Neural Network

Wallet reputation scores decide who receives an airdrop, who can borrow, and who enters an allowlist across decentralised finance. They almost always begin as hand-written formulas: compositions of clamped logarithmic, linear and square-root transforms over behavioural features, with every threshold and point award set by hand. Such a formula is readable and deterministic, but it is piecewise and non-differentiable, it cannot improve as data accumulates, and it cannot distinguish a feature that is genuinely zero from one its pipeline failed to capture. We present zScore-N, the neural network that replaced ours in production. The formula served as its teacher: calibrated against 5,208,952 wallets sampled across 2019-2024 and verified to reproduce production output to within 2.3e-13, it supplies unlimited labelled training data at zero label noise. The trained network reproduces it to 0.58 points RMSE on the 1000-point scale (R^2 = 0.99997), against 2.25 for gradient-boosted trees and 28.04 for linear regression on identical features and splits. Trained with missing-value masks against uncorrupted targets, it halves the error that incomplete data introduces: at 10% feature-level missingness the formula drifts 51.4 points from its own complete-data output with a systematic -12.5 point bias, while the network drifts 17.9. The network carries the score at production scale, across a population of millions of wallets spanning six orders of magnitude in size and activity.

The learning-rate schedule is a consequential choice in training deep networks, yet the policies in common use are heuristic, and published comparisons are hard to read, because architecture, dataset, and budget tend to vary alongside the schedule. We study BrachistoneLR, a schedule built by mapping the vertical coordinate of the brachistochrone, the curve of fastest descent under gravity, onto the range between a peak and a floor rate. Expanding the definition shows it to be cosine annealing with the half-period set to E - 1 instead of E, the configuration a standard implementation gives when its period argument is one less than the number of epochs. The rate therefore reaches its floor at the last epoch trained rather than one epoch later, and we show this difference decays as E^-2, making it a short-horizon effect. We then benchmark six schedules over 72 runs on three image classification datasets (MNIST, Fashion-MNIST, CIFAR-10) and four architecture families (fully connected, convolutional, recurrent, residual), fixing the optimizer, data pipeline, and evaluation protocol so that only the schedule varies. Schedules that fall smoothly from peak to floor beat the constant rate and calendar-based decay by margins that grow with task difficulty, reaching 2.5 points of dataset mean on CIFAR-10. Within that leading group, BrachistoneLR, cosine annealing, and warmup-cosine lie within 0.06 accuracy points and 0.17 of a mean rank, which one seed per configuration cannot separate. BrachistoneLR is best on both residual networks and has the highest CIFAR-10 mean, and it sets no milestones, decay factor, warmup length, or restart period. We conclude that the shape of a schedule matters more than its parameterization, that the choice of whether to use a smooth schedule matters more than the choice among them, and that the terminal-rate distinction is worth attention only over short horizons.

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a meshfree alternative that integrates physical principles into the learning process. This presents a new paradigm compared to traditional discretization methods and purely data-driven machine learning techniques. While promising, PINNs are not a panacea; they inherit challenges such as spectral bias and unstable convergence. Moreover, their potential in seismology remains largely unexplored. In this work, we provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology. We investigate the performance of PINNs on problems with varying degrees of complexity across various seismic sources and parameter models, from constant to highly heterogeneous settings. A pivotal aspect of our work involves investigating whether embedding physical principles directly into the network architecture enhances convergence and accuracy. We test an extensive range of neural architecture designs, from unrestricted, uninformed PINNs to highly specialized ones. We find that integrating an understanding of wave physics into the network design significantly improves accuracy. For instance, introducing a custom wavelet or plane wave layer, coupled with encoder and decoder layers, consistently yields a relative $L_2$ error approximately half that of the standard PINN, as evidenced across numerous experiments. We further demonstrate that this novel architecture enhances accuracy when applied to the acoustic wave equation, underlying the versatility of our network. Another key contribution of our research is the successful conditioning of PINNs on seismic source locations. This signifies a considerable advancement towards rapid seismic hazard detection and seismic analysis.

Physics-informed neural networks (PINNs) struggle on PDEs whose governing physics varies across the domain. We trace this to a structural property of standard coordinate networks: their neural tangent kernel (NTK) is translation-variant and lets training points of large coordinate magnitude disproportionately influence predictions elsewhere, producing long-range coupling and gradient conflict during training. We show analytically and empirically that mixture-of-experts (MoE) architectures with centered, compact-support routers yield a uniformly banded NTK whose kernel-regression weights decay exponentially with distance, localizing the learning. Building on this, we propose \emph{Latent-MoE}, which interleaves domain-aware MoE blocks within a shared backbone. Unlike FB-PINNs or X-PINNs, which rigidly partition both the domain and the parameters so that the parameters on different subdomains are updated independently, Latent-MoE is designed to preserve the localization benefit of domain-aware routing while allowing capacity to flow across regions through the shared backbone. On standard homogeneous-physics benchmarks Latent-MoE is competitive with established baselines; on benchmarks with multi-stage time-variable physics, where global models and rigid domain decompositions both fall into spurious solutions, it improves over them by more than an order of magnitude, with markedly reduced gradient conflict during training.

Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the $\ell^2$ loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.

We study the use of graph neural networks (GNNs) for finding approximate ground states of Ising models. Efficiently finding these ground states is of broad significance because many combinatorial optimization problems can be formulated as an Ising model with the appropriate choice of couplings and fields. Exactly solving these problems is hard but there are many good heuristic methods. A lineage of these heuristics build from mean-field approximations: one approach uses the leading eigenvector of an appropriately defined matrix, another is the min-sum algorithm, also known as the energetic cavity method. Without modification, GNNs perform worse than both of these methods. We consider small modifications to the GNN to incorporate these heuristics and find that this considerably improves performance. While the modified approach is competitive against other deep-learning approaches, we still find that simulated annealing is reliably at least as good as deep learning methods for the same computational cost.

Segmentation of complex structures in X-ray tomographic data is a fundamental task in biomedical research, but it often requires large amounts of precisely annotated data, making fully supervised approaches costly and difficult to scale. In this study, weakly supervised deep learning is investigated as a strategy to reduce annotation effort while maintaining accurate segmentation. A two-dimensional convolutional neural network based on the nnU-Net framework was adapted to a weak supervision setting using sparse dot-based annotations, complemented by a limited number of fully segmented images. The approach was evaluated on high-resolution microCT slices of rat kidneys, targeting the segmentation of renal glomeruli, which are small, low-contrast anatomical structures. Results indicate that weak supervision provides a meaningful learning signal, enabling reliable localization of glomeruli even in the absence of dense labels. Incorporating a small set of high-quality annotations substantially improves segmentation performance, approaching that of a fully supervised model. These findings highlight the potential of weakly supervised learning as an annotation-efficient strategy for the analysis of complex structures in X-ray tomographic data, and suggest that alternative loss formulations tailored to sparse annotations may further enhance performance.

