Neural Network

Normalization layers such as BatchNorm and LayerNorm have long been considered essential for stable training in deep networks. This work demonstrates that they can be fully replaced by a single learnable activation mechanism. We identify a plasticity suppression effect induced by standard normalization: learnable activation parameters rapidly lose adaptability when paired with normalization layers. Motivated by this observation, we introduce SALU (Saturated Adaptive Linear Unit), $$\operatorname{SALU}(x;a,b) = \frac{a x}{\sqrt{1 + a b x^2}},\quad a>0,\; b>0$$ a bounded, learnable activation that provides intrinsic signal stabilization without relying on batch statistics or external affine parameters. Building on SALU, we propose SaluNet, a paradigm grounded in total plasticity: SALU replaces normalization layers, while SWALU and GALU replace standard activations. With ResNet-18, SaluNet-C-18 achieves 97.35% on CIFAR-10 and 83.25% on CIFAR-100 without normalization, maintaining 93.44% and 76.23% at batch size 1 where normalized architectures fail. For transformers, SaluNet-T improves over LayerNorm-GELU from 90.92% to 91.01% on CIFAR-10 and from 66.54% to 68.10% on CIFAR-100. SaluNet-C-50 reaches 78.67% Top-1 on ImageNet-1K at $224\times224$, and $79.23\%$ at $288\times288$. These results suggest normalization layers suppress total plasticity, a property biological neurons inherently possess, enabling deep networks to learn effectively.

Deep learning weather models now match numerical weather prediction accuracy while running orders of magnitude faster, but produce deterministic forecasts without uncertainty estimates, a critical gap for high-stakes decisions during extreme weather events. This paper proposes Neural Tangent Kernel-based uncertainty quantification (NTK-UQ) using last-layer empirical features. Theoretical analysis predicts that UQ quality is architecture-dependent through two mechanisms. First, a variance collapse mechanism explains when UQ fails: when the eigenvalue truncation rank approaches the effective rank of the feature space, the GP correction term consumes nearly all prior variance, destroying discrimination between tropical cyclones and routine conditions; architectures with concentrated spectra (spectral operators) require aggressive truncation ($k \leq 10$), while attention-based models tolerate full-rank computation. Second, decomposition performance depends on the non-Gaussian, heavy-tailed structure of extreme weather: Independent Component Analysis exploits higher-order statistics (kurtosis, negentropy) to isolate heavy-tailed extreme-event features, achieving higher discrimination than singular value decomposition, which captures only second-order variance. A data-driven selection rule chooses ICA or SVD from the feature eigenspectrum concentration ratio, correctly prescribing the superior decomposition for all four evaluated architectures. Compared to split conformal prediction (the natural post-hoc baseline), NTK-UQ achieves 31--37% sharper prediction intervals at 90% coverage, and uniquely produces \emph{adaptive} intervals that scale with extreme event severity, which conformal prediction cannot achieve by construction. The framework requires no retraining; inference-time uncertainty requires only a single matrix-vector product per sample.

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.

Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency.

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.

Dictionary learning with sparse autoencoders (SAEs) produces overcomplete bases from neural network activations that are often interpretable and reduces polysemanticity. However, features from SAEs vary substantially across random seeds -- a problem known as instability. Archetypal SAEs (Fel et al., 2025) were proposed as a general dictionary-learning intervention for more reliable concept extraction, and report more stable dictionaries at the end of training. We demonstrate that the stability claimed by archetypal SAEs is a result of setting identical initialization across multiple runs. Through our analyses, we attempt to clarify two distinct notions in mechanistic interpretability that may be ambiguously used: stability is agreement between two independently trained models, whereas stabilization is the convergence of independently initialized runs toward a common solution. This distinction is critical for mechanistic interpretability of natural language processing (NLP), where feature stability is increasingly used as evidence that SAE features are reusable units of analysis. Experiments from archetypal SAEs share a deterministic k-means decoder initialization, setting inter-run dictionary distance to zero before training begins. When this initialization is removed, the archetypal constraint provides no stabilization advantage in our setting. We further identify a preprocessing-dependent cosine geometry issue that complicates interpretation of endpoint stability metrics. Overall, our study supports the value of studying SAEs within the larger dictionary-learning tradition while showing that stability claims require trajectory diagnostics and initialization ablations.

