Neural Network

We study information bottlenecks in modern deep-learning architectures -- RNNs, softmax transformers, linear-attention transformers and state-space models -- through the lens of the indexing primitive. In this primitive, the input consists of $n$ bits and one integer $i$ from $1$ to $n$ called the index, and the output equals the value of the $i$-th bit. We introduce causal complexity for masked architectures. We show that architectures with low causal complexity cannot solve the indexing primitive in any constant number of layers when the index appears at the end of the input. In particular, this limitation applies to low-parameter RNNs, SSMs and masked linear-attention transformers. In contrast, small softmax transformers can solve it in one layer, while non-masked linear-attention transformers can solve it in 2, which separates them from their masked counterparts. In turn, when the index appears at the beginning, we show that small RNNs are capable of solving this task in 1 layer, while all the other architectures require 2. All our impossibility results are unconditional and apply even to models that employ infinite-precision real arithmetic. Moreover, experiments for up to $n=64$ qualitatively align with our theory: configurations with low-parameter theoretical solutions learn the indexing task easily, while configurations that do not admit such theoretical solutions struggle to learn as the sequence length grows.

Models trained on long-tailed data using standard softmax tend to exhibit higher training error and a larger generalisation gap for classes with fewer training samples. We characterise this class-wise disparity as the preference issue and quantify it using a new metric, the model imbalance level $I$. To understand this issue, we analyse how imbalanced training data adversely affects class-wise gradients under standard softmax training. This paper then develops a finite-data Generalised Balanced Softmax (GBS) framework for analysing and mitigating the preference issue. The framework uses the training-time logit adjustment $z_{nc}+β\log|N_c|$, which is algebraically identical to the training-time logit-adjusted loss of Menon et al. (2021) when $τ=β$. The case $β=1$ also coincides with Balanced Softmax and with the unit adjustment supported by the Fisher-consistency argument under the true data distribution, corresponding to an idealised infinite-data setting. Building on this existing loss family, this paper uses a heuristic power-law assumption to motivate the adjustable coefficient and studies how $β$ affects trained models. Across the evaluated long-tailed benchmarks, $β=1$ does not attain the highest average testing recall on most datasets, showing that a different coefficient can be preferable when training on finite data. The selected values of $β$ reduce $I$ and improve average testing recall relative to the $β=1$ reference, while retaining negligible computational overhead and compatibility with existing representation-learning frameworks.

High task performance does not show whether a model retains prediction-relevant structural information in its internal representation. Temporal graph models, for example, can achieve high future-link AUC while basic graph statistics remain difficult to recover from the same representation. We identify one source of this gap in the weighted averaging used by standard attention: when an evidence pattern is repeated, the numerator and denominator grow at the same rate, so inputs with different amounts of accumulated evidence can produce the same aggregate. We propose Mass-Aware Attention (MAA), which generalizes standard L1 normalization to an Lp family. Under repetition, MAA makes the numerator and denominator scale at different rates, retaining the effective number of contributing inputs in the representation magnitude. It adds no supervision, parameters, hidden dimensions, or explicit count features, and recovers standard attention at p=1. Across four continuous-time dynamic graph models and three datasets, MAA improves future-link AUC in 11 of 12 model-dataset cells. Linear recovery from the same hidden representation increases by 4.49% on average, and preferential-attachment recovery improves in all 12 cells after family-wise correction. We also observe consistent evidence in marked temporal point processes, temporal knowledge graphs, retrieval-augmented generation, and spatio-temporal point processes. Information accessibility and task utility remain distinct: NLL improves in MTPP, ranking is largely preserved in TKG, additional information in RAG does not improve the diagnostic head, and downstream LayerNorm can erase the signal in STPP. These results position MAA as a general normalization principle for improving predictor-facing representation informativeness by controlling repetition invariance in standard attention.

Accurate pixel-level classification of coronary angiograms is critical for cardiovascular disease assessment, yet the field lacks standardized evaluation protocols. In this work we demonstrate a new benchmark for the assessment of deep learning models which densely classify pixels of coronary angiograms to one of SYNTAX classes (or background). The evaluation covers 24 distinct architectures starting with classic convnets to recent state-space-based vision algorithms. We release CARDIAG - a multi-center, multi-label dataset which we carefully split to reliably compute metrics, accounting for diameter error, overlap, centerline quality and calibration. The data contains SYNTAX labels, binary, uncertainty and segmentation masks as well as intermediate frames together with the selected non-sensitive DICOM metadata. From the multitude of algorithms, we nominate ConvNeXt V2 encoder with DeepLab V3 Plus decoder as the best performing, achieving macro $F_1=0.456$, which we then ensemble with Mamba U-Net and Feature Pyramid Network, for an increased $F_1=0.479$. We demonstrate all the architectures to be well calibrated and determine the generalization of the top 5 methods, together with the data efficiency of these architectures. We highlight the importance of both high-resolution and low-resolution features in encoding. We also demonstrate the model correctness in the context of patient demographic, vessel sides and projection angle configurations. Overall the released benchmark allows for future studies to robustly and rigorously assess the proposals, not only for SYNTAX segmentation, but lesion detection and many more.

We investigate whether symbolic regression can discover explicit neural network weight-update rules that outperform standard hand-designed optimizers on small symbolic regression benchmarks. Candidate update rules are represented as fixed-depth symbolic expressions over operands derived from common optimizers, including gradient, momentum, adaptive-gradient, and moment-estimate quantities. Across 30 benchmark/neural network combinations, the symbolic regression procedure found an update rule outperforming the best hyperparameter-tuned established optimizer in 25 cases, with an aggregate MSE reduction of 44.47% over the improved cases. The discovered rules do not all share a single common symbolic form, but many combine adaptive normalization, momentum-like quantities, nonlinear transformations, and rational expressions. These results suggest that symbolic regression can serve as a lightweight mechanism for discovering compact optimizer variants, while also highlighting the need for larger-scale validation.

Deep neural networks encode complex representations, but deconstructing this internal knowledge remains a challenge. Given the link between learning and compression, network compression offers a promising lens to analyze this knowledge. However, standard compression heuristics often suffer from scale symmetries and architectural biases. To resolve these, we introduce Hilbert Operator for Progressive Encoding (HOPE), a mathematical framework to gradually deconstruct the representations in trained network weights. HOPE shifts network compression from the discrete domain into a Hilbert space of continuous functions. By modeling individual neurons as rank-1 Hilbert-Schmidt operators, HOPE unifies pruning and neuron merging as low-rank subspace projection. Extending this formulation, HOPE introduces macro block eviction to encompass multi-layer structures like entire residual pathways under the same unified metric. This unified approach enables unbiased architectural decisions across layers with different types and sizes. HOPE is a data-free and hyperparameter-free framework. We present proof-of-concept experiments in model compression and fine-tuning to highlight the practical potential of our theory.