Accurate city-level IP Geolocation is an important enabler for the modern digital ecosystem, underpinning services ranging from local content delivery and targeting to digital rights enforcement. However, traditional heuristic and database-driven methods often struggle to resolve the complex, non-linear allocation patterns of modern network infrastructures, particularly within the exploding IPv6 address space and transient mobile networks. In this paper, we introduce IPGeoAI, a novel deep learning model architecture that reframes geolocation from a static lookup problem to a sequential modeling task. Our approach utilizes the Transformer Encoder to capture hierarchical dependencies inherent in IP subnet structures. We propose a method to resolve geographic ambiguity by integrating unstructured semantic context via a Zero-Shot LLM Feature Extraction pipeline. We utilize Large Language Models to transform raw, noisy Autonomous Systems (AS) descriptions into structured, domain-specific metadata (such as 'University' vs. 'ISP' or 'Global' vs. 'Local') via an offline pre-computation process. By fusing these semantic signals into the network via a Multi-Head Cross-Attention module, we bridge the gap between numerical network topology and real-world semantic identity. Extensive offline evaluation on a proprietary dataset spanning 200,000 cities demonstrates that IPGeoAI significantly outperforms a leading external vendor in city-level granularity. By adopting a hierarchical inference strategy that refines coarse-grained country signals, our model achieves a 6% improvement in city-level accuracy while extending coverage to 100% of the traffic. Furthermore, in large-scale online production tests, the model drove a statistically significant +0.35% improvement in our 1st-tier downstream use cases metric.

Neural fields (NFs) map continuous coordinates to signals such as color or density, but fast high-quality reconstruction from sparse observations remains difficult. Classical Neural Tangent Kernel (NTK) regression gives closed-form fits, yet it is fundamentally linear and cannot accumulate reusable task priors. We develop three algorithms that address these gaps. NTK-KIP learns a distilled support set of coordinates (and optional labels) so that a finite NTK can inpaint large missing regions from little observed data, yielding a compact non-linear representation instead of a raw kernel solve. MetaQuill meta-learns a shared initialization for an INR so that new scenes can be adapted by updating only a small task-specific weight offset, which provides true feature learning and a reusable prior. Finally, MetaQuill-KIP fuses both ideas: it seeds the task with a KIP-style non-linear warm start, then refines only that small offset around the meta-learned initialization. MetaQuill-KIP achieves high-PSNR reconstructions and semantically plausible inpainting under very sparse observations, while requiring only lightweight per-instance adaptation, whereas diffusion-style baselines typically depend on large pretrained generative priors and costly per-image tuning. This shows that NTK-driven neural fields can be made both non-linear and meta-learnable, narrowing the gap between analytic kernels and practical few-shot reconstruction.

Graph neural networks (GNNs) are a class of neural networks suitable for learning on graph-structured data. Their application to spatial data is a natural extension, however its relatively unclear which message-passing operations, architectural configurations, and graph representation is best suited for classifying changes to objects in electronic navigational charts (ENCs)--geospatial vector datasets used for marine navigation. Maintaining these datasets is a challenge, and categorizing changes to objects in the ENC based on their significance to navigational safety is of particular importance. Here, we propose to represent these vector navigation datasets as a graph structure where the spatial objects serve as nodes and their spatial and semantic relationships form edges. We encode both the old ENC dataset and new ENC dataset into a pair of graphs and frame the task as a graph-pair classification problem. Building on this representation, we investigate the use of GNN architectures to classify whether the encoded graphs constitutes a critical or non-critical risk to navigational safety. We train and evaluate several GNN architectures and model configurations on ENC changes reviewed by maritime experts. Our results demonstrate that graph-based representations improve the classification of ENC updates, providing a scalable approach for automating or improving ENC maintenance workflows.

Physics-informed neural networks (PINNs) offer a promising framework for modeling semiconductor devices, yet standard architectures struggle with severe numerical stiffness and multiscale spatial discrepancies inherent to oxide heterostructures. Here, we demonstrate a cascaded PINN architecture coupled with a custom second-order Chebyshev second kind polynomial spectral optimizer (DSO V2 Hybrid) to model ion-electronic drift-diffusion transport in Pt/SrTiO$_3$/Si memristive heterostructures across a 20 nm STO film on a 380 $μ$m Si substrate. By isolating potential, carrier density, and vacancy transport into four sequentially trained sub-neural-networks, our model circumvents condition numbers exceeding $10^{16}$ without operator splitting. The trained surrogate reproduces experimental conductive-AFM current-voltage hysteresis ($R^2 > 0.96$) while ensuring strict Poisson consistency across continuous space. Compared to conventional finite-element solvers (e.g., COMSOL), the PINN surrogate enables differentiable inverse parameter estimation and linear time inference.

Bitcoin's pseudonymous nature makes it challenging to analyze user-level activity, since a single user may control multiple identifiers (addresses). Existing heuristic-based methods attempt to identify addresses belonging to the same user, but they often produce flat cluster assignments with limited modularity and are prone to errors such as merging different users together. In this work, we propose a method for refining heuristic-obtained clusters by grounding our clustering on contrastive embeddings yielded by graph neural networks. Our contributions are threefold: (i) we release a publicly available dataset of Bitcoin transaction graphs containing a substantial number of clusters; (ii) we propose a methodology for learning address embeddings consistent with heuristics, and back it up with theoretical guiding intuitions; (iii) through hierarchical clustering, we enable a finer analysis of heuristic clusters and provide a quantitative criterion for flagging suspicious merges.

Higher-order interaction components are important for signal, image, and video modeling, but explicit high-order operators often suffer from rapidly increasing parameter and computational costs. This paper presents kVNN, a learnable kernelized Volterra Neural operator for compact higher-order filtering. The motivation is to use kernelization to improve the efficiency of Volterra-type neural operators while providing a structured interpretation of their higher-order components. The proposed formulation combines the order-wise structure of Volterra filtering with learnable polynomial-kernel atoms, allowing different interaction orders to be represented by separate learnable centers and coefficients. This order-decoupled representation avoids explicit high-order tensor parameterization and can be implemented as a CNN-compatible layer. Experiments on representative vision tasks show that kVNN achieves a favorable accuracy--efficiency trade-off.

We describe a neural network architecture and training procedure designed to model electronic ground and excited states of arbitrary molecular systems. By indirectly learning a latent, implicit basis representation of the electronic-state Hamiltonian, the model offers a unified treatment of multiple electronic states, conical intersections, and non-adiabatic couplings. The formalism can be further extended to learn consistent latent representations of additional operators such as transition dipole moments, for example. To demonstrate the general capabilities of our architecture, we train and evaluate networks on two realistic photochemical systems, thymine and azobenzene. The resulting models accurately reproduce energies and oscillator strengths for the ground- and low-lying excited states relevant to the photochemistry of these systems. We highlight the performance of the trained networks by studying critical molecular geometries, including conical intersections and excited state minima. By construction, the proposed framework also recovers the emergence of Berry phase accumulation around conical intersections. By pairing key mathematical structure from quantum chemistry with the representation learning power of transformers, the presented architecture offers a qualitatively new path toward fast and accurate ground- and excited-state simulations.

Automatic Target Recognition (ATR) in Synthetic Aperture Sonar (SAS) is a task largely dominated by deep neural networks (DNNs). Most SAS-ATR models use convolutional neural network (CNN) architectures whereas transformer-based architectures have had much less representation in the literature despite being state of the art in general computer vision (CV) research. Additionally, researchers have had mixed results in attempting to overcome challenges presented by a scarcity of labeled training data by using methods such as data augmentation and the use of pretrained weights from a variety of imaging modalities. In this work, we compare the performance of modern CNN and transformer-based DNNs to determine which architecture and training configurations elicit the highest performance in SAS-ATR. We investigate how network size, architecture, pretraining method, data augmentation and other forms of regularization affect SAS-ATR performance with a focus on producing the highest-performing model and providing a roadmap for training state-of-the-art SAS-ATR models.

