli posteriors can saturate on long non-stationary streams, wiping out epistemic uncertainty and freezing plasticity. We propose BiMU, derived from a bounded-memory variational objective that balances stability, plasticity, and forgetting. BiMU combines a data term with controlled relaxation toward the prior and an uncertainty-dependent step size that prevents saturation and sustains informative uncertainty. This non-degenerate posterior enables fully online, buffer-free active querying via Monte Carlo disagreement, reducing label queries and backpropagation updates under imbalance. BiMU sustains learning and strong OOD detection on 1000-tasks Permuted-MNIST, and on OpenLORIS-Object achieves up to 32$\times$ label/update savings at matched accuracy under class imbalance and feature compression.