Artificial intelligence provides a practical framework for crop damage assessment from imagery data, supporting early decision-making in agricultural management. In peach orchards, climate change increases abiotic stress and biotic pressures, including pests and diseases, which often produce visually similar foliar symptoms. This overlap makes manual diagnosis difficult, especially across multiple fields with varying environmental conditions, highlighting the need for automated models with strong generalization ability. We propose an image-based classification approach for peach leaf damage detection. A benchmark dataset was created through manual annotation of publicly available images, consisting of 1,366 peach leaves across six damage categories. Several deep learning architectures were evaluated. EfficientNet models achieved the best results, with EfficientNetB0 reaching 92.9 percent accuracy, EfficientNetB3 achieving 91.5 percent, and EfficientNetB5 showing the strongest performance on minority classes. DenseNet121 reached 92.6 percent accuracy. The integration of the Convolutional Block Attention Module (CBAM) improved performance in several backbones, particularly EfficientNetB5 and InceptionV3, while showing limited or negative impact in others. The CBAM-enhanced EfficientNetB5 achieved the best overall accuracy of 93.3 percent. To evaluate robustness under realistic conditions, a local dataset of 180 images across four classes was collected, and transfer learning strategies were applied to address domain shift. Three fine-tuning strategies were tested. EfficientNetB3 combined with CBAM achieved the best performance in the local domain, reaching a 93 percent macro F1-score after transfer. Overall, attention-based models showed improved robustness for minority classes and better generalization across different field conditions.

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.

We study optimal learning-rate selection in two-layer and three-layer linear neural networks trained to learn linear target functions. In particular, we derive the exact closed-form expressions for the gradients and test loss after one and two steps of gradient descent, enabling a precise characterization of early training dynamics. We characterize how learning rates should scale under the gradient approximation in the first two steps, and prove that performing updates with this approximation yields a tractable surrogate loss with a tight, small approximation error. This formulation enables the theoretical analysis of layer-wise learning rates and reveals a distinct early-training regime: test loss can be minimized by unequal learning rates at the initial step, while equal learning rates become optimal in subsequent steps. Our numerical experiments validate the theory and demonstrate the importance of balancing layer-wise learning rates early during training. The code is available at: https://github.com/TDCSZ327/Layer-Balancing.

Deep neural networks often contain redundant hidden units. Removing individual weights can reduce parameter count, but unstructured sparsity is not always easy to exploit in standard dense implementations. This paper develops a structured pruning framework in which complete neurons are removed using multi-armed bandit (MAB) algorithms. Each candidate neuron is treated as an arm; pulling an arm temporarily masks that neuron, measures the change in loss on a sampled mini-batch, restores the neuron, and updates an estimate of its safe-removal reward. The framework supports stochastic policies, including Epsilon-Greedy, Softmax, UCB1 and Thompson Sampling, and multiplicative-weight policies, including Hedge-style multiplicative weights and EXP3. We evaluate the method on tabular classification, tabular regression and deep neural-network benchmarks covering image, text and reasoning tasks. Statistical comparisons using the Friedman test followed by the Nemenyi post-hoc test show significant differences between methods. On tabular classification tasks, UCB1 obtains the highest mean rank among pruning policies and improves on the unpruned neural network. On regression tasks, UCB1 obtains the highest mean rank and is statistically competitive with, or superior to, several standard regression models according to R^2. On deep-learning tasks, UCB1 and Thompson Sampling obtain the strongest ranks, and several MAB policies significantly outperform the unpruned model, magnitude-based neuron pruning and greedy activation-variation pruning. The results show that MAB-based neuron pruning is an effective and computationally practical approach for structured model reduction.

InfoNCE is the standard contrastive learning objective, but its softmax form is not only a computational convenience: it also encodes a statistical assumption about how the top-scoring example is selected. Using extreme value theory, we show that this assumption is often misaligned with the normalized embedding setting used in modern contrastive learning. Motivated by this mismatch, we propose \textsc{WEINCE}, a simple modification of InfoNCE that uses anchor-wise online batch statistics to blend the usual softmax logits with an endpoint shortfall correction, adding no trainable parameters. Across five vision benchmarks, \textsc{WEINCE} yields consistent improvements in frozen-feature evaluation. These results show that a more faithful statistical treatment of hard negatives can improve contrastive objectives.

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.

Recent analyses of multi-pathway Deep Linear Networks use Gradient Flow to predict a "winner-takes-all" specialization in which path symmetry breaks and each feature concentrates in a single pathway. In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.