Training Physics-Informed Neural Networks (PINNs) requires jointly optimizing physics residual and initial/boundary condition loss terms, which often induce conflicting gradients. Gradient surgery methods mitigate this issue by constructing directions from loss-specific gradients to reduce conflict before optimizer transformation. However, even when the constructed direction is conflict-free, this property may not be preserved after optimizer transformation. Let $a_t$ denote the direction constructed by gradient surgery, $u_t$ the optimizer proposal, and $\mathcal{C}_t$ the conflict-free cone induced by the loss-specific gradients. We show that modern optimizers can transform $a_t$ through mechanisms such as historical state, adaptive scaling, preconditioning, or decoupled weight decay, so $a_t \in \mathcal{C}_t$ does not generally imply $u_t \in \mathcal{C}_t$. We refer to this optimizer-induced discrepancy in conflict-freeness between $a_t$ and $u_t$ as Gradient-Update Mismatch (GUM). Accordingly, we propose Gradient-Update Alignment (GUA), which projects $u_t$ onto $\mathcal{C}_t$ to obtain the aligned update $p_t$ and applies $p_t$ to the parameters. When the optimizer maintains internal state, GUA further adjusts this state toward targets reconstructed from the applied update. We conduct extensive experiments and find that GUM is widespread across momentum, adaptive, and curvature-based optimizers, with conflict rates reaching up to 86.3%. Across all PINN settings, GUA achieves conflict-free applied updates and consistently improves various gradient surgery methods, reducing the relative $L_2$ error by up to 98.2% in individual settings. Data and code are available at https://github.com/JingXiao10/GUA.

Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): two graphs it cannot tell apart receive identical representations, however deep or wide the network. A common remedy augments node or edge features with precomputed structural descriptors, most often counts of a fixed small subgraph such as triangles or longer cycles, but such counts require committing in advance to the size of the substructure counted, a choice usually made blind to the data. We study a descriptor that avoids this choice. The edge-girth of an edge is the length of a shortest cycle through it, and its multiplicity is the number of such shortest cycles; together they form a per-edge invariant that reports cycles of arbitrary length, computable exactly by a single breadth-first search per edge. Injected into a gated message-passing architecture, EGAGNN, it reaches a test MAE a factor three below the closest gated comparator on the ZINC-12k regression benchmark at 104k parameters; against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary counting cycles up to length eight, using twice as many channels, while a dictionary capped at length four performs no better than no structural information at all. On graph discrimination we prove a matching limitation: on graphs where every edge sees the same number of shortest cycles of the same length, the descriptor becomes constant and any model built on it collapses back to the 1-WL bound. This holds without exception across all 400 pairs of the BREC benchmark: not one of the 90 such pairs is distinguished.

The research explores the pioneering integration of Physics-Informed Neural Networks (PINNs) into the domain of Ground-Penetrating Radar (GPR) data prediction. This research presents a detailed development framework for a specialized PINN model, proficient at interpreting and forecasting GPR data, much like how medical imaging models predict tumor behavior. By harnessing the synergy between deep learning algorithms and the physical laws governing subsurface structures or in medical terms, human tissues the model effectively embeds the physics of electromagnetic wave propagation into its architecture. This ensures that predictions not only align with fundamental physical principles but also mirror the precision needed in medical diagnostics for detecting and monitoring tumors. The suggested deep learning structure comprises three components: a CNN, a spatial feature channel attention (SFCA) mechanism, and ConvLSTM, along with temporal feature frame attention (TFFA) modules. The attention mechanism computes channel attention and temporal attention weights using self-adaptation, thereby fine tuning the visual and temporal feature responses to extract the most pertinent and significant visual and temporal features. By integrating physics directly into the neural network, our model has shown enhanced accuracy in forecasting GPR data. This improvement is vital for conducting effective assessments of bridge deck conditions and other evaluations related to civil infrastructure. The use of Physics Informed Neural Networks (PINNs) has demonstrated the potential to transform the field of Non-Destructive Evaluation (NDE) by enhancing the precision of infrastructure deterioration predictions. Moreover, it offers a deeper insight into the fundamental mechanisms of deterioration, viewed through the prism of physics-based models.

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

Neural networks are increasingly employed to identify both well-defined and ambiguous concepts, yet output-level metrics reveal little about how those concepts are represented internally. Our study asks if these networks exhibit \textit{conceptual separation}: if examples of the same concept form coherent representations, and whether related concepts lie closer together in the representation space. We examine this conceptual organisation in Convolutional Neural Networks (CNNs) and Large Language Models (LLMs) through geometric and distributional analysis of their internal activations. In CNNs, familiar ImageNet concepts form coherent and semantically ordered representations, while this coherence weakens for unseen concepts and suffers within-class domain shift. In LLMs, clearly distinct domains remain well separated, related subdomains move closer together, and the distinction between ambiguous topics collapses at both the mean and covariance level. These results suggest that conceptual separation can reveal structure that output accuracy alone cannot, and may serve as a useful diagnostic of how robustly a model represents the concepts it is asked to identify. Code and data available on \href{https://github.com/JaeeRoshniCapstoneProject/Are-You-Thinking-What-I-m-Thinking-Examining-Conceptual-Separation-in-Neural-Architectures}{GitHub}.

Local learning methods offer an alternative to end-to-end backpropagation, but their unstructured local objectives can produce weak feature learning in deep networks. We study whether the order of hidden-state optimization can address this limitation. We propose a boundary-first inference schedule that partitions a model into chunks, first coordinates hidden states at chunk boundaries, and then refines representations within each chunk. We instantiate this schedule in predictive coding networks (PCNs), a local-learning framework in which hidden activities and prediction errors are explicitly exposed during inference. On CIFAR-10, the resulting boundary-first predictive-coding instantiation improves accuracy over standard predictive coding by $9.77\%$ under a standard parametrization and by $5.51\%$ under a $μ$-parametrization. Diagnostic analyses further show more non-trivial early-layer updates, lower initial-to-final CKA, and more diverse layerwise gradients, consistent with stronger feature learning. These results support boundary-first, chunk-based inference as a practical design principle for predictive-coding training and motivate its study in broader local-learning systems.

Conventional neural networks learn predominantly through affine transformations followed by nonlinear activations, while elapsed time is often treated as an auxiliary feature or assumed to be uniformly sampled. This paper introduces Wave Function Backpropagation (WFB), a wave-parameterized learning formulation in which neural responses are represented by learnable amplitude, wavenumber, angular frequency, and phase. The formulation associates an observed state with its temporal interval Delta t through the phase of a differentiable spatiotemporal wave. We derive standard WFB gradients and a spatial-curvature correction based on the Laplacian of the wave response. WFB is instantiated in a deliberately feed-forward trajectory predictor to provide a controlled proof of concept; sequence learning is outside the scope of the present evaluation. With motion features, STD-WFB using real intervals reduces average displacement error (ADE) by 20.4% relative to the original FFN baseline. In a new position-only evaluation that removes temporal leakage through precomputed velocity and acceleration, real-interval WFB reduces ADE by 10.4% relative to the original FFN and remains competitive with parameter-matched ReLU controls, obtaining 2.1% lower mean ADE than the matched FFN with explicit Delta t. Shuffled-interval WFB attains the lowest mean ADE, indicating that the present evidence supports the effectiveness of the wave representation but does not attribute the gain to interval alignment. These results establish WFB as a viable structured feed-forward learning formulation and define a clear basis for subsequent architectural studies.