Numerous studies have analyzed the use of hybrid quantum-classical convolutional neural networks as a promising alternative to classical deep learning. However, network components on quantum hardware impose fundamental limitations, while the scalability of quantum circuits leads to trainability issues. In this work, we investigate whether small, classically-emulated quantum circuit components can play a meaningful role within complex models, offering an alternative to purely classical convolutional architectures. To this end, we present a systematic study of the effectiveness of a Hybrid Quantum-inspired Convolutional Neural Network (HQiCNN) compared with a parameter-matched classical Convolutional Neural Network (CNN) that differs only in an intermediate dense neural layer. Both models are evaluated on two real-world medical datasets while systematically varying the different hyperparameters, ensuring a fair model comparison that is both dataset and hyperparameter independent. The results show that no architecture consistently dominates the other: the HQiCNN achieves its largest gains in intermediate-data regimes, whereas the CNN reaches the highest accuracies for the largest training sets in both datasets. Furthermore, removing entanglement produces comparable performance while enabling substantially better scalability of quantum simulations, and richer observable sets become beneficial only when sufficient training data are available. Finally, we propose two SHAP-based explainability tools for comparing the predictions between both models, $|SHAP|$IoU and $EMD_{pos}$ metric, to demonstrate that both architectures consistently attend to anatomically plausible regions. Thus, we provide a comprehensive benchmark showing that, under certain conditions, hybrid quantum-inspired models are an alternative that can offer benefits in practical tasks such as medical image classification.

Neural network compression and interpretability remain open challenges in modern deep learn- ing, where billion-parameter architectures deliver impressive accuracy at the cost of trans- parency, computational efficiency, and reliable uncertainty quantification. This paper introduces Neural Atom Prevalence (NAP), a principled Bayesian framework for structured node-level model selection in feedforward neural networks. NAP introduces the neural atom (activation unit) and functions as a hybrid method operating through a four-phase pipeline: Bayesian Lottery Ticket (BLT) identification via Iterative Magnitude Pruning (IMP), soft variational training of the Spike and Slab Independent Gaussian (SS-IG) model, Poisson-Binomial (PB) optimal layer-size selection, and Bayesian fine-tuning to produce a sparse, stable, interpretable, and accurate model. Extensive empirical validation across simulated nonlinear regression, two UCI benchmark datasets (Concrete, YearPredictionMSD), and the MNIST image classification task demonstrates that NAP achieves state-of-the-art structural sparsity, reducing active nodes to as few as 8% of the original dense architecture on MNIST, while well-calibrated probabilisti- cally: the aleatoric-epistemic uncertainty decomposition reveals that model ignorance accounts for only 3 to 4% of total predictive variance across all experiments, and regression reliability diagrams confirm a near-nominal predictive interval coverage (93.4% observed against a 95% target). These results establish NAP as a reliable, theoretically grounded, and computation- ally tractable solution to the simultaneous pursuit of sparsity, accuracy, interpretability, and uncertainty quantification in Bayesian neural networks.

In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains incomplete even when dealing with the simplest kinds of activation functions. Indeed, while there has been a number of very recent results that establish ever-tighter lower bounds for the problem under linear and ReLU activation functions, less progress has been made towards the identification of novel polynomial-time tractable network architectures. In this article we obtain novel algorithmic upper bounds for training linear- and ReLU-activated neural networks to optimality which push the boundaries of tractability for these problems beyond the previous state of the art. In particular, for ReLU networks we establish the polynomial-time tractability of all architectures where hidden neurons have an out-degree of $1$, improving upon the previous algorithm of Arora, Basu, Mianjy and Mukherjee. On the other hand, for networks with linear activation functions we identify the first non-trivial polynomial-time solvable class of networks by obtaining an algorithm that can optimally train network architectures satisfying a novel data throughput condition.

Unique and rapid classification of knots and links is an open mathematical problem that is relevant to a range of (bio)physical systems, including polymer melts, DNA, and proteins. In this paper, we explore a data-driven approach to the classification problem of link topology. Extending the framework introduced in Ref. 1 (Sleiman et al, 2024 Soft Matter, 20(1), pp.71-78), we show that a feedforward neural network trained on the writhe density matrix classifies thermally equilibrated configurations of the first six prime links with 97% accuracy. We demonstrate that this accuracy remains high across a range of temperatures and lengths of link components, while rapidly deteriorating with the addition of topology-altering Gaussian noise; a result consistent with the writhe density matrix containing features sensitive to topology. Our results show that neural networks based on the writhe density matrix efficiently classify two-component links, establishing machine learning as a promising tool for rapid classification of more complex link topologies, e.g. Borromean rings and multi-component links, as the computational cost of exact numerical calculation of topological invariants becomes prohibitive.

Sparse autoencoder (SAE) features are used to interpret and steer large language models, yet whether a feature's causal role is stable across SAE families remains untested. Single-token features that activate on one vocabulary item provide the diagnostic case where ground truth permits direct comparison. We analyze 3.9M features across six models and three SAE families using zero-ablation at full layer depth. Single-token features cluster 4.7x tighter in decoder space and concentrate in early layers (Layer 0 in GPT2-Small; L0-L4 in Gemma). Ablating them yields Benjamini-Hochberg-significant logit reductions in 178 of 208 full-layer conditions, with depth controlling whether damage cascades downstream or shapes the output directly. Cross-family causal differences exceed within-family scale effects: on the same base model, GemmaScope and BatchTopK features remain causally anchored, while LlamaScope features are locally redundant. The target token's rank recovers to within 2x baseline 96-98% of the time after the same ablation, and a controlled activation-function comparison reverses sign within the same model, leaving training recipe as the residual candidate. Cross-family interpretability claims are therefore sensitive to training methodology, not just activation function or scale.

We study the depth required by ReLU networks to exactly represent piecewise linear functions, focusing specifically on the maximum function. This problem has recently received significant attention in both the ML and TCS literature. We prove that $\max_n(x)=\max\{x_1,\ldots,x_n\}$ is exactly representable with two hidden layers for every $n\leq 12$. Previously, this was only known up to $n\leq5$ [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. We obtain our constructions through an exact computer-assisted search within a space of candidate solutions: After a symmetry reduction, we obtain a finite system of linear equations over $\mathbb{Q}$ such that any solution yields a valid representation of the maximum function. The resulting constructions have a structured first hidden layer, which enables recursive substitution into deeper networks. This yields an exact ReLU representation of $\max_n$ with at most $\lceil \log_6(n/2) \rceil+1$ hidden layers. Consequently, every continuous piecewise-linear function on $\mathbb{R}^d$ admits an exact representation with at most $\lceil\log_6((d+1)/2)\rceil+1$ hidden layers; in particular, two hidden layers suffice for $d\leq 11$. Again, these results improve upon [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26], who proved analogous logarithmic bounds with base three.

