Neural Network

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.

This paper proposes a Gaussian-Based Shift-Variant filtered backprojection (FBP) neural network, which is designed for the efficient reconstruction of non-circular trajectory cone beam computed tomography. The traditional differentiable shift-variant FBP model consists of a filtering component and a backprojection process. The filtering component includes operations such as weightings, differentiations, a 2D Radon transform, and a 2D backprojection. The proposed methods build on this framework by introducing a trainable 2D Gaussian model to represent the trajectory-related part in the filtering process, achieving a substantial reduction in the number of trainable parameters. Experimental results demonstrate that the proposed model reduces the parameter count by 99%, while only sacrificing a slight amount of reconstruction quality. Furthermore, the training time for each trajectory is reduced to one-fourth of the original, significantly accelerating convergence. These enhancements demonstrate a considerable augmentation in the model's practicality and effectiveness, making it a valuable asset for real-world applications.

We introduce a straightforward yet effective method to empirically study memorization in deep neural networks for classification tasks. Our approach augments each training sample with auxiliary random labels, which are then predicted by a random label prediction head (RLP-head). RLP-heads can be attached at arbitrary depths of a network, predicting random labels from the corresponding intermediate representation and thereby enabling analysis of how memorization capacity evolves across layers. By interpreting the RLP-head performance as an empirical estimate of Rademacher complexity, we obtain a direct measure of both sample-level memorization and model capacity. We leverage this random label accuracy metric to analyze generalization and overfitting in different models and datasets. Building on this approach, we further propose a novel regularization technique based on the output of the RLP-head, which demonstrably reduces memorization. Interestingly, our experiments reveal that reducing memorization can either improve or impair generalization, depending on the dataset and training setup. These findings challenge the traditional assumption that overfitting is equivalent to memorization and suggest new hypotheses to reconcile these seemingly contradictory results. The source code is available at https://github.com/MarlonBecker/RandomLabelHeads

In this paper, we propose TECO, a multi-dimensional pruning framework to collaboratively prune the three dimensions (depth, width, and resolution) of convolutional neural networks (CNNs) for better execution efficiency on embedded hardware. In TECO, we first introduce a two-stage importance evaluation framework, which efficiently and comprehensively evaluates each pruning unit according to both the local importance inside each dimension and the global importance across different dimensions. Based on the evaluation framework, we present a heuristic pruning algorithm to progressively prune the three dimensions of CNNs towards the optimal trade-off between accuracy and efficiency. Experiments on multiple benchmarks validate the advantages of TECO over existing state-of-the-art (SOTA) approaches. The code and pre-trained models are available at https://github.com/ntuliuteam/Teco.

Neural networks increasingly guide decisions in high-stakes domains such as medical diagnosis, credit approval, and energy bidding. Audit in these settings requires case-level evidence: which training cases support an action and what outcomes they carried. Case-based decision theory (CBDT) formalizes this reasoning by aggregating outcome support from remembered cases. We show that an OLS action readout fitted on a fixed neural representation admits an exact case-based decomposition. Each action score is a weighted sum of training-case returns, with coefficients determined by empirical Gram geometry. We identify a sufficient regime for CBDT similarity semantics; outside it, the coefficients should generally be treated as signed Gram-geometric influence. The decomposition yields audit signals that trace scores to training cases, measure action coherence, and identify weak support. Across synthetic CBDT, PJM, Adult Income, and Default Credit tasks, the method recovers case-level preference structure and achieves the highest mean Top-30 consistency among compared attribution baselines, while remaining competitive on support reconstruction. The audit requires only fitting an OLS top-layer probe, without retraining the representation or accessing the original optimization trajectory; probe fidelity is measured by score reconstruction.

Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point {equation}; they include as special cases neural network controllers. We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map, thereby making well-posedness and Lyapunov/IQC analysis mathematically easy to handle. For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron--Frobenius and norm conditions for well posedness, LMI/IQC certificates for exponential stability, and LMIs for discounted infinite-horizon quadratic performance. We then formulate synthesis as a certification-compatible heuristic search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas provide gradients, and the resulting controller is accepted only after independent post-training LMIs or regional admissibility checks are feasible. Finally, we establish constrained-control separation results: for a specific scalar unstable plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. Additional results cover quadratic state-input costs, comparison with linear static output feedback, and computable upper/lower-bound certificates. Numerical examples illustrate the mechanism and the resulting certified performance.

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs); however, their synchronous optimization treats residuals of different regions and constraints equally, which is inconsistent with the progressive "from source to response" physical information propagation path, degrading training stability and accuracy. Existing causal training methods focus mainly on the temporal dimension, lacking a unified characterization of spatial and boundary dimensions. To address this, we define a unified class of training priorities according to the physical information propagation path: premise regions should be learned before dependent regions; temporal, spatial, and boundary priorities are instances of this principle. Using neural tangent kernel (NTK) dynamics, we theoretically analyze why standard PINNs do not obey this priority: their residual convergence order is governed by the NTK spectrum and is independent of the propagation path. Accordingly, we propose a unified multi-dimensional priority-constraint framework that partitions the domain along the propagation path and constructs negative-exponential residual weights, converting the physical propagation order into a training priority. For cases with coexisting priorities, we introduce a directional compatibility coefficient to clarify that "orthogonal directions can be coupled multiplicatively in synergy, whereas coaxial opposite directions cannot." Benchmark cases show that this method consistently improves the convergence behavior and prediction accuracy of PINNs on problems with clear propagation paths or constraint-dominated structures, without modifying the network architecture and with controllable additional computational cost.