This work studies hybrid quantum-classical neural networks for learning routing heuristics. Specifically, this paper asks whether small quantum neural networks can replace parameter-heavy modules inside a competitive attention-based routing model while maintaining solution quality. For the capacitated vehicle routing problem, encoder feed-forward replacement emerges as the most promising design: it reduces the number of model parameters by 56.6% while keeping the hybrid model close to the classical neural baseline at small and medium instance sizes, although the gap grows for larger instances. This work also compares to classical routing algorithms, which remain highly competitive and often superior on the fixed Euclidean test sets. Our results therefore do not indicate quantum advantage or solver dominance, but identify encoder feed-forward replacement as a viable hybrid-module compression strategy for neural combinatorial optimization.

Machine learning interatomic potentials bridge the gap between quantum chemical precision and classical computational speed, enabling molecular dynamics simulations with first-principles accuracy. Their reliability is often improved through active learning, which iteratively expands the training set by identifying uncertain, out-of-distribution configurations. Existing uncertainty-quantification methods often involve a trade-off between computational cost and reliability, and generally cannot account for redundancy as an acquisition batch is assembled. Here, we introduce AdaptNTK, a single-model framework that measures uncertainty as a regularized Mahalanobis distance in empirical neural tangent kernel (NTK) feature space. With the NTK features fixed during acquisition, the uncertainty depends on the acquired configurations but not their reference labels. This allows the uncertainty to be updated recursively after each selection without retraining, reducing redundancy within an acquisition batch. On held-out rMD17 data, AdaptNTK achieves the highest mean correlations with force errors (Spearman 0.68, Pearson 0.71) and matches a three-member ensemble in error retention. In active learning experiments, AdaptNTK achieves the lowest force errors across rMD17 and Transition-1X, with particularly strong performance on transition-state configurations in Transition-1X. AdaptNTK provides a 2.6-fold speedup per Transition-1X cycle relative to the ensemble, providing efficient single-model uncertainty estimation with sequential updates for data-efficient active learning.

Recent Relational Deep Learning architectures have been proposed as foundation models for multi-table relational data, yet they impose constrained neighborhood budgets that force row truncation when an entity has many related records. We introduce Animus, a synthetic financial dataset in which predicting customer income requires aggregating up to tens of thousands of transactions. On the raw representation, three recently proposed models (RT, Griffin, RelGT) achieve $R^2 \le 0.18$; a single, routine, temporal pre-aggregation step recovers $R^2$ up to $0.65$. This questions whether current relational foundation models are ready for high-cardinality real-world data.

Observations increasingly reveal the coupled radiative, chemical, and dynamical processes that shape exoplanet atmospheres. Interpreting these atmospheres requires models that can capture this complexity. However, multidimensional models remain fundamentally limited by computational cost, and answering key questions requires simulating the governing physical mechanisms at speeds classical methods cannot achieve. As a result, models often rely on simplifying approximations, such as equilibrium chemistry, even when those assumptions miss important effects. There is a pressing need for fast and accurate chemical kinetics solvers to model planetary atmospheres. Here we present a machine learning local-box chemical kinetics solver for exoplanet atmospheres using a residual flow-map architecture. We demonstrate that this surrogate model is several orders of magnitude faster than a classical solver, achieving microsecond-scale inference while retaining percent-level accuracy. The surrogate model covers a parameter space that spans $T=300$-$3000$ K, $P=10^{-6}$-$10^{4}$ bar, $Δt=10^{-3}$-$10^{8}$ s, and compositions ranging from $10^{-2}$ to $10^{3}$ times solar in both C/O ratio and metallicity. Our model outperforms several commonly used machine learning architectures and performs robustly under the extreme stiffness characteristic of atmospheric chemistry. The machine learning framework presented here is a flexible and efficient approach to emulating state-to-state flow-map problems that commonly arise in numerical simulations.

The linear recurrent neural network (LRNN) is a simple model for studying how much memory a network builds up as it trains. For uncorrelated inputs, earlier work found that training itself settles the network between keeping the past and reacting only to the present. Real sequences are correlated, and we solve the learning dynamics exactly for correlated inputs. In the solution, keeping the past carries a cost. The whole effect of correlation lands on that cost. This cost reduces to the earlier one when inputs are uncorrelated and grows once they are positively correlated. Three findings follow. (1) Correlation reshapes the course of learning, not only its end. Memory builds, overshoots, and is partly removed, and the settled network keeps less of the past. (2) Memory switches off at a threshold set by one number, how much each input resembles the one just before it. Neither sequence length nor longer-range correlation moves this threshold. Memory is worth keeping only when the task needs the previous input more than the current input already supplies it through correlation with the past. (3) The best network changes too. Zero error demands a feedthrough, a path that passes the current input straight to the network's output and remembers nothing, and training builds it unprompted when given one spare hidden dimension. Our work turns one property of the input into a prediction of whether a network learns memory and explains why correlated data turns recurrent networks into change detectors.

Many parameter-efficient methods generate the parameters of a large neural network from a low-dimensional latent representation. Given an architecture $Φ$ with $P_Φ$ parameter slots, we write $\boldsymbolθ_f=\mathcal{G}(\boldsymbolξ_f)$, where $\mathcal{G}\colon\mathbb{R}^M\to\mathbb{R}^{P_Φ}$ is a parameter generator and $\boldsymbolξ_f\in\mathbb{R}^M$ is a latent representation of the target function $f$. The architecture $Φ$ and the generator $\mathcal{G}$ are shared across the entire target class, while each target $f$ is represented by its own latent vector $\boldsymbolξ_f$, with $Φ_{\mathcal{G}(\boldsymbolξ_f)}$ approximating $f$. This framework encompasses hypernetworks, low-dimensional parameterizations, parameter-efficient adaptation, and model compression. Understanding the tradeoff between the latent dimension $M$ and the network budget $P$ is therefore fundamental to characterizing the expressive efficiency of these methods. We study this tradeoff for affine generators and fully connected ReLU architectures. More precisely, optimizing jointly over architectures $Φ$ satisfying $P_Φ\leq P$ and affine generators $\mathcal{G}:\mathbb{R}^M\to \mathbb{R}^{P_Φ}$, we prove that the optimal worst-case uniform approximation error over the unit ball of $α$-Hölder functions on $[0,1]^d$, where $0<α\leq1$, has the sharp order $\bigl(P\min\{M,P\}\bigr)^{-α/d}.$ In particular, our result shows that even a fixed-dimensional latent space suffices to achieve vanishing approximation error as the network budget increases.

Neural network mixed-effects models (NMMs) have gained traction by combining the strong representation and predictive power of artificial neural networks with the capacity of mixed-effects modeling to capture complex correlation structures. However, existing estimation approaches rely heavily on manual derivations of objective functions and gradients, which inherently forces simplifying approximations and severely constrains the complexity and accuracy of NMMs. In this work, we introduce a general framework for implementing NMMs using Template Model Builder (TMB). By leveraging automatic differentiation and Laplace approximation, TMB requires users to specify only the negative joint log-likelihood and any regularization terms. The framework automatically integrates out random effects and evaluates the marginal objective function alongside its exact gradients, eliminating the need for manual derivations or ad hoc approximations. We demonstrate the efficiency, flexibility, and statistical performance of TMB-based NMMs across two numerical examples, including an application to monotonic NMMs. Reproducible code is provided to facilitate broader adoption.