Constitutive modeling under uncertainty remains a central challenge for reliable mechanics simulations, particularly when the available stress-deformation data are sparse, noisy, or heterogeneous. We propose interval and fuzzy physics-augmented neural networks (iPANNs and fPANNs) for uncertainty-aware hyperelastic constitutive modeling. iPANNs learn sparse lower, mean, and upper free energy density branches whose stresses, obtained by automatic differentiation, ultimately enclose noisy stress observations. In contrast to this deterministic interval description, fPANNs embed the learned iPANN branches into a fuzzy-set representation through alpha-cut interpolation, yielding a nested family of admissible responses. iPANNs and fPANNs encode mechanistic constraints - preserving objectivity, consistency and promoting polyconvexity - and smoothed L0 regularization promotes interpretable energy representations. The bound models are trained through a two-stage transfer-learning procedure in which a sparse mean constitutive response is learned first and then fine-tuned into lower and upper energy branches. We evaluate the framework on synthetic isotropic hyperelastic data with heteroscedastic noise, varying random realizations, shifted noise means, and varying noise magnitudes. The results show that the learned bounds enclose noisy stress observations while generalizing to the test set. Further, we examine the propagation of uncertainty through the mean, upper and lower bound predictions of the learned iPANN models in a finite element setting. The proposed framework provides a compact, physics-consistent route for distribution-free aleatoric uncertainty quantification in hyperelastic constitutive modeling, and propagation in downstream finite element simulations.

Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, the FVM-loss model achieves an all-field normalized root-mean-square error (nRMSE) of 2.3-2.8%. It demonstrates close agreement with the CFD references, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model outperforms the supervised baseline in terms of accuracy, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.

We study Gaussian averaging as a smooth surrogate for quantized neural models. Under bounded local oscillation, we derive a local dimension-dependent bound on |f-g|, linking Gaussian smoothing to the stability analysis of discontinuous networks. We compute closed-form Gaussian averages of the rectified linear unit (ReLU) and sign activation functions, and illustrate the mechanism on a high-dimensional binary perceptron, where layer-preactivation aggregation under an explicit quantization-noise surrogate yields the Gaussian envelope used in inference-side smoothing and training-side smooth surrogate gradients.

We study vector-valued affine refinement operators of the form [ (Wγ)(t)=\sum_{j\in\mathbb{Z}} A_jγ(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data. Building on the homogeneous realization theorem for (B\equiv 0), we prove that, for (M\ge 3), every finite affine iterate (W^nγ) admits an exact fixed-width ReLU realization whose depth is (O(n)). The main new ingredient is a residual memory controller. It replaces the noninvertible residual dynamics by an injective skew-product and permits exact backward replay of the residual states required by a Horner-type evaluation of the affine forcing sum. Offset frames align the forcing atoms away from residual seams, allowing complementary loop readouts to recover their values exactly. The remaining branch-selection ambiguity occurs only where the accumulated affine state has already vanished. For (M\ge 3), the result applies to arbitrary compactly supported continuous piecewise linear forcing terms. For (M=2), the same construction applies to ordinary-frame seam-separated forcing. We also prove a stage-dependent extension for forcing terms in a fixed finite-dimensional continuous piecewise linear span and record the resulting linear-depth upgrade for open-curve, finite-state, and Hilbert- and Morton-type recursive constructions.

Physical noise in near-term quantum hardware is usually treated as a nuisance to suppress. We ask whether it can instead act as a hardware-native regularizer for photonic hybrid quantum-classical neural networks (PHQCNNs), analogous to noise-injection regularization in classical deep learning. Using Quandela's Perceval simulator and the MerLin framework, we build PHQCNNs for Iris, Digits, and MNIST and inject Perceval's seven-parameter physical noise model directly into training. A genetic algorithm searches the six continuous noise dimensions and 1 boolean parameter to find, per dataset, the configuration maximizing validation accuracy, compared against a noiseless baseline across five seeds. GA-tuned noise yields modest accuracy gains on Iris (+0.82pp) and Digits (+1.45pp), but a clear degradation on MNIST (-1.21pp). Per-parameter sweeps show that no individual noise parameter is consistently beneficial, motivating the joint search, while a second-order loss expansion shows that physical noise induces a Tikhonov-like regularization term whose effect is dataset-dependent. Physical photonic noise can thus act as a free regularizer, but not universally.

AI researchers describe state-of-the-art models as one thing repeated at scale: the Transformer, wired identically for text, pixels, or speech. Neuroscientists describe the cortex as a mosaic - dense Layer 4 in visual cortex for spatial encoding, thick Layers 5/6 in motion cortex for temporal integration - different jobs solved by different structures. This paper argues the gap is a structural error, not a stylistic one, and is measurable. A century of cytoarchitecture, from Brodmann to single-cell Patch-seq, shows distinct cognitive functions are implemented by qualitatively different structures, not by rescaling one template. The convolutional neural network is the field's own proof: local receptive fields and hierarchical depth encoded this prior directly, reaching strong image recognition on far less data than later architectures needed. The paper traces how this lesson was discarded: the "Hardware Lottery" made the Transformer the path of least resistance, not the principled choice, and Mixture-of-Experts, often cited as diversity, in fact partitions parameters among identical experts. A functionalist analysis shows the Transformer is best understood as a functional analog of the hippocampal formation, not a general-purpose cortex - the same mistake as treating cortex as one giant Broca's area, except the field has now standardized on a giant hippocampus, applied to tasks it was never built for: audition, executive gating, working memory. The paper closes with an alternative: a Heterogeneous Topological Network, a System of Systems in which distinct modules keep the inductive bias their computation demands and communicate through standardized interfaces. This is a design discipline for AI architects, not cognitive science: specify modularity before training, using structural evidence as a design input rather than reverse-engineering architecture from a trained model's behavior.

Geometric architectures are often justified by internal mechanisms such as rotations, yet task performance alone cannot show whether those mechanisms drive predictions. Using sheaf neural networks (SNNs) as a testbed, we introduce the first basis-independent measurement of trained triangle-loop products, separating rotation, stalk-space area, and orientation. In a custom high-homophily GraphUniverse regime, Neural Sheaf Propagation (NSP) increases the triangle-weighted mean two-dimensional SO(2) loop rotation from 0.010 to 0.388 radians for triangle counting, while the community-detection comparison ends at 0.029 radians. Across the training-set-size experiment, replacing all learned SO(2) transports with identities sharply increases test error, establishing post-training sensitivity to the complete learned connection. However, a graph-summary ridge predictor is more accurate, diagonal maps also improve, and fixed-degree graphs develop increasing rotation without outperforming the training-mean predictor. This measure-intervene-control study separates geometric change, connection sensitivity, and evidence for triangle-specific computation.

Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.

Associative emotional learning enables organisms to adaptively link pleasant or unpleasant outcomes to the presence of predictive stimuli. Whereas computational models such as the Rescorla-Wagner model have shed light on this important function, the limitations of these models are also known, especially when they are applied to neural data. The advent of deep neural networks has opened another avenue for modeling associative emotional learning. In this work we proposed a deep neural network model of visual valence processing, consisting of a visual module that encodes complex natural scenes and a module that recognizes their emotional significance in terms of valence, a key dimension of emotion, and tested a novel Pavlovian learning paradigm on the model. The results showed that with learning, the model reproduced several observations from human associative learning studies, including association formation and generalization, and that the neural representations of the conditioned and the unconditioned stimuli became increasingly aligned both at the single unit and at the neural population level. Comparison between the model and human experimental data provided further validation of our approach. This study thus suggests that deep neural network models, when combined with appropriate learning algorithms, can be used to model behavioral and neural signatures of associative emotion/valence learning.