Neural scaling laws describe predictable power-law relationships between model size, dataset size, compute, and performance. While these laws guide the development of modern foundation models, the mechanisms underpinning them remain poorly understood, in part due to the absence of scalable analysis tools. To close this gap, we introduce "spectral position": a scalable measure of which eigenvalues of the empirical neural tangent kernel (eNTK) currently drive loss reduction. Applying this measure to scaling experiments, we find that spectral position decreases throughout training: learning shifts from dominant eigenmodes into the spectral tail. Larger models reach further into the tail than smaller models, revealing a size-dependent capacity we call "spectral reach". This suggests why larger models achieve lower losses: they sustain learning on weak spectral signals inaccessible to smaller models. We further identify feature learning as a key enabler of spectral reach. It adaptively amplifies gradient magnitudes as learning advances, sustaining progress where frozen representations stall. This points to concrete interventions through architecture and optimizer design.

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.

Training deep spiking neural networks (SNNs) remains challenging due to sharp loss landscapes and temporal inconsistency caused by surrogate gradients. To address these challenges, we propose a unified framework: adaptive and asymmetric surrogate gradients A2SG. The adaptive gradients adjust an effective window for spatio-temporal adaptation, reducing spatial gradient variation and maintaining directional consistency of gradients over time. The asymmetric gradients reflect neuronal dynamics by assigning larger gradients to neurons with higher membrane potentials, and we prove that they yield lower variation than symmetric surrogates. Our analysis further establishes a direct connection between local gradient variation and the curvature of the loss landscape, providing a principled explanation for how A2SG promotes convergence to flatter minima and improves generalization. We conduct extensive experiments on diverse models, including CNN-based and Transformer-based SNNs, across various tasks such as image classification using both static and neuromorphic datasets, as well as segmentation. The results demonstrate that A2SG consistently improves accuracy and energy efficiency, establishing it as a general and reliable solution for training deep SNNs. Our code is available at https://github.com/KIST-NCL/A2SG.git.

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.

Mislabeled samples in training datasets severely degrade the performance of deep networks, as overparameterized models tend to memorize erroneous labels. We address this challenge by proposing a novel approach for mislabeled data detection that leverages training dynamics. Our method is grounded in the key observation that correctly labeled samples exhibit consistent entropy decrease during training, while mislabeled samples maintain relatively high entropy throughout the training process. Building on this insight, we introduce a signed entropy integral (SEI) statistic that captures both the magnitude and temporal trend of prediction entropy across training epochs. SEI is broadly applicable to classification networks and demonstrates particular effectiveness when integrated with contrastive language-image pretraining (CLIP) architectures. Through extensive experiments on four medical imaging datasets -- a domain particularly susceptible to labeling errors due to diagnostic complexity -- spanning diverse modalities and pathologies, we demonstrate that SEI achieves state-of-the-art performance in mislabeled data identification, outperforming existing methods while maintaining computational efficiency and implementation simplicity. Our code is available at https://github.com/MedAITech/SEI.

Reinforcement learning with verifiable rewards (RLVR) and group-based policy optimization methods such as GRPO update a stochastic policy by sampling multiple completions per prompt and increasing the policy's probability on those with higher reward, regularized by a KL penalty toward a reference policy. These updates do not include explicit mechanisms that track epistemic uncertainty. This paper studies a stylized explanation for why such uncertainty-agnostic updates can nevertheless be effective. We analyze an annealed softmax (Boltzmann) policy that selects actions according to a softmax of empirical mean rewards in a many-armed Bayesian Bernoulli bandit. Under a linear upper-tail condition on the prior (the $β=1$ case of $β$-regularity), which implies an abundance of near-optimal arms, we prove that annealed softmax greedy achieves Bayes regret $\tilde{O}(m + T/m)$, and in particular $\tilde{O}(\sqrt{T})$ when the number of arms scales as $m = Θ(\sqrt{T})$. This is the near-optimal Bayes regret rate in this regime, attained also by empirical-mean greedy. Under $β$-regularity, many arms maintain empirical means close to the optimum throughout learning, so when softmax samples an arm other than the empirically best, that arm tends to be another near-optimal one rather than a clearly inferior one. By contrast, with a small number of arms, the same kind of softmax policy can suffer linear regret. The result also provides a structural analogy to RLVR, where a base policy with a non-negligible probability of producing a correct completion plays the role of $β$-regularity.

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).