Gradient flow is widely used as a continuous-time surrogate for gradient descent, but state convergence does not imply convergence of differentiated training maps in nonsmooth networks. We characterize the fixed-horizon, vanishing-step limit of exact automatic differentiation through hard-ReLU gradient descent. Under a stable finite itinerary of separated, same-direction transverse activation events, gradient-descent states converge at first order to the corresponding piecewise-smooth gradient flow, while the exact derivative of every nonresonant discrete program converges to an event-free regional propagator. The true flow derivative instead interleaves classical saltation matrices that encode event-time sensitivity. For globally convex objectives, any strict activation event prevents complete cancellation of these missing transfers. Moreover, minimal globally 1-strongly convex residual-ReLU risks can realize arbitrarily large reciprocal sensitivity gaps, subject to an explicit transversality-scale tradeoff, and a coupled strongly convex construction yields an open set on which the largest initialization-gradient coordinate is reversed. In a controlled 17-parameter ReLU MLP, state and regional-AD errors vanish under mesh refinement while AD-to-flow errors remain between 0.18 and 0.39; an event-aware corrected product restores convergence. Resolved smoothing likewise recovers the flow sensitivity when the transition layer is sufficiently resolved. These results show that the gradient-flow limit of hard-ReLU training need not remain valid after differentiation.

A central question in modern machine learning is how much a trained model can be compressed without changing its behavior, to reduce the memory, compute and energy required to deploy it. To study this, we quantify functional degeneracy through the behavioral recovery rank, defined as the number of leading behavioral-Hessian eigendirections required to recover a trained model's performance. Using the behavioral recovery rank as a geometric benchmark for compression, we find that structural and magnitude pruning retain more degrees of freedom, even after the task is saturated. This gap suggests that functional redundancy is distributed across parameter directions and is not exposed by individual weights or neurons.

Representational similarity is foundational to analyses of deep networks, yet distances between point-valued representations are not intrinsically tied to downstream function: nearby states may produce different behaviors, while distant states may behave similarly. We instead give representations volume, turning similarity into statistical distinguishability. Overlapping stochastic representations necessarily induce overlapping downstream distributions, grounding latent comparison in model function and bringing it under information-theoretic tools such as the data-processing inequality. We realize this idea in pretrained transformers through a light-touch modification to LayerNorm: at each residual-stream read, we normalize the state, add isotropic Gaussian noise, and renormalize. During distillation fine-tuning, one learned allocation parameter per residual-stream read distributes a fixed global rate budget across the processing stack. The resulting model can be viewed as transformer blocks reading the residual stream with learned finite precision under a shared global rate budget. Using the Bhattacharyya coefficient, we trace which counterfactual distinctions are preserved through MLP blocks or selectively exposed to the query, key, and value computations of individual attention heads. Experiments on ViT-S and GPT-2 small reveal the depthwise propagation of continuous visual perturbations and head-specific sensitivity to token distinctions aligned with known attention motifs. These results establish distinguishability as a functionally grounded lens on transformer computation that complements existing interpretability approaches.

Optimizing absorption, distribution, metabolism, and excretion (ADME) is an important part of small molecule drug discovery. Many machine learning models have been built to predict ADME properties to facilitate this optimization process, but explaining model predictions is challenging. We propose a new graph neural network architecture with built-in meaningful per-atom attributions. Our model MolLedger outputs predictions that are the sum of per-atom scores. MolLedger's additive framework obtains exact interpretability at no cost to performance because the global context vector gives the additive head enough context to produce good per-atom scores. Furthermore, MolLedger produces attributions that are more faithful to chemical properties than other interpretability methods because the auxiliary loss in MolLedger anchors the atom scores to chemical properties. Our case studies comparing interpretations from multiple methods on molecular pairs reveal that MolLedger is much better at producing sensible explanations for predicted property changes.

In this research, a novel framework is proposed for the SOH estimation, which employs a hybrid deep learning architecture of a concatenation of a Convolution Neural Network (CNN) and a Bidirectional Long Short-Term Memory (BiLSTM) Neural Network (NN) with the integration of Bayesian Optimization-based hyperparameter tuning for the network. Three different deep learning architectures are being evaluated: standalone recurrent models, CNN-RNN architectures and CNN-RNN combinations enhanced with intermediate Fully Connected (FC) layers. Among the three, the model with the intermediate FC layers demonstrated the highest predictive accuracy. A comprehensive feature engineering approach combines capacity (Q), voltage (V), Incremental Capacity Analysis (ICA), and Differential Voltage Analysis (DVA), with systematic evaluation of multiple combinations to identify the optimal input representation. To validate the proposed method, three publicly available datasets were utilized, ensuring reproducibility of the results, two from external sources and one developed by the author of this study using a unique experimental setup. The comparison study was performed using the Mean Absolute Error (MAE), the Root Mean Squared Error (RMSE) and the FLoating-point OPerations (FLOPs) as evaluation metrics.

Systematic generalization remains a significant challenge in deep learning. In particular, combinatorial generalization - generalizing to new configurations of known factors of variation - is effortless for humans but difficult for standard neural architectures that rely on statistical correlations rather than explicit structural representations. We introduce a new architectural component that embeds structured inductive bias into deep learning: an attention mechanism operating over tensor-product representations (TPRs). Through controlled experiments on compositional tasks, we show that this TPR-attention mechanism outperforms existing architectural components in combinatorial generalization. These results highlight the value of integrating explicit compositional structure into neural attention and point toward a promising path for models capable of systematic generalization.

We prove a depth hierarchy for ReLU neural networks in which every additional ReLU layer can save exponentially many neurons. For all $k\geq2$, we construct a globally $[0,1]$-valued, $1$-Lipschitz function realized by a depth-$(k+1)$ network of width $\mathcal{O}(d^4)$, whereas any depth-$k$ network with unrestricted weights and width at most $\frac{2^d}{2d(k-1)}$ has squared $L_2$ error at least $1/24$ under an absolutely continuous distribution supported at exponential distance from the origin. To the best of our knowledge, this is the first exponential hierarchy across all adjacent fixed depths, and the first exponential separation for ReLU networks between two fixed depths whose shallower network has depth at least $3$. The lower bound also immediately yields the corresponding hierarchy for exact computation. Moreover, the case $k=2$ gives a compactly supported separation between depths $3$ and $2$ with unrestricted shallow-network weights, answering a question raised by Safran, Eldan, and Shamir (2019). The distribution used in our construction nevertheless has all its mass at exponential radius, placing the hierarchy outside the regularity regime in which such a separation would imply major threshold-circuit lower bounds. We also prove an exact separation for a more regular target, which is globally $[0,1]$-valued and $\mathcal{O}(\sqrt d)$-Lipschitz and maps the unit hypercube onto $[0,1]$. It is computed by a polynomial-width depth-$4$ network, whereas any depth-$3$ network agreeing with it on the unit hypercube requires exponentially many first-layer neurons, even with unrestricted weights.