Functional magnetic resonance imaging (fMRI)-based visual decoding aims to recover visual information from measured brain activity, commonly by mapping fMRI responses into latent visual features for downstream decoding tasks. Most existing methods learn mappings from fMRI responses to visual features extracted by artificial neural networks (ANNs), yet it remains unclear whether ANN-derived features provide suitable targets for brain decoding. In this study, we investigate spiking neural network (SNN)-derived visual features as alternative targets for fMRI-based visual decoding. We compare an ANN baseline with four SNN variants from the same architectural family, which differ in their spiking dynamics. To isolate the effect of the target features, all models use the same L2-regularized linear fMRI-to-feature decoder, while only the feature vectors used as regression targets are varied. Compared with the ANN baseline, SNN-derived features exhibit stronger alignment with fMRI responses and improve visual semantic decoding performance. For instance, on the GoD dataset, SNN-derived features reduce feature-prediction error from 0.7707 to 0.0282 and improve top-1 semantic decoding accuracy from 0.1800 to 0.4400. Ablation results further indicate that both spiking neural dynamics and temporal simulation steps contribute to the observed advantage. These findings support SNN-derived features as effective brain-decodable visual representations and highlight target feature design as an important component of fMRI-based visual decoding.

Refinement operators generate many functions used in wavelet constructions, subdivision schemes, and geometric modeling. Their finite iterates can develop rapidly increasing numbers of linear pieces, making them a natural test case for the expressive power of deep neural networks. Earlier work showed that, for scalar binary refinement with a finitely supported mask, every compactly supported continuous piecewise linear seed has finite refinement iterates that admit exact ReLU realizations of fixed width and depth growing linearly with the number of refinement steps. The present paper gives a new construction of this known theorem. The difficulty is that the refinement cascade is driven by discontinuous binary digit choices, whereas ReLU networks produce continuous piecewise linear maps. We represent the residual dynamics on a polygonal model of the circle and describe each residual position in two overlapping coordinate systems, one ordinary and one shifted by one half. Their discontinuities occur at different points. The network switches between the two descriptions only where both are valid and the corresponding fixed linear cascade updates agree, so the switch is exact and requires no multiplication by a variable selector. The construction also gives exact readout of every continuous piecewise linear circle function satisfying the natural endpoint compatibility condition. Localized seeds are handled by a two-pass network, and translation covariance, finite decomposition, and gluing extend the result to arbitrary compactly supported continuous piecewise linear seeds in a preserved support window.

Neural hypergraphs are a natural generalization of neural networks, the reference models in modern machine learning. Yet, their deployment has proven demanding: the number of weighted hyperedges required leads to an intractable parameter explosion. However, a novel parametrization that leverages spectral attributes for neural hypergraphs has been recently proposed, that enables to recycle parameters via a weight sharing scheme and consequently yields a significant reduction of the associated computational cost. Preliminary tests carried out on spectral higher-order architectures pointed to meaningful improvements in both performance and interpretability. Building on these results, we advance the benchmarking efforts by evaluating the spectral higher order framework on N-bit parity tasks, a well-established testbed known to be particularly challenging. As we will convincingly argue, Spectral Higher-Order Neural Networks (SHONNs) possess a versatile and highly tunable hypothesis space.

Information Extraction (IE) from Architecture, Engineering, and Construction (AEC) drawings remains hindered by manual inefficiency, while Layout Detection, a vital 'middleware' organizing graphical and textual hierarchies, is underexplored. General document layout models, optimized for text-centric content, lack validation on engineering drawings. This study constructs a custom AEC-specific layouts dataset and benchmarks five deep learning architectures. RF-DETR achieves state-of-the-art performance with an $mAP_{50}$ of 0.949, while the Vision-Language Model Qwen3-VL attains a leading F1-score of 0.911. Conversely, models pre-trained on general document datasets suffer from "domain interference", causing performance degradation. This establishes a robust technical foundation for automated IE in AEC.

Reinforcement learning is conventionally divided into model-based and model-free methods. In this taxonomy, model-based methods perform lookahead planning over a learned world model, whereas model-free methods learn a reactive state-action mapping. Recent work, however, has shown that planning can emerge from model-free reinforcement learning alone. The conditions under which this behavior emerges from a pure reward-maximization objective have so far remained unclear. In this paper, we present evidence that, in the observed cases, the hidden-state structure of the neural architecture is the deciding factor. We find that a network of relational hidden states, each anchored to an environment state and exchanging messages along learned relations, acquires a planning mechanism. These hidden states recover the environment's transition structure in their learned relations, and improve the policy at decision time by planning over the learned graph. In a matched control agent that must additionally discover which cells represent which states, no such binding arises, and no planning follows from it. We argue that this explains the observed phenomenon of emergent planning in model-free reinforcement learning and raises the question of how common such emergent planning might be more generally. Finally, we hypothesize that the discovered mechanism could describe how planning emerges from pure reward maximization in the human brain through a neural architectural prior.

Benchmarks for Geospatial Foundation Models (GFMs) increasingly rank models by aggregate score, but such rankings obscure why models differ: how much of the gap is architecture, how much is decoder capacity, and how much is a use-case-specific artefact? This study addresses that gap through a controlled comparison of two GFMs developed under European Space Agency's $Φ$-lab with contrasting design philosophies: THOR, which introduces a compute-adaptive architecture supporting variable patch sizes and unifies Sentinel-1, -2, and -3 data at their native resolutions; and TerraMind, a multimodal generative GFM pretrained with a dual-scale token/pixel objective that enables any-to-any cross-modal generation (Thinking-in-Modalities) to infer missing sensors at inference time. Rather than reporting a single leaderboard, we investigate the axes along which the two architectures actually differ - patch size, decoder complexity, finetuning regime, input modality, and model scale - across ten use cases spanning segmentation and regression in diverse domains, including climate disaster response, methane leak detection, snow monitoring, or sea ice mapping. We find that architectural design choices - patch size and decoder type in particular - explain more performance variance than model identity itself, that the two models embody complementary investment strategies (pretraining-time scale for TerraMind versus inference-time tokenisation for THOR), and that correctly interpreting results requires dataset-level characterisation. The resulting picture is not a single winner but a set of hypotheses and a diagnostic ablation methodology that we expect to generalise to future GFMs beyond THOR and TerraMind.

Wireless physical neural networks (WPNNs) embed neural computation directly into analog hardware, offering lower energy consumption and latency than conventional digital implementations. In this paper, we propose a deep WPNN in which nonlinear activations are realized by a multi-hop multiple-input multiple-output (MIMO) relay network, in which each relay implements a trainable complex linear gain and bias, followed by the power amplifier's intrinsic nonlinearity acting as an activation function. The cascade of multiple relays therefore realizes an over-the-air fully connected network whose parameters can be trained end-to-end. We develop two transceiver designs for different channel state information (CSI) availability scenarios: a least squares (LS)-based scheme requiring only receiver-side CSI, and a singular-value-decomposition (SVD)-based scheme requiring both transmitter-side and receiver-side CSI. Simulation results show that the proposed architecture enables accurate over-the-air inference for image classification. In particular, the results highlight the advantage of exploiting hardware nonlinearity for enhanced inference capability.