Predictive coding (PC) is a local-learning alternative to backpropagation (BP), training deep networks via local energy-minimization dynamics rather than a global backward pass. We introduce Augmented Lagrangian Predictive Coding (PC-ALM), which maintains PC's inference budget but aligns each weight update toward BP by accumulating per-layer constraint errors into a layer-local Lagrange multiplier. In linear PC networks, PC-ALM converges to an equilibrium with exact BP gradients distributed across the network via only layer-local updates. We analyze PC-ALM in nonlinear PC networks up to depth 128 and show that it matches BP performance across all width-depth regimes, notably in deep narrow networks where PC underperforms. PC-ALM introduces recurrent dynamics in each layer's activations. Compared to PC's heat flow on a scalar energy, PC-ALM dynamics are driven by dual ascent on the augmented Lagrangian. We observe "ballistic" credit propagation across very deep networks, with credit signals evenly distributed across layers, compared to PC's slow, diffusive credit propagation. Beyond the algorithm itself, the augmented Lagrangian framework offers a generalization of PC, and may yield insights into how distributed systems could compute and propagate BP-like credit signals through purely local dynamics.

Modern deep networks remain fragile under distribution shift and adversarial perturbations, often due to excessive or poorly structured input sensitivity. We introduce TASER (Task-Aware Stein Regularisation), a training-time regularisation framework derived from Langevin Stein operators. By penalising pointwise Stein residuals under the training distribution, TASER encourages geometric compatibility between predictors and data density, inducing anisotropic, data-aware smoothness. We provide theoretical links between Stein regularisation and reduced first-order shift sensitivity, develop scalable implementation variants compatible with modern architectures, and demonstrate improved robustness and stability across regression and vision benchmarks. Across CIFAR-10 experiments, TASER consistently improves the adversarial robustness of established training methods without incurring statistically significant clean-accuracy degradation.

Large deep neural networks are costly to store and deploy because inference must move and evaluate many parameters. This paper studies \emph{Automatically Differentiable Nonlinear Tensor Networks} (ADNTNs), compact differentiable weight generators for replacing selected dense, convolutional, and attention layers. An ADNTN maps a small set of trainable tensor cores to a full weight tensor through hierarchical contractions and learnable nonlinearities; the generated layer is then used as an ordinary linear or convolutional operator. We investigate three multilayered topologies: Tree Tensor Networks, augmented Tree Tensor Networks with boundary disentanglers, and MERA-style multi-scale decoders. Compared with flat brick-wall automatically differentiable tensor networks, these hierarchies provide logarithmic-depth communication between tensorised modes and optional lateral mixing, which can improve long-range structure without large increases in stored parameters. We give a unified forward--adjoint formulation showing how reverse-mode automatic differentiation computes pre-activation adjoints and contracted-environment gradients for all trainable cores. The formulation supports task losses, reconstruction losses, distillation, quantisation-aware terms, batching, and modern optimisers. Proof-of-concept experiments on selected AlexNet and VGG-16 layers on CIFAR-10 datasets achieve per-layer parameter-compression ratios from about $2{,}000\times$ to $430{,}000\times$. Several VGG-16 compressed models match or slightly exceed the dense baseline, whereas AlexNet shows moderate degradation under more restrictive redundancy. These results indicate that nonlinear tensor-network generators are a promising structured route to compact pattern-recognition models, while also showing that contraction schedules and hardware-aware implementations remain essential for practical speedups.

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.

Deep networks often exhibit a preference for "simple" solutions, and such a simplicity bias is widely believed to play a key role in generalization. Yet a broadly applicable, quantitative measure of simplicity remains elusive. We introduce polynomial representations as a distribution-aware, low-dimensional surrogate for neural functions: we approximate a network's predictive behavior along data-dependent interpolation paths using orthogonal polynomial bases, yielding a compact functional representation. We show that the effective degree of this representation serves as a practical simplicity metric that is predictive of generalization across tasks and architectures, and consistently outperforms existing generalization proxies such as sharpness. Finally, polynomial representations naturally yield a differentiable simplicity regularizer, which consistently improves generalization in image and text classification, fine-tuning contrastive vision-language models, and reinforcement learning.

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

The purpose of this article is to provide validation to my deep neural network alternative in the context of LLMs. Very recently, there has been a significant interest by Chinese researchers in a model called RBF network, as a substitute to standard DNNs, with increased explainability and higher accuracy. It turns out that my new model, discovered independently, is based on the exact same machinery. But with a major twist: it does not need DNN as it finds the global optimum of the loss function in closed form, in one iteration, thus eliminating the tedious training step. Here I provide a high-level overview of my technology, with case study and comparison to similar methods.