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under $μ$P, and its extra compute stays between $22.2\%$ and $25.0\%$ of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to $46.39 \pm 4.32\%$ fewer steps at 67M and $22.12 \pm 0.80\%$ at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

We consider for an arbitrary fixed $ρ$ and for each positive integer $n$ a multilayer feedforward artificial neural network with $ρ$ layers, $n$ neurons in the first layer (the input layer) and only one neuron, the output neuron, in the last layer. Very roughly formulated, the main result is that if the distribution of weights of connections from a layer to the next are, for all large $n$, approximated well by a fixed continuous (but otherwise arbitrary) curve which does not depend on $n$, and if the values of the $n$ input neurons are independently and identically distributed with a continuous probability density function, then there is a number $ψ$ such that for all $\varepsilon > 0$ the probability that the value of the output neuron is in $[ψ- \varepsilon, ψ+ \varepsilon]$ tends to 1 as $n$ tends to infinity.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Forecasting the long-horizon evolution of mechanical systems from position-only observations is a pivotal yet difficult task, as hidden velocities and trajectory-specific physical properties must be inferred simultaneously. Although physics-guided neural networks like Lagrangian Neural Networks (LNNs) guarantee physical plausibility, they generally require complete state inputs and lack adaptability to changing system parameters. To break these limitations, we introduce History-informed Lagrangian Neural Networks (HiLNN). Grounded in the insight that temporal position sequences implicitly encode underlying dynamics, HiLNN employs a recurrent encoder to extract a latent context from history. This context not only reconstructs the unobserved initial velocity but also adaptively modulates the mass matrix, potential energy, and damping coefficients of a structured Lagrangian system. By leveraging a differentiable RK4 rollout scheme, the entire pipeline is optimized end-to-end under multi-step trajectory supervision and energy-consistency regularization. Empirical evaluations across conservative, dissipative, and heterogeneous variable-parameter systems show that HiLNN delivers superior long-term prediction accuracy and maintains precise energy profiles compared to state-of-the-art baselines. The source code is publicly available at https://github.com/yingtian22/History-informed-LNN.

Verification of neural networks against relational specifications, such as global robustness, is crucial for safety-critical applications of cyber-physical systems (CPS), given their increasing adoption of AI components. Compared to simple trace properties (e.g., local robustness), verifying relational specifications requires reasoning about the relationship between multiple network inferences, which brings significant technical challenges. Existing research has explored abstraction techniques based on sound and convex over-approximation of neural network outputs; however, since these approaches are inherently incomplete and may raise false alarms, they further underscore the need of effective abstraction refinement. In this paper, we propose a branch-and-bound (BaB) framework to mitigate the issue, which iteratively splits the problem until all sub-problems are verified. Specifically, our BaB framework features splitting of relational neurons rather than individual neurons as prior works do, and as the core of our technique, we devise a relational neuron selection strategy based on the dual formulation of the verification problem, which allows us to efficiently select the (most likely) optimal relational neuron that maximizes the refinement brought by problem splitting. We evaluate SaBRe on 817 verification problems across ACAS Xu, MNIST-F, MNIST-C, CIFAR and GTSRB. The results show that SaBRe outperforms different baseline approaches, in terms of the number of solved instances and verification efficiency, which demonstrates the effectiveness of our proposed techniques.

In this research work, we are constructing the sensing matrix, which is essential for the success of the compressive sensing technique. We have chosen a learning-based technique for the construction of the sensing matrix. The novelty and uniqueness of the proposed technique is that it does not use any data set and also does not use a specific application. It uses the mathematical property/constraint for the construction of the sensing matrix for the perfect recovery of the signal. The perfect recovery of signals is an old and still very challenging problem in real-world applications. In late 2000, compressive sensing became a popular mathematical tool for the perfect recovery of sparse signals. The core of the compressive technique is the construction of the sensing matrix, which satisfies certain special properties such as restricted isometry property (RIP), null space property (NSP), and spark property (SP). All these properties are NP-hard problems and hence computationally challenging to solve. For all practical purposes, the construction of the sensing matrix needs to achieve low mutual coherence to achieve the perfect recovery of the signals. We have used a neural network for the construction of the sensing matrix, and this framework constructs a binary sensing matrix with low mutual coherence. The entries in the matrix are generated through a shared underlying rule. The proposed architecture is simple and does not use large-scale training data sets. Such uniqueness and novelty bring a drastic reduction in computational cost, and also, for the first time in literature, the use of a mathematical property for defining the loss function. In this proposed research work, the mutual coherence property has been used in the neural network framework. Such a neural network framework brings generality, robustness, and reduces storage requirements.

McCoy & Griffiths (2025, henceforth M&G) suggest that a Bayesian prior can be distilled into Artificial Neural Networks (ANNs) through Model-Agnostic Meta-Learning (MAML, Finn et al., 2017). They support this empirically by showing that meta-trained networks demonstrate formal language learning abilities comparable to Yang & Piantadosi (2023)'s Bayesian learner, significantly outperforming standard ANNs. We point out that under the standard interpretation of a prior, M&G's procedure does not actually instill one; it merely initializes network weights favorably, leaving the objective function unchanged. We then consider a more permissive interpretation, where the system as a whole can be seen as implementing a Bayesian learner even without an explicit prior in the objective. We show that this interpretation faces nontrivial challenges. Finally, we assess how well MAML approximates the empirical results of Bayesian learning, showing that unlike genuine Bayesian learners, M&G's model overfits and generalizes poorly to unseen data.

A flat training curve does not reveal whether a neural network has reached a global optimum, is locally trapped, is representation-limited, or is mismatched to its trainer. We introduce Training Under Challenge, an executable-certificate framework in which predeclared, architecture-valid procedures construct complete alternatives in the same certified class and reevaluate the same objective. Any lower-valued candidate is a replayable witness that lower-bounds the checkpoint's empirical global-optimality gap. Passing a finite suite is only suite-relative; global-gap conclusions require a separately justified coverage mechanism. We define a resource-indexed challenge-power modulus that characterizes the largest gap compatible with passage. For squared loss, current block-decrease operators make coverage checkable and yield uniform and realized-residual bounds. We prove the converse frontier: without coverage, a first-order ReLU trainer can reach infinitely many exact conditional head optima while converging to a non-global point. On a channel-gated ResNet-18 distillation problem with known optimum, eight internal challenges cover all 240 audited output directions, and realized-residual bounds lie within factors of 1.74--3.02 of the true gap. Paired predictive certificates separate decoder under-use from representation insufficiency, while quantized-denoising studies demonstrate diagnosis, repair, and current-state recertification.

We prove that, on finite simple undirected graphs equipped with a single Boolean node feature, the Boolean queries expressible in $Σ$-MPLang, for any collection $Σ$ of eventually constant activation functions and with arbitrary real coefficients, form a strict subclass of the Boolean queries expressible in ReLU-MPLang. We thereby settle a recently posed open problem: whether ReLU-MPLang is more powerful than trReLU-MPLang when it comes to Boolean queries. In particular, this implies that ReLU-GNNs are strictly more expressive than {TrReLU,id}-GNNs with respect to Boolean queries on Boolean-featured graphs.