Structured pruning is essential for making neural network inference feasible under homomorphic encryption (HE), yet its impact on model reliability has remained unexplored. This paper presents a systematic reliability characterization of pruned CKKS-encrypted neural networks and introduces Polynomial-Sensitivity-Aware Pruning (PSAP), a structured pruning method that is inherently reliability-aware. PSAP scores filters jointly by weight magnitude, polynomial activation sensitivity, and rotation cost, which concentrates pruning in fault-tolerant regions. Across two architectures, two datasets, two numerical representations, and five bit-error rates (40 full-model and 108 per-layer experiments), PSAP-pruned models limit catastrophic (>10 pp accuracy drop) layers to at most two versus 5--14 for magnitude-pruned baselines, reducing worst-case vulnerability by up to 29 times under int32 bit-flip injection. Direct CKKS encrypted fault injection indicates a safe operating boundary near BER~ 10^{-5}, supporting int32 injection as a conservative reliability proxy. The fault-critical structural layers account for only 1.1% of parameters, enabling selective hardening at minimal overhead. These reliability gains are obtained alongside competitive efficiency: PSAP reduces Halevi--Shoup rotations by up to 45.2% on ResNet-32, and an adaptive mixed-degree allocation scheme lowers multiplicative depth from 66 to 56 levels, enabling leveled inference without bootstrapping.

In robotic manipulation studies, grasping is often treated as a binary success or failure problem, usually defined by whether the object simply stays in the hand. For forceful tool use, however, this view is insufficient because grasp compliance becomes a critical factor governing how the hand and tool behave under load. Compliance arises from coupled kinematics, grasp configuration, passive mechanics, and contact conditions, producing nonlinear behavior in which deformation and interaction forces influence each other. Understanding this relationship is essential for predictive models of how a grasped tool and a compliant hand jointly respond to external loading. In underactuated hands, these effects are amplified: such designs offer low cost and adaptive grasping, but make compliance behavior more difficult to model and predict. Our goal is therefore to develop a predictive model for grasped tool behavior during forceful interactions. To address this challenge, we introduce an analytical model informed neural network (AMINN), a hybrid predictive model that combines an analytical mechanics layer with data driven learning to estimate grasp stability and in hand tool displacement under external loading. The model is evaluated on a three finger underactuated robotic hand and shows strong predictive capability with mechanically meaningful outputs across diverse loading conditions. Compared with a black box multilayer perceptron baseline, AMINN also achieves better energy based physical consistency. Beyond prediction accuracy alone, this framework advances physically interpretable learning for robotic manipulation and supports more reliable, safer, and more trustworthy autonomous tool use in safety critical settings during forceful interaction.

Medical image segmentation models require both high accuracy and lightweight design to accommodate real-world medical applications. The deployment of these models on resource-limited medical platforms remains a significant challenge due to their high computational and parameter requirements. Existing pruning methods for model compression mostly overlook the intrinsic connections and similarity between the internal structures of complex deep neural networks. As a result, compressed models may not effectively retain the basic features of the pretrained network. To solve this problem, we propose a hierarchical clustering compression method for medical image segmentation models (MIS-HCC). This approach employs hierarchical clustering to partition channels and fuse their parameters efficiently. Specifically, it leverages the Wasserstein distance to represent similarity of channels within layers of pre-trained network, forming a similarity matrix that guides the clustering process. Channels within each cluster are then fused to produce a compressed network. Experimental results on three medical image datasets application demonstrate that MIS-HCC outperforms the state-of-the-art methods in both accuracy and compression efficiency, offering an effective solution for deploying medical image segmentation models on resource-limited medical platforms.

In this work, we investigate the effect of lookahead branching strategies in neural network verification. We present a general recipe to integrate lookahead into any branch-and-bound verifier and demonstrate how one of the current state-of-the-art branching heuristics, FSB, can be viewed as a special instantiation of the lookahead branching strategy. We also describe how, in addition to improving the quality of branching decisions, lookahead can generate additional lemmas that accelerate verification. We instantiate the method in two representative branch-and-bound-based verifiers (Marabou and $α$-$β$-CROWN), and demonstrate that lookahead leads to consistent speedups in verification time and up to $57\%$ more solved instances. Code is available at https://github.com/ai-ar-research/lookahead-branching.

We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.

Graph Neural Networks (GNNs) have emerged as a powerful, differentiable class of learning models for graph-structured systems. Their ability to generalize across topologies opens the prospect of a surrogate for combined structural and parametric optimization, which classical metamodels cannot offer. Supply chains are a natural target, yet the use of GNN surrogates for supply chain problems is largely unexplored. This paper lays the foundation, presents initial steps, and discusses key research directions. As a foundation, we formulate the problem and create a large public training dataset of programmatically generated supply chain graphs with input parameters and steady-state performance metrics obtained using our SupplyNetPy simulation library. As initial steps, we explore GNN architectures that work well as surrogates for node- and network-level predictions, and analyze their accuracy-compute trade-off against simulation. Most importantly, we outline the exciting directions this opens, namely gradient-based optimization over topology, fast design-space exploration, and sensitivity analysis.

Understanding deep neural networks remains a central challenge in machine learning. In particular, the theoretical properties of even two-layer ReLU networks, especially in the presence of weight decay, remain poorly understood. To this end, we derive a sufficient condition on the hyperparameter settings under which the global minima collapse to the zero solution. Interestingly, our experiments reveal that using AdamW as an optimizer prevents the collapse of the learned parameters, whereas using SGD does not, which may help explain the success of AdamW in deep learning training. In addition, when restricting the input dimension to one, we derive an analytical solution for the globally optimal parameter sets of two-layer ReLU networks and show that $\ell_2$-regularization has a width-invariant effect on connectivity, but its dimensionality-reducing effect becomes stronger as the network width increases. These results provide insight into how width-dependent hyperparameters influence the geometry of regularized loss landscapes.

The widespread adoption of high-level deep learning libraries, while accelerating model development, has increasingly abstracted away the internal mechanics of neural networks, creating a gap between practical usage and fundamental understanding. To address this, the paper presents a self-contained neural network framework implemented entirely from scratch without relying on automatic differentiation or pre-built deep learning modules. The implementation encompasses all essential components, including multi-layer architectures, diverse activation functions, regularization techniques, and state-of-the-art optimizers. Beyond serving as a pedagogical instrument that demystifies forward/backward propagation, gradient dynamics, and optimization landscapes, the framework demonstrates robust performance when applied to a multi-class classification task, successfully validating its correctness, numerical stability, and generalization across varied configurations. The extensible design and clean modularity further position it as a reliable baseline for educational purposes and future research exploration.