Given the wide range of deployment targets, flexible model selection is essential for optimizing performance within a given compute budget. Recent work demonstrates that stitching pretrained models within a model family enables cost-effective interpolation of the accuracy-efficiency tradeoff space. Stitching transforms intermediate activations from one pretrained model into another, producing a new interpolated stitched network. Such networks provide a pool of deployment options along the accuracy-efficiency spectrum. However, existing stitching approaches often yield suboptimal tradeoffs and lack generalizability, as they primarily rely on heuristics to select stitch configurations. We argue that constructing improved accuracy-efficiency tradeoffs requires explicitly capturing and leveraging the similarity between pretrained models being stitched. To this end, we introduce KLAS, a novel stitch selection framework that automates and generalizes stitch selection across model families by leveraging KL divergence between intermediate representations. KLAS identifies the most promising binary stitches from the $O(k^2n^2)$ possibilities for $k$ pretrained models of depth $n$. Through comprehensive experiments, we demonstrate that KLAS improves the accuracy-efficiency curve of stitched models at the same finetuning cost as baselines. KLAS achieves up to $1.21\%$ higher ImageNet-1K top-1 accuracy at the same computational cost, or maintains accuracy with a $1.33\times$ reduction in FLOPs.

Tabular data in knowledge-rich domains often carries a latent prior in the form of Boolean implication relationships (BIRs) between pairs of features. We mine such relationships with a sparse-exception binomial test. The mined implications form a typed directed graph, equivalent to a propositional rule base of 2-literal clauses. We encode this graph as the connectivity of a layered neural network, called BIRDNet, in which each hidden unit corresponds to one mined rule and binds only to its two features. We show two consequences of this design: First, the architecture is sparse by construction: at most $2/d$ of the weights in each BIR layer are active, where $d$ is the input dimension. Second, the model is interpretable: every trained unit keeps a stable symbolic identity, so rules can be read off the network without surrogate models. Unlike most neurosymbolic models, BIRDNet does not consume an external rule base; its structural prior is mined from the data. We evaluate BIRDNet on six transcriptomic and proteomic benchmarks. Our results show that BIRDNet stays within 0.02 AUROC of the strongest dense baseline, at a small accuracy cost, while using up to $96\times$ fewer active parameters than an architecture-matched dense MLP. First-layer rules recover known biological signatures across multiple cancer subtypes and tissue types, including canonical amplicons, lineage-defining co-expression modules, and immune-infiltration markers. Data and code are available at: https://github.com/MAHI-Group/BIRDNet.

Most existing expressivity theories for neural networks assume exact real arithmetic, whereas practical neural networks are executed under finite-precision floating-point arithmetic with implementation-dependent execution semantics. Recent works have begun studying the expressive power of floating-point neural networks, but existing results are limited to highly restricted activation functions and idealized assumptions such as fixed left-to-right reduction orders and correctly rounded activation implementations. In this work, we study the expressive power of floating-point neural networks under generalized floating-point execution semantics, including arbitrary reduction orders and inexact activation implementations with bounded ulp errors. We investigate when floating-point neural networks can represent arbitrary functions between floating-point domains exactly. To this end, we introduce a general distinguishability framework and show that the ability to distinguish every pair of distinct inputs in the first layer is necessary for universal representability. This characterization yields broad classes of activation implementations that are not universal representators, extending previous isolated counterexamples such as the correctly rounded cosine activation. We further prove that a suitable form of distinguishability is also sufficient for universal representability under mild conditions on the activation implementation. Using this framework, we establish universal representability results for a broad class of practical activation functions, including implementations of $\mathrm{Sigmoid}$, $\tanh$, $\mathrm{ReLU}$, $\mathrm{ELU}$, $\mathrm{SeLU}$, $\mathrm{GeLU}$, $\mathrm{Swish}$, $\mathrm{Mish}$, and $\sin$, under significantly more realistic floating-point execution models than previously known.

Parity functions are fundamental Boolean operations with critical applications across machine learning, cryptography, and error correction. Yet, learning high-dimensional parity functions poses significant challenges: in a general setting, standard neural network architectures typically require exponential sample complexity, making gradient-based optimization intractable for large number of inputs $N$. We demonstrate that compact product-based neural architectures combined with stochastic data sparsity (Bernoulli inputs with $p_e \leq 1/N$) and appropriate hyperparameter choice enable efficient parity learning, with theoretical guarantees of convergence. Experiments validate our theory across dimensions up to $N = 100{,}000$, with empirical evidence showing optimal hyperparameter choices for $p_e$ and learning rate $α$, as well as polynomial complexity scaling laws. This work establishes fundamental connections between architectural inductive bias and data sparsity, opening new possibilities for neural arithmetic, structured reasoning, binary neural networks, and machine learning applied to automated protocol discovery.