Neural networks can often be trained or fine-tuned through random low-dimensional reparameterization, where a small latent vector is mapped into a full parameter update by a frozen random map. This raises a practical question: how large must the latent search space be to reach a low-loss region? We first express the known accessibility transition in an equivalent conic form, centered for compact convex targets at the statistical dimension of the polar cone. Our main theoretical contribution is an orientation-resolved quadratic master formula that predicts the random-slice residual from both the curvature spectrum and the reference-to-solution displacement profile. It yields a self-consistent isotropic-orientation predictor and, in a conservative radius-only specialization, recovers the earlier Gaussian-width quadratic bound. Building on this analysis, we introduce Random Mapping Networks (RaMaN), which instantiate the predicted latent dimension using structured Hadamard or seed-regenerated Gaussian maps. These constructions avoid the O(dP) storage of dense random maps and reduce optimizer-state memory from O(P) to O(d). We also develop matrix-free curvature approximations and sweep-free dimension selection. Across controlled quadratic and neural-curvature experiments, the orientation-resolved predictor closely tracks measured transition locations and outperforms orientation-agnostic approximations when displacement direction matters. End-to-end experiments further show sharp, protocol-dependent training transitions across image and language models.

Financial volatility is regime dependent, yet incorporating regime information into neural networks can also destabilize training. This paper asks where such information should enter a neural cross-sectional volatility forecasting model. We study five-day realized-volatility forecasts for 1,027 U.S. equities using a rolling walk-forward evaluation framework in which information, model capacity, hyperparameter tuning, and random seeds are matched across architectures. We propose RG-ResMoE, a regime-gated residual mixture-of-experts architecture in which regime information is used only for expert routing rather than for direct forecasting. The base predictor models volatility from stock features, while a gating network uses regime state variables to route residual corrections. RG-ResMoE consistently outperforms a capacity-matched MLP in both forecasting accuracy and training stability in the main U.S. study. Similar gains are observed on an independent Japanese panel. The integration pathway is decisive: appending the same regime variables directly to the forecasting input degrades both predictive performance and training stability, whereas restricting them to the routing gate improves accuracy and Value-at-Risk calibration. Hard routing consistently underperforms soft routing. The results suggest that, in compact neural volatility forecasting models, the primary value of mixture-of-experts models lies less in increasing model capacity than in controlling how nonstationary regime information influences prediction.

The parity problem--deciding whether the number of ones in a binary vector is odd or even--remains challenging for standard neural networks due to linear inseparability and the need for global interactions. We propose TESLA, an activation defined as a learnable combination of sine and cosine terms, enabling explicit control over polynomial degree and selective amplification of high-order components. Theoretically, we show that constraining TESLA's coefficients yields Lipschitz/Rademacher complexity bounds and shapes the training dynamics to emphasize higher-frequency structure. Empirically, on parity with input length n = 32, TESLA attains strong generalization with 100K training samples (approximately 0.002% of the 2^32 input space) and remains robust under heavy corruption, retaining high accuracy with up to 30% label noise. We also compare against periodic and frequency-based baselines (SIREN, SNAKE, and Fourier feature embeddings) on parity and Forrelation. Beyond synthetic structure, TESLA delivers comparable performance on ImageNet-100, indicating that activation-level degree control transfers to more general vision workloads. Code: https://github.com/KAU-QuantumAILab/TESLA

We establish convergence guarantees of gradient descent for general feedforward neural networks of arbitrary width or depth, with no special requirements on the initialization or dataset. We only assume that the activation functions are Lipschitz smooth, Lipschitz continuous, and linearly bounded--- properties that hold for linear, tanh, softplus, and sigmoid activation functions. For the loss function, we require that it is Lipschitz smooth in the model outputs, which is true for mean-squared error. The key theoretical insight is that the Lipschitz properties of the activation functions are partially preserved even through repeated compositions, leading to a novel generalized Lipschitz smoothness condition where the change in gradient is upper bounded by the change in the parameter space, multiplied by polynomial terms of the parameter norms at both endpoints. This type of condition holds for both the model function and the loss function, enabling a descent lemma where the loss decreases as long as the learning rate is small enough with respect to the parameter norms. By ensuring that the parameter norms do not grow too quickly to infinity, we prove that the minimum squared gradient norm converges to zero in $T$ iterations at rate $O(1/T^{1/L})$ for an $L$-layer neural network.

Direct teleoperation of vehicles faces critical technical bottlenecks: communication latency and the operator's inability to physically perceive unmodeled environmental disturbances (e.g., aerodynamic drag, bank angles) coupled with highly nonlinear tire-road dynamics. To address these challenges, we propose a tailored unilateral teleoperation framework. The system integrates the Wave Variable (WV) approach to passively guarantee stability under stochastic delays, and an adaptive Radial Basis Function Network (RBFN) to actively compensate for vehicle-specific uncertainties. Unlike existing WV-neural network architectures designed for bilateral robotic arms, our framework features decoupled adaptive laws specifically designed for vehicle longitudinal and lateral dynamics. Furthermore, compared to model-heavy predictive controllers, the model-free RBFN offers rapid online adaptation without heavy computational overhead. Building upon our preliminary theoretical formulation, this brief paper presents comprehensive comparative analyses and real-world hardware validations. Simulation benchmarks against PID, LQR, MPC, and NMPC demonstrate that the RBFN achieves superior robustness against unmodeled disturbances while requiring orders of magnitude less execution time than MPC and NMPC, making it ideal for resource-constrained vehicle edge computing. Finally, hardware-in-the-loop experiments using a 1/10th scale vehicle over a 4G network validate the system's practical feasibility, safety, and robust trajectory tracking under physical road uncertainties.

In this work we present a method to accelerate the optimization of learning high dimensional functions using deep neural network (DNN). This optimization procedure introduces contextual features into the first layer of a DNN. The parameters of DNN are optimized via standard gradient descent while keeping the input-feature basis fixed. After optimization of the DNN parameters, the feature layer is provided a chance to update and change before DNN optimization resumes. The feature layer has two types of functions: those that can be evaluated quickly in a matrix-free way on the domain (i.e. rank-1 features) and more complex features that must first be decomposed using tensor network (TN) decomposition strategies (tensor features). In particular, we study the effect of adding features which distill pretrained DNN into TNs using a discretize and decompose strategy. To efficiently decompose high-dimensional functions constructed from discretized DNN, we leverage a randomized tensor decomposition strategy. Using randomization, we are able to reduce the storage cost of decomposing high dimensional functions by at least 8 orders of magnitude. Using this approach, we are able to efficiently train models between 5 and 40 dimensions.

Competitive artificial-life systems can rank trained controllers differently under training and ecological evaluation. We present Neuroevolution Arena, a GPU-accelerated spatial ecology of independently parameterized neural-network cells, and an audit-tracked nested evaluation protocol. Three implementation-specific update-and-inheritance regimes (EvoEvo, EvoRL, and RLRL) are crossed with two neural architectures for 50,000 generations in three independent training runs per condition. One saved elite-controller artifact from each of the 18 runs enters an aligned-run frozen-evaluation design comprising 198 computational jobs. Pairwise effects average three seed-defined ecological contexts (two cooperation-permitting and one attack-permitting) within each aligned training-run block; the independent level remains n = 3 runs per condition. RL-enabled regimes attain higher recorded training fitness than EvoEvo, whereas pairwise outcomes show architecture-conditioned majority patterns and substantial artifact dependence. Six-way winners vary across artifacts and contexts, and the prespecified survival endpoint has a complete floor. We contribute a nested protocol that separates training-run artifacts from evaluation contexts and exposes, rather than conceals, their different sources of variation.