Backpropagation makes training deep networks memory intensive because it must store intermediate activations. Forward-mode methods avoid this cost, but their gradient estimates become increasingly noisy as the number of trained parameters grows. We introduce Split Forward Gradient (Split-FG), which splits a network at an intermediate representation: it computes the output head gradient exactly and estimates only the trunk gradient with a Jacobian--vector product. This reduces estimator variance and requires no backward pass through the trunk, while retaining an Adam-style convergence guarantee. Our experiments reveal an important practical failure mode. On WikiText-103, naive forward-gradient training of the trunk performs worse than leaving a randomly initialized trunk frozen, likely because Adam updates every noisy, under-determined trunk coordinate too aggressively. Simply using a much smaller learning rate for the trunk reverses this result: a $16$M-parameter GPT-2-style model reaches validation perplexity $387$, compared with $668$ for the frozen-trunk control and $2{,}885$ for a matched pure forward-gradient baseline (backpropagation reaches $150$). Split-FG also produces the strongest backprop-free results on our tabular benchmarks and reaches $60.5\%$ on CIFAR-10 and $35.2\%$ on CIFAR-100 with a heavy-head design. It reduces peak memory by up to $35\%$ relative to matched backpropagation, although the performance gap widens as the forward-mode trunk grows.

Oscillatory Neural Networks (ONNs) present an attractive physics-based computing paradigm rooted in the dynamics of a network of typically fully coupled oscillators aiming to minimize an underlying energy function. In this paper, we propose an ONN-based solver for one well-known constrained combinatorial optimization problem, namely a Sudoku, by formulating the problem as a Graph Coloring problem. By modifying the already existing Graph Coloring solver to a computationally cheaper version and introducing an additional term ensuring the fulfillment of the Sudoku constraints, our solver is shown to significantly outperform the existing HNN- and ONN solvers in terms of accuracy. In particular, we are able to achieve nearly flawless accuracies on $4 \times 4$ as well as rather high accuracies on $9 \times 9$ Sudoku puzzles for different numbers of unknown digits.

Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.

Deep Neural Networks (DNNs) used in safety-critical applications are vulnerable to hardware and memory faults that corrupt network weights and degrade reliability. In this paper, we propose a Center of Gravity (CoG) guided weight correction method that restores faulty weights based on their spatial characteristics within each layer. The proposed approach detects and corrects weight faults using distance-aware correction rules, eliminating the need for retraining or architectural modification. The effectiveness of the proposed method in terms of the capability of tolerating hardware faults has been evaluated through performing fault injection at different Bit Error Rates (BERs). Experiments on safety-critical LSTM-based Networks, including StageNet for disease progression tracking and MTFNet for cardiac anomaly detection, demonstrate fault tolerance improvements of up to 230x and 6.41x, respectively, at a BER of 10^{-3}, with negligible accuracy loss. When extended to Convolutional Neural Networks (CNNs), the method achieves up to 49.55x and 20.79x improvements under comparable fault conditions on ResNet-18 and VGG-16, respectively. To the best of our knowledge, this is the first work to apply the CoG concept to neural network weight tensors for enhancing model reliability.

Physics-informed neural networks (PINNs) are unusually sensitive to interacting choices of architecture, activation, loss weighting, collocation, optimization, and constraint enforcement. Large language models (LLMs) can propose these choices, but independent recommendations do not accumulate experience from previously trained PINNs. We propose a closed-loop evolutionary algorithm that guides an LLM to generate complete, executable PINN configurations across generations, using measured training outcomes to determine subsequent search decisions. The algorithm maintains an evaluated population and lineage, applies parent-conditioned mutation and crossover, preserves elite and diverse solutions, rejects effective duplicates, and converts parent-relative successes and failures into the next-generation context supplied to the LLM. Every proposed configuration is executed directly under an exact optimizer-step budget. On a one-dimensional multiscale wave equation, two independent ten-generation runs trained 60 PINNs for 600,000 optimizer steps. In both runs, the best configuration appeared in the final generation, with best mean-squared error reduced by 2.97% and 95.38% relative to the initial population. The stronger run validated residual connections and increased depth on separate branches, combined them in a later generation, and then refined width and collocation density. It also revealed that low solution error can coexist with a high PDE residual. These results demonstrate the feasibility of evolutionary-algorithm-guided LLMs for PINN design on a controlled PDE while motivating broader, physics-aware evaluation.

We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particular, we employ a decoupled neural network architecture explicitly separating the subgrid corrections into two distinct components: a conservative Flux Potential network and an Eddy Viscosity network. We demonstrate that this reduced-order framework maintains high physical fidelity, accurately reproducing the energy spectrum, spatial and temporal correlation functions, and dynamical characteristics of the full-scale system. Furthermore, we show that our approach is robust and applicable to parameters outside the training regime.

Continuous-time spiking neural networks (SNNs) provide an event-driven framework for temporal computation, computational neuroscience, and neuromorphic hardware. However, training deep continuous-time SNNs is severely constrained by the memory required for exact spike-time computation, which evaluates and retains candidate firing times over intervals determined by presynaptic spike ordering. Here we introduce a memory-efficient training framework based on differentiable spike-time discretization (DSTD) for leaky integrate-and-fire neurons with general membrane and synaptic time constants. DSTD maps irregular presynaptic spikes onto differentiable weighted events at fixed time points, replacing the input-dependent candidate dimension with $M$ fixed time intervals while accurately approximating continuous-time membrane-potential dynamics. This reduces candidate-related activation memory from $O(N_{\mathrm{out}}N_{\mathrm{in}})$ to $O(N_{\mathrm{out}}M)$ in the case of time-to-first-spike (TTFS) coding, where $N_{\mathrm{in}}$ and $N_{\mathrm{out}}$ denote the numbers of presynaptic and postsynaptic neurons, respectively. We further introduce synfire-chain-inspired temporal regularization that organizes layer-wise firing windows, mitigates dead-neuron failures, and enables pipeline-like processing. In dense LIF layers, DSTD reduced peak memory consumption by up to approximately 100-fold and training time by up to approximately 20-fold compared with exact spike-time computation. Together, these methods allowed us to train 9-layer convolutional SNNs on CIFAR-10 and 20-layer convolutional SNNs on Fashion-MNIST on a single GPU.

Estimating spatially heterogeneous elastic properties from low-resolution displacement measurements is a severely ill-posed inverse elasticity problem because low resolution obscures spatial details needed to distinguish heterogeneous property variations, and small measurement perturbations or fitting errors are amplified through inverse estimation. Existing inverse methods often rely on high-fidelity observations and manually prespecified loss weights, limiting their adaptability and making them sensitive to noise and resolution degradation. We propose a Probabilistic Inverse Elasticity Physics-Informed Neural Network (PIE-PINN) framework for robust estimation of Young's modulus and Poisson's ratio from noisy, low-resolution displacement data. PIE-PINN models displacement observation, strain-discrepancy, and equilibrium residuals using Laplace distributions within a unified probabilistic model. To improve robustness, the framework combines a B-spline-guided displacement network with a hierarchical half-Cauchy model for displacement residual scales. The B-spline provides a smooth global representation of the displacement field, while the neural network correction captures local variations. The hierarchical scale model adaptively downweights severe displacement fitting errors, enabling more robust recovery of the latent mean displacement field. An alternating maximum-likelihood training strategy updates the mean through weighted residual minimization and updates the scales to adjust the loss weights. Systematic case studies across varying noise levels and observation resolutions demonstrate the robustness of PIE-PINN.