This note provides an interesting observation on casting partial least square (PLS) as a linearized self-attention so that PLS may be studied within the neural network paradigm. On the other hand, the dimensionality reduction and selection of predictors in PLS may indicate that self-attention includes certain degree of dimensionality normalization toward improved learning.

We propose the Tikhonov layer, a graph neural network layer that is interpretable by design: once trained, its learned parameters directly reveal which node features and which aspects of the graph topology were leveraged for prediction. In practice, the layer's propagation matrix takes the closed-form $R = (p(L)+Q)^{-1} Q$, where $L$ is the normalized graph Laplacian, $Q = diag(q_1,...,q_n)$ a learnable diagonal matrix of positive node-importance scores, and $p(\cdot)$ a learnable polynomial. For any input feature $x$, the layer output $Rx$ is the exact minimizer of a generalized graph Tikhonov problem that trades off node-level data fidelity against a topology-driven regularization penalty. The learned pair $\{\{q_i\},p\}$ constitutes a built-in explanation: large $q_i$ indicates that node $i$'s own features drive the prediction, while small $q_i$ signals reliance on the local graph topology; the shape of $p$ reveals whether homophily, heterophily, or a band-pass response is exploited. Expressivity is preserved by routing complexity through a dedicated, arbitrarily deep Q-network that produces the importance scores, while the Tikhonov layer itself remains transparent. We prove that distinct node-importance matrices yield distinct propagation operators, structurally coupling the explanation to the computation. Additionally, the Tikhonov layer provides, in a single layer, a global receptive field, mitigating both oversmoothing and oversquashing. Experiments on standard graph classification benchmarks confirm that the model matches (and sometimes outperforms) opaque baselines while producing interpretable and faithful explanations.

This paper addresses the problem of identifying parsimonious explicit piece-wise polynomial relationships that might involve a relatively large number of raw features. The algorithm leverages a recently proposed identification algorithm that yields parsimonious implicit relationships enabling to derive normality characterization in the context of anomaly detection and localization. The algorithm proposed in this paper goes a step further by deriving explicit piece-wise representations that are built using the set of polynomials involved in the implicit representations. The framework is illustrated on the problem of identifying parsimonious explicit representations of the inverse model of a 6-axis manipulator robot. Moreover, further experiments on a 4-axis robot are also shown which are designed to investigate the generalization capability of parsimonious models compared to state-of-the-art DNNs structures, when models face unseen contexts of use.

Traditional network analysis focuses on single-layer networks, real-world systems often form multilayer networks with multiple relationship types. However, existing methods typically fail to capture complex inter-layer dependencies by treating layers independently or aggregating them. To address this, we propose T-GINEE (Tensor-Based Generalized Multilayer-graph Estimating Equation), a statistical regularization framework combining tensor-based generalized estimating equations with task-specific loss to model cross-network correlations explicitly. Key innovations include: (1) CP tensor decomposition capturing structural dependencies via shared latent factors; (2) a generalized estimating equation framework modeling inter-layer correlations through working covariance matrices; and (3) a flexible link function accommodating characteristics like sparsity. Our theoretical analysis establishes consistency and asymptotic normality under mild conditions. Extensive experiments on synthetic and real-world datasets validate T-GINEE's effectiveness for multilayer network analysis.

Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span. Such linear methods struggle, however, for data with slowly decaying Kolmogorov $n$-widths, such as advection-dominated and turbulent flows, which require many modes for accurate reconstruction; moreover, energy-based truncation can discard low-energy modes needed to capture small-scale features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy mode selection achieve better reconstruction with fewer modes, but fix the form of the nonlinear mapping a priori, limiting expressivity. In contrast, neural network (NN) manifolds offer greater expressivity yet employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through a residual connection with a linear skip layer, to simultaneously select informative modes and learn a nonlinear mapping that minimizes reconstruction error. On benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number $Re_τ= 5200$, our method reduces reconstruction error by 51-78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