Deep neural networks are increasingly deployed in safety-critical domains as perception modules, where failures are often caused due to rare and under-represented scenarios. This necessitates the need to evaluate the semantic robustness of perception models; conformance of behavior to high-level requirements over real-world perceptual variability. To address this, we propose SeFaR, a framework for systematic semantic-feature-centric testing of vision models. Given a natural-language requirement and a set of satisfying inputs, SeFaR evaluates robustness with respect to diverse realistic semantic variations that preserve requirement satisfaction. The approach employs a novel hierarchical concept model enabling structured exploration of the feature space and incorporation of domain knowledge via user-defined concepts. State-of-the-art diffusion and vision-language models are leveraged to generate photorealistic semantics-preserving perturbations and identification of previously unknown features impacting behavior. A feedback-driven adaptive process is adopted to generate interpretable failure-inducing semantic concepts along with corresponding test inputs. Evaluation on case studies demonstrates that the proposed framework effectively satisfies requirement preconditions while identifying requirement-independent features that influence model decisions, enabling it to both uncover faults and relate them to such features.

We consider federated parameter efficient fine-tuning of large neural networks with low-rank adaptation (LoRA,~Hu et al.\ 2022). Combining LoRA with federated PEFT introduces challenges absent from either setting alone: clients may use different LoRA ranks, making their factor matrices dimension-incompatible, and factor-wise averaging suffers from a bilinear mismatch. We propose SeFoRA, a sketch-aggregated federated LoRA algorithm in which each client transmits a linear sketch of its local updates, enabling direct aggregation at the federator. As a result, SeFoRA alleviates the bilinear mismatch, and allows for aggregation in a small subspace of the full model. We introduce a rank-homogeneous version called SeFoRA-Ho which allows for direct adapter aggregation in this setting. We prove convergence to a neighborhood of the first-order stationary point at rate $\cO(1/T)$ for the rank-homogeneous setting. Numerical experiments on fine-tuning RoBERTa-Large on GLUE datasets show how our algorithms outperform the state-of-the-art.

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.

Recurrent neural networks (RNNs) are widely used for sequence learning, yet their application is commonly associated with temporal data, although recurrent computation fundamentally operates on ordered sequences rather than on time itself. Building on this observation, we introduce the Ordered Structural Dependency Hypothesis (OSDH), which proposes that multiple admissible orderings of the same observations may reveal complementary structural dependencies inaccessible through a single sequential organization. To operationalize this hypothesis, we propose the Independent Structural Expert Principle (ISEP), whereby projection-specific sequence models are trained independently before their learned representations are integrated through a dedicated fusion model. As a concrete realization, we present Structural Evolution RNNs (SE-RNNs), which employ conventional RNNs as projection-specific structural experts while preserving the underlying recurrent computation unchanged. Proof-of-concept experiments on three synthetic datasets with substantially different levels of structural complexity demonstrate that the proposed architecture consistently benefits from multiple ordered projections when hidden structural dependencies are present, while remaining competitive on simpler datasets. Since OSDH is independent of the underlying sequence-processing model, the proposed framework naturally extends beyond recurrent networks and may be instantiated using alternative architectures. The results suggest a general computational perspective for exploiting complementary ordered representations across diverse structured learning problems.

Human-scene interaction is an active research topic with several industrial applications in virtual reality, gaming, robotics, and surveillance. Despite significant progress in network architectures to improve the results or optimize models' parameters for fast inference speed, the efficient representation of contact between humans and their environments remains an open challenge. In this paper, we propose a new efficient human-contact representation for human-scene interaction. Our primary contribution is the introduction of sparse contact masks that strategically select essential contact information, significantly reducing redundant data in high-dimensional inputs. Leveraging this efficient contact representation, we propose a suite of sparse operators to replace traditional dense operators within deep network layers for faster computation. Our approach not only enhances computational speed but also filters out non-essential contact data, thereby improving the precision of human-scene interaction models. To validate the effectiveness of our method, we conduct intensive experiments across three public benchmark datasets, focusing on two critical tasks for human-scene interaction: contact prediction and scene synthesis. The experimental results show that our approach outperforms state-of-the-art models in reconstruction accuracy and achieves a computation speed-up of at least 12 times over recent baselines.

We introduce Motif 3, a decoder-only Mixture-of-Experts language model with 314 billion total parameters and 13.2 billion activated per token. Each sparse MoE layer contains 384 routed experts, with eight selected per token. This fine-grained sparsity provides substantial expert capacity while limiting computation. Motif 3 is built around Grouped Differential Latent Attention (GDLA), which integrates grouped differential attention with the compressed key-value representation of Multi-head Latent Attention. The architecture further incorporates modified manifold-constrained hyper-connections, Expert Specific PolyNorm activations, and multi-token prediction to improve optimization stability, expert specialization, and inference efficiency. We pretrain Motif 3 on approximately 12.5 trillion tokens spanning web documents, STEM, code, mathematics, multilingual content, and domain-specialized corpora. Expert-balancing and numerical-stabilization techniques support stable training at scale, while selective MXFP8 computation and communication, memory-efficient fused kernels, and window-aware context parallelism enable training with context lengths up to 256K tokens. Our post-training pipeline combines general supervised fine-tuning, six specialist teachers trained with reinforcement learning, a software-engineering teacher trained with supervised fine-tuning, and Multi-teacher On-Policy Distillation. The resulting unified model consolidates complementary capabilities in reasoning, coding, tool use, professional work, long-context understanding, calibrated abstention, and instruction following. Across a broad evaluation suite, Motif 3 demonstrates competitive performance against leading open weight models, including strong results on long-horizon agentic tasks, mathematical reasoning, scientific knowledge, and hallucination-sensitive evaluation.

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

Spiking neural networks (SNNs) offer a promising pathway to energy-efficient AI and brain-inspired computing. However, their widespread adoption is hindered by a lack of fast, accessible, and versatile simulation frameworks. In this paper, we introduce SuperNeuroMAT, an open-source, scalable, and highly efficient Python-based SNN simulator. We devise a novel matrix-based approach to model the leaky integrate-and-fire (LIF) neuron dynamics and natively support dense and sparse execution modes. This enables fast simulation of approximately 10,000 neurons in dense mode and 100,000 neurons in sparse mode on standard laptops and desktops without requiring specialized hardware. We demonstrate that SuperNeuroMAT consistently outperforms four established SNN simulators---NEST, Brian2, BindsNET, and snnTorch---on two performance metrics (execution speed and peak resident memory) and across various network sizes and connection probabilities. Furthermore, we demonstrate SuperNeuroMAT's applicability across a diverse set of problems. SuperNeuroMAT can efficiently handle conventional machine learning benchmarks such as the Digits and citation network datasets as well as neuromorphic event-based vision tasks such as N-CARS and ASL-DVS. Moreover, it can be extended beyond machine learning workloads and facilitate general-purpose workloads. We validated this by implementing the neuromorphic shortest path algorithm and two arithmetic primitives (addition and multiplication). SuperNeuroMAT can be installed via the Python Package Index (PyPI), thereby lowering the barrier to entry into the field of neuromorphic computing and accelerating the broader development of neuromorphic algorithms.