The interconnected nature of global financial systems makes them vulnerable to systemic risks, where the failure of a few institutions can trigger catastrophic cascading defaults. Traditional risk models often fail to capture the complex, non-linear dynamics of these networks. While Graph Neural Networks (GNNs) have shown promise in modeling relational data, they primarily learn correlative patterns and function as black boxes, offering little insight into the causal mechanisms of shock propagation. This limitation is critical for regulators who require explainable models to perform stress tests and devise effective interventions. We introduce CausalGraphX, a novel framework that integrates GNNs with counterfactual reasoning to provide explainable assessments of systemic risk. CausalGraphX employs a Graph Attention mechanism to learn representations of institutional vulnerability and uses an adversarial regularization technique to ensure these representations capture causal drivers rather than spurious correlations. Furthermore, we propose an optimization-based approach to generate counterfactual explanations, answering questions such as, "What minimum capital injection would have prevented Bank A's default under a specific stress scenario?" We validate CausalGraphX on large-scale synthetic financial networks. Our results demonstrate that CausalGraphX significantly outperforms traditional and deep learning baselines in predicting cascading defaults while providing sparse, plausible, and actionable counterfactual explanations.

We study exact verification of ReLU networks in an adversarial smoothed model. Every network weight and bias is independently perturbed by Gaussian noise, clipped to $[-2,2]$, and rounded to the exact dyadic grid determined by the input bit complexity. We show that, under the standard assumption $\mathrm{NP}\not\subseteq\mathrm{BPP}$, there is no sound and complete verifier whose expected running time is polynomial in network size, bit complexity, and inverse noise level for every base instance. The conclusion already holds at the fixed noise level $σ_\star=2^{-11}$ for one-hidden-layer networks over a unit box, with hidden fan-in at most three and base coefficients in $[-1,1]$. The proof combines an exact gap embedding with a quantitative robustness argument. For every E3SAT formula $Φ$ with $m$ clauses, a four-ReLU-per-clause construction satisfies $\max_{x\in[0,1]^n} g_Φ(x)=(m-\operatorname{unsat}(Φ))/3$, and coordinatewise threshold rounding never decreases the objective. A weighted parameter-sensitivity inequality and Gaussian concentration then show that a verification gap linear in $m$ survives the aggregate perturbation of all coefficients with probability at least $1-e^{-m/8}$. The proof includes clipping, exact dyadic rounding, output-layer perturbations, polynomial-bit sampling of the rounded Gaussian law, and the conversion from expected smoothed running time to a BPP algorithm. Computational checks test the exact identity and illustrate the different scaling of extensive and constant gaps; they are diagnostics rather than evidence for the complexity theorem. The result concerns worst-case base networks in the stated absolute-noise model, but it shows that parameter nondegeneracy alone does not yield a universal smoothed-polynomial guarantee for exact verification.

Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE). However, PINNs have been shown to perform poorly, sometimes even converging to trivial solutions, in challenging PDE domains, or when generalizing to unseen but related PDE domains. Previously proposed solutions detail hyperparameter tuning to reduce loss imbalance between data-driven and physics guided losses, curriculum learning based training strategies, or dynamic re-sampling of hard collocation points. These methods face certain pitfalls: hyperparameter tuning is expensive, designing a training curriculum is ambiguous in multi-parameter PDE settings, and dynamic resampling still fails in complex PDE settings. Complementary to this line of thinking, we believe the initial PINN network weights also play a crucial role in the emergence of catastrophic failures during training, yet the effect of PINN weight initialization has been surprisingly under-investigated. To this end, we propose a framework for Learned Initialization via Gated Layerwise Optimization (LIGO-PINN) to overcome PINN convergence failures. Through rigorous evaluation on 1D and 2D PDE domains, including a challenging 2D fluid dynamics setting, we demonstrate that our methodology outperforms state-of-the-art methods designed to alleviate PINN failures, achieving a 91.5% average performance improvement across six baselines and 81% over the strongest baseline. We also verify that LIGO-PINN generalizes to 3D unstructured domains. Finally, we analyze training dynamics across all three PDE domains to explain both LIGO-PINN's improvement and the convergence failure of traditional PINNs. Code: https://github.com/scailab/ligo-pinn Keywords: Machine Learning, Physics-Informed Neural Networks, Deep Learning, PDE Modeling

We investigate how each component of the Transformer feedforward block architecture design determines how much rank survives across depth at initialization. We reinterpret skip connections and normalization, long understood as controlling magnitude, as mechanisms for preserving gradient rank across depth, since the very matrix multiplications and nonlinear activations that make the network expressive also reduce the rank. We show that skip connections trade off rank collapse against ensemble-like behavior, controlled by the relative scales of the branch and the skip: skip connections route the gradient around the residual branch, where rank is lost, rather than along the long gradient paths that encourage the layers to compose. The placement of the normalization layer controls this same tradeoff by setting the branch-to-skip ratio across depth, unifying much of the normalization placement and depth scaling literature, in particular why rank collapses for Post-Norm but plateaus for Pre-Norm. Other aspects of the architecture, like the two-matrix structure that expands and contracts the width, use additional parameters to preserve the representation or branch Jacobian rank. The second matrix decorrelates a coherent mean spike that would grow across blocks with a single matrix and uncentered activation, preventing the residual representation from collapsing. The width expansion between the two matrices keeps the branch Jacobian full rank: applying the rank-reducing activation in this expanded space leaves enough directions to span the original, at a width that follows a Marchenko--Pastur law. The initialization rank of the input--output Jacobian predicts which networks train on CIFAR-10. Taken together, we recast architecture design for deep networks as navigating an intrinsic tradeoff among rank collapse, ensemble-like behavior, and parameter count.

Neural networks trained on modular arithmetic exhibit grokking, a delayed transition from memorisation to generalisation known to depend on model capacity: too little and the network memorises slowly or not at all, too much and it generalises almost immediately. What happens at the extreme of this spectrum, when the architecture's expressible function class collapses to a finite-dimensional algebraic variety? We study two-layer networks with a holomorphic monomial activation sigma(z)=z^k, trained on modular tasks encoded via roots of unity. Here the network output, regardless of hidden width, is confined to a (k+1)-dimensional subspace of characters of (Z_p)^2, an O(k/p^2) slice of the full function space. We give a complete algebraic characterisation of this subspace: a task is representable if and only if its discrete Fourier support lies on the diagonal u+v = k (mod p), which for linear-phase targets reduces to the arithmetic criterion m+n=k. This is not merely a constraint on eventual generalisation but on memorisation itself: because the outputs are algebraically confined, a non-representable target cannot be fit even on the training set, and we prove a positive lower bound on the training loss, independent of width. Across 585 runs the algebraic prediction matches the observed outcome with 99.8% accuracy, with no memorisation regime and no grokking; outcomes split cleanly into instant success and outright failure. This binary behaviour is the limiting case of the capacity-grokking relationship: when the expressible class shrinks to a fixed algebraic object, the question of when a network will grok dissolves into whether it can represent the target at all. A bottleneck ablation connects this extreme to standard networks, tracing a continuous path from representational failure, through memorisation without generalisation, to grokking with a shrinking gap as capacity grows.