Complex-valued Neural Networks (CVNNs) are often motivated by domains where information is naturally encoded in magnitude and phase. Yet complex-valued inputs alone do not determine when complex arithmetic improves learning: the label signal may lie in amplitude, phase, their coupling, or a symmetry that real-valued models can also represent under suitable coordinates. We study this through a representation-first evaluation of CVNNs against Cartesian real, polar, phase-only, magnitude-only, parameter-matched real, and FLOP-matched real baselines. Across synthetic RF tasks, complex representations are useful but not universally superior. PSK-only tasks favor phase-aware and complex-valued models, QAM-only tasks favor magnitude-based models, mixed PSK+QAM gives only a small complex-valued advantage, and unseen carrier-phase rotations break coordinate-dependent models without augmentation. Similar patterns appear beyond RF: in quantum-wavefunction prediction, momentum is invisible to $|ψ|$ but recoverable from phase, while EEG analytic-signal experiments show that phase locking, amplitude bursts, and phase-amplitude coupling each favor different coordinate views. We also identify a benchmarking artifact on RadioML 2018.01A. Under matched-shared-trial selection, a CReLU complex model exceeds the best real baseline by 22.94 PP; under independent per-family tuning on the same data and 16-trial search space, the gap collapses to 2.46 PP. Gradient analysis traces the inflated gap to high-learning-rate first-step instability in real baselines, while complex parameter coupling distributes the loss signal more robustly. A learning-rate $\times$ activation factorial confirms the failure is primarily hyperparameter-driven. Overall, CVNNs are best viewed as structured inductive biases whose gains depend on representation, symmetry, and optimization, not as universally superior architectures.

Equivariant neural networks encode geometric symmetries by construction, yet they are often difficult to optimize and can underperform less constrained architectures. A growing body of work addresses this through architectural modifications such as constraint relaxation or approximate equivariance, while the role of the optimizer remains comparatively underexplored. We study this direction by comparing Muon and Adam across several equivariant and geometric architectures under pointcloud and molecular learning settings. On ModelNet40, where the comparison is clearest, Muon consistently improves over Adam across all architectures considered. We then analyze the trained ModelNet40 checkpoints through Hessian estimates, loss surface visualizations, and spectral properties of learned weights and intermediate representations. The checkpoints reached by Muon have larger Hessian curvature summaries but more regular loss surfaces, and their learned weights and representations have higher stable and effective ranks. These observations suggest that the interaction between optimizer design and geometric inductive bias deserves further attention from the community.

The successful training of neural networks hinges on the use of first order optimization methods, yet the theoretical characterization of these methods remains incomplete. This is especially true in settings with mild overparameterization. In this work, we study the gradient flow dynamics of two-layer ReLU networks from small initialization with orthogonal training data. We prove the limiting flow converges to a saddle-to-saddle jump process as the initialization scale tends to zero, revealing an incremental learning phenomenon in which a new neuron activates at each saddle. This analysis recovers the known result of Dana et al. (2025, arXiv:2502.16977) that the network interpolates the training data with high probability as soon as $m \gtrsim \log(n)$, where $m$ is the network width and $n$ is the number of training samples. This incremental process characterization also allows us to derive a novel implicit bias result: the learned interpolator has a squared $\ell_2$-norm scaling as $\sqrt{n}$, which is within a constant factor of the minimal $\ell_2$-norm interpolator. More broadly, our work provides the first rigorous proof of an incremental learning process for ReLU networks, whilst suggesting mildly overparameterized networks can converge to interpolating solutions whose complexity is of the same order as that of the optimal interpolator.

Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise process. We show that the signal-to-noise ratio (SNR) and the training sample size influence the alignment in qualitatively similar ways in networks trained on real-world datasets and in an extremely simple $linear$ network with a single hidden layer, for which the alignment can be estimated analytically. Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error. These findings reveal a non-trivial dependence of alignment on data quality and quantity, decoupled from generalization performance.

Traditional Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM) units operate on discrete time steps, often failing to capture the fluid temporal dynamics of real-world physical processes. Liquid Neural Networks (LNNs), specifically Closed-form Continuous-time (CfC) networks, address this by modeling the hidden state evolution as a continuous differential equation. In this paper, we conduct a comprehensive benchmarking study across four distinct sequential modalities: neuromorphic event-based data (N-MNIST), stroke-based drawing (QuickDraw), visual handwriting (IAM), and physiological time-series (PhysioNet Sepsis-3). Furthermore, we perform a rigorous stress test using temporal dropout to evaluate model robustness against missing data. Our findings reveal that LNNs consistently provide superior parameter efficiency and significantly higher robustness in natively temporal domains and clinical environments where data sparsity is prevalent. This extended preprint provides additional background on related datasets and the LNN theoretical lineage, supplemented with a detailed appendix documenting our full implementation and experimental settings.

Feedforward network (FFN) layers account for a large fraction of parameters and nonlinear expressivity in Transformer-based large language models (LLMs). Despite the evolution from ReLU and GELU to gated variants such as SwiGLU, most FFN designs still use a single fixed activation function, applying the same nonlinear transformation to all tokens. In this work, we propose Mixture of Activations (MoA), a token-adaptive FFN design that mixes a dictionary of activation functions using lightweight input-dependent gates while sharing the same linear projections. As an input-independent counterpart, we also introduce learnable activations (LA), which form linear combinations of activation functions for both ReLU-type and SwiGLU-type FFNs. Theoretically, we establish strict finite-width expressive separations among fixed-activation FFNs, LA, and MoA: LA strictly contains fixed-activation FFNs, while MoA strictly contains LA, with the additional expressivity arising from input-dependent nonlinear hybridization. Empirically, we evaluate MoA through extensive pre-training experiments on dense and MoE language models ranging from 0.12B to 2B parameters under different token budgets, optimizers, and learning rate schedules. MoA consistently achieves lower terminal loss and exhibits more favorable scaling behavior than well-tuned baselines, with minimal parameter and computational overhead. These results suggest that token-adaptive activation mixing is a simple and effective mechanism for improving FFN expressivity in LLMs.

The detection of unmanned aerial vehicles (UAVs) is important for the protection of civilian and military infrastructure. In this paper we propose a cost effective UAV detection system using sound signals obtained from microphones. The recorded signals are passed through a signal processing pipeline which employs interpretable adaptive feature extractors using so-called rational Gaussian wavelets. These adaptive wavelet transformations are embedded into and trained together with an underlying small neural network which detects and classifies UAVs based on the obtained features. This leads to a physically interpretable machine learning algorithm that in addition to classifying UAVs is also capable of detecting UAV swarms. We demonstrate our results using data collected in indoor studio and noisy outdoor environments. We conclude that the proposed method outperforms traditional machine learning approaches for detecting and classifying single UAVs as well as drone swarms, while retaining a high degree of interpretability. Our implementation of the proposed methods is made publicly available for reproducibility.

Temporal signed networks (TSNs) model the time evolution of cooperative and adversarial relationships that arise in applications such as social media analysis, trust and reputation systems, and financial transaction networks. While graph neural networks (GNNs) perform well for static or unsigned link prediction, effective learning in temporal signed graphs remains challenging due to the interaction of signed relations, evolving structure, and balance-theoretic constraints. To address this gap, we propose a \emph{modular} temporal enhancement framework for signed GNNs that integrates historical context into otherwise static architectures. The framework introduces a Historical Context Integration Module (HCIM) that combines learnable recency-aware temporal weighting, LSTM-based embedding trajectory modeling, and multi-head temporal attention to capture both short- and long-term signed interaction dynamics. Historical information is fused with current node representations using either global or node-adaptive weighting, allowing the architecture-agnostic framework to accommodate heterogeneous temporal behaviors. We instantiate the approach on the Self-Explainable Signed Graph Transformer (SE-SGformer), preserving interpretability while extending it with temporal awareness. Experiments on real-world and synthetic TSNs, including Bitcoin OTC, Bitcoin Alpha, Reddit, and small-world network models, demonstrate consistent and statistically significant improvements over the static baseline.

Graph neural networks (GNNs) achieve strong performance on relational data, but real-world graphs are often distributed across organizations that cannot share raw data due to privacy and policy constraints. Existing federated GNN methods either ignore cross-client links, leading to degraded accuracy, or require frequent embedding exchanges, incurring substantial communication and privacy costs. We propose CE-FedGNN, a communication-efficient and privacy-preserving federated GNN framework for learning over such coupled graphs. Our approach avoids sharing raw data or per-round embeddings by infrequently exchanging aggregated node representations. To handle cross-client dependency and staleness, we introduce a moving-average estimator that continuously tracks node representations and enables their stable reuse across rounds. To provide formal privacy guarantees for the released representations, we adopt the metric differential privacy (metric-DP) framework, which measures privacy with respect to distances in the learned embedding space rather than worst-case input perturbations. This yields meaningful guarantees at noise levels where standard differential privacy becomes overly conservative. We establish convergence to a stationary point at a rate of $O(1/\sqrt{T})$ with $O(T^{3/4})$ communication complexity. In addition, we derive $(\varepsilon,δ)$-metric-DP guarantees via Rényi differential privacy composition under a public-cohort threat model. Experiments on synthetic interbank anti-money laundering benchmarks and citation networks demonstrate that CE-FedGNN achieves strong performance while significantly reducing communication and maintaining robustness under privacy-preserving noise.