We investigate conditional invertible neural networks (cINNs) as probabilistic inverse-dynamics models for multirotor control. For a planar X8 coaxial multicopter, we learn $p(u \mid s_t, c_t)$ from an incremental nonlinear dynamic inversion (INDI) teacher using rational-quadratic spline coupling and invertible linear mixing. Open-loop reproduction reaches $R^2 = 0.944$, mean CRPS 0.0915, and log-probability-error correlation $ρ= -0.60$. Over 15 closed-loop scenarios, position RMSE matches INDI (9.7 vs. 9.5 m), with 47 percent tracking acceptably; failures separate into attitude divergence under aggressive steps and phase lag under high-frequency references, isolating command bandwidth and data coverage as dominant failure mechanisms.

Interfacing with Biological Neural Networks (BNNs) requires encoding information into stimulation patterns that can be effectively processed and that enable the underlying system to adapt. Nevertheless, the role of stimulation encoding remains poorly understood. In this work, we compare multiple encoding strategies, including rate-based, phase-based, burst-based, and time-to-first-spike temporal encodings, in a closed-loop neural classification task using cultured BNNs. We encode visual inputs as spatiotemporal stimulation patterns delivered via a Multi-Electrode Array (MEA) and evaluate classification performance for each encoding scheme. We find that burst-based temporal encoding yields the highest observed performance, achieving up to 95.6 % accuracy in a binary classification task, compared to substantially lower performance from rate- and phase-based approaches. We further show that performance is highly sensitive to the spatial distribution of stimulation, with suboptimal electrode selection significantly degrading accuracy. These findings indicate that effective interfacing with biological neural systems requires the joint optimization of temporal and spatial encoding strategies, and highlight temporal encoding as a key design dimension for bio-digital computing.

The Hessian matrix is an important quantity of interest when it comes to studying the loss landscape and optimization dynamics in deep learning, as well as designing measures of generalization, second-order learning algorithms, etc. Prior works have focused on empirical results or pursued a theoretical treatment under overly simplified settings. In this work, we derive the eigenvalues of the Hessian of linear networks with arbitrary widths and depths, and datasets with an arbitrary number of samples, features, and labels. Importantly, for classification tasks with MSE loss, we identify that the sharpness of the solution is directly related to the maximum proportion of samples belonging to any class. We empirically validate our predictions and systematically analyze the effects of shedding the impractical assumptions one at a time, as well as incorporating nonlinearities. We observe that our predictions are considerably robust in most cases, allowing us to extend our conclusions to more practical learning setups.

Centroid neural network (CentNN) is an unsupervised competitive learning algorithm in which centroid splitting is triggered only after strict local stabilization, often leading to prolonged low-movement training phases before model expansion. This report proposes FastCentNN, an accelerated variant that addresses this inefficiency by introducing an early splitting strategy based on the total centroid movement per epoch, which serves as a training entropy proxy. As a result, FastCentNN reduces unnecessary reassignment epochs while preserving the original winner-loser learning dynamics. FastCentNN supports both absolute and stage-relative movement thresholds, allowing the splitting criterion to remain either fixed or adaptive throughout training. Experiments on some benchmark datasets show that FastCentNN consistently achieves clustering quality comparable to CentNN while reducing runtime by up to 16% on synthetic 2D datasets and about 5% on high-dimensional datasets. FastCentNN therefore provides a practical and efficient drop-in replacement for CentNN, retaining its online adaptive learning behavior while offering a simple and interpretable speed-stability trade-off through configurable splitting thresholds.

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Predictive Coding (PC) offers a biologically motivated alternative to backpropagation via local weight updates, yet routing error between layers still relies on an autograd Jacobian-transpose ($J^\top$) product - the last non-local operation in PC. We show that this dependency is largely avoidable. For any layer $f(x)=\mathrm{Act}(\mathrm{Norm}(L(x)))$ with frozen normalization statistics, the exact $J^\top$ factors into three locally available terms, $J^\top v = L^\top(s \odot σ'(z) \odot v)$, where $σ'$ is the activation derivative, $z$ is the pre-activation, and $s=γ/σ_{\mathrm{run}}$ is the normalization gain. Prior weight-feedback methods omitted both corrections; restoring them closes the transport gap for this layer class. Locality here holds up to three assumptions, which we state upfront: weight symmetry ($L^\top$ mirrors the forward operator, as assumed by all PC), a soft spectral-norm control that is not synapse-local, and a nearest-neighbour approximation for MaxPool. Substituting the identity into PC yields WF-Act-PC, which removes the autograd backward pass from error transport. On CIFAR-10/100 (50 epochs, 5 seeds), WF-Act-PC is the only PC method whose accuracy improves with depth, surpassing iPC - the strongest classical PC baseline - by 2.7-22.3 pp on CIFAR-10. With both methods tuned per architecture, it matches or exceeds a comparably-tuned backpropagation baseline on the deeper CIFAR-10 architectures (VGG-9: 93.57% vs. 92.43%; ResNet-18: 92.76% vs. 91.54%) and on the harder Tiny-ImageNet benchmark, while trailing tuned BP on the deeper CIFAR-100 VGG cells. Our WF-Act-PC implementation is publicly available at https://github.com/jlshen025/pcax

Muon has recently emerged as a strong optimizer for large-scale deep learning, where it reshapes gradient updates through approximate orthogonalization and has been reported to outperform Adam and AdamW in large language model training. Its empirical success has motivated a growing body of theoretical work that interprets Muon as steepest descent under the spectral norm. Yet it remains unclear which of Muon's advantages stem from its update rule itself and which are artifacts of the scale, architecture, and data of modern deep networks. In this work, we isolate the optimizer from these confounding factors by studying Muon on a simple, well-understood, and spectrally structured problem: low-rank matrix factorization. Through a controlled comparison against carefully tuned adaptive baselines, we find that Muon does not consistently outperform AdamW in this setting and that several previously reported advantages are sensitive to hyperparameter choices. Our results provide a more nuanced picture of when spectrum-aware orthogonalization is beneficial and argue for evaluating modern optimizers on controlled problems in addition to end-to-end benchmarks.

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from $69.2\%$ to $96.2\%$ across five-seed synthetic-corruption experiments and $94.4\%\pm1.9\%$ on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from $28.88\pm1.95$ under clean training to $97.09\pm1.78$ at $60\%$ corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.

We study the gradient flow dynamics of diagonal linear networks for regression tasks under infinitesimal initialization. Extending Theorem 1 from Pesme & Flammarion (2023), we generalize the analysis to both deep diagonal linear networks and a broader class of two-layer diagonal linear networks (as defined in Definition 4.1). Specifically, we demonstrate that the training trajectories of these models can be equivalently characterized by the proposed Algorithm 1. We further prove that this algorithm converges to the solution of a modified $\mathcal{l}_1$ norm minimization problem. As a result, we establish that the implicit bias of both network architectures corresponds to a modified $\mathcal{l}_1$ norm in the regime of infinitesimal initialization. Additionally, we provide insights into the underlying mechanisms governing these dynamics by identifying the Structural Invariant Manifold (SIM) (Zhao et al., 2026) as the key geometric structure that shapes the learning process.