We study the population gradient flow of an infinitely wide two-layer neural network learning a misspecified single-index model in high dimension. The two layers are optimized jointly, with a perturbative parameter tuning the relative training speed between the first and second layer. This setting was considered by Berthier, Montanari and Zhou in \cite{berthier2024learning}, who conjectured a hierarchical learning scenario with explicit timescales as the second layer is trained faster than the first. In this paper, we prove that the constant and linear components of the hidden link function are indeed recovered within the predicted timescales, at sharp explicit thresholds. We then analyze the onset of learning of the quadratic component and show that the components learned at earlier stages continue to influence the dynamics in an essential way. Our proof is based on quantitative approximation results for singularly perturbed flows evolving near a manifold defined by integral constraints. At a phenomenological level, we also show that the empirical measure of the weights displays singular behaviour when reaching the quadratic component of the hidden link, with a small fraction of neurons growing significantly while the remaining ones rearrange to preserve the components already learned.

In contrast to most studies on neural network approximation theory that characterize results through a single parameter, such as the total number of network parameters, \cite{shen2020deep} pioneered the characterization of approximation rates as a joint function of the width parameter $N$ and the depth parameter $L$, thereby granting greater architectural flexibility. Existing works using the $(N,L)$-characterization focus on function classes with finite smoothness $s$, establishing a typical approximation rate of $\mathcal{O}\left(N^{-2s/d}L^{-2s/d}\right)$ with $d$ denoting the input dimension, which indicates that network depth and width play symmetric roles for these classes. In contrast, this paper establishes upper bounds for the approximation of analytic functions, which possess infinite smoothness, via ReLU networks under the $(N,L)$-characterization. Specifically, we derive approximation rates of $\mathcal{O}\left(N^{-C L^τ}\right)$, where $C>0$ is some constant and $τ>0$ is a parameter influenced by the relation between $L$ and $N$. In particular, $τ=1$ if $N$ scales roughly as $L^d$. Our findings reveal that depth plays a more critical role than width in the context of analytic function approximation. The main technical difficulty of obtaining such upper bounds lies in the trade-off between the smoothness parameters and the approximation accuracy. To overcome this difficulty, we employ refined constructions of several ReLU networks to approximate power functions, multivariate multiplication, and polynomials, which may be of independent interest.

Dimensionality reduction has proven powerful for identifying neural manifolds, which are low-dimensional structures underlying high-dimensional neural activity. These low-dimensional representations have improved the interpretability of population-level coding. Yet whether such low-dimensional representations are biologically relevant and confer functional advantages in learning systems, or merely reflect neuron-level activity, remains contested in neuroscience. We show that an explicit information bottleneck forcing a recurrent neural network to learn a low-dimensional representation is necessary for rotational and out-of-distribution generalisation in a time-series prediction task. Using information-theoretic measures of causal emergence, we characterise the dynamics of this representation across the memorisation-to-generalisation transition, finding a non-monotonic trajectory which shows an initial decrease, a minimum, and a subsequent rise to a maximum, even as prediction loss falls monotonically. This trajectory scales with task complexity, and the magnitude of emergent structure reliably predicts generalisation performance. Analysis of CA1 hippocampal activity in mice learning an alternating maze task reveals analogous non-monotonic emergence dynamics that track behavioural performance. Together, these findings indicate that the ability of neural networks to learn compact, distributed and emergent representations confers a functional advantage for generalisation, supporting a causal role for learned representations in cognition.

The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.

We show how an adversarial model trainer can plant backdoors in a large class of deep, feedforward neural networks. These backdoors are statistically undetectable in the white-box setting, meaning that the backdoored and honestly trained models are close in total variation distance, even given the full descriptions of the models (e.g., all of the weights). The backdoor provides access to invariance-based adversarial examples for every input, mapping distant inputs to unusually close outputs. However, without the backdoor, it is provably impossible (under standard cryptographic assumptions) to generate any such adversarial examples in polynomial time. Our theoretical and preliminary empirical findings demonstrate a fundamental power asymmetry between model trainers and model users.

The current state of the art in AI/ML rests on deep neural architectures, which, in general, suffer from a lack of interpretability. Interpretability is crucial to gleaning insights while analyzing experimental data, where quantitative predictions may not be adequate for a scientist. We present a three layer neural architecture, SAMPAT (Smooth Approximation via Multivariate Polynomials and Analytic Transformations), that can provably learn a continuous, everywhere differentiable function, that can approximate any smooth function arbitrarily closely. SAMPAT's approximant can be expressed as a closed and compact algebraic, analytic expression, providing complete interpretability. Experiments on synthetic and benchmark datasets indicate that SAMPAT yields competitive performance with simpler representations. For many tasks, a two layer SAMPAT suffices. By imposing restrictions on the connectivity between neurons, SAMPAT may be used to provide a range of approximants, including regular and trigonometric polynomials, rational expressions, Gaussians, mixtures of Gaussians, as well as arbitrary combinations of the same; without restrictions, it learns a suitable structure. SAMPAT may be used to factorize polynomials and model nonlinear systems. With the addition of skip connections, a 4 to 6 layer SAMPAT is adequate to represent a substantive range of methods widely used in AI/ML, allowing the choice of a model's family, not just its parameters, to also be optimized as part of the learning process.

Over the last few years, there has been an increased interest in making machine learning models more interpretable. Although a great deal of effort goes into developing techniques for interpreting the interactions learned by a given model, fewer studies focus on assessing the quality of such explanations. Even fewer focus on how to adjust the model to produce explanations faithful to prior knowledge, a process known as explanation-guided learning. Furthermore, most approaches in this area focus on classification problems and usually assume prior knowledge about which input features or regions are most important. In this work, we introduce a new approach to steering neural networks based on partial dependence, such that their average response to certain features aligns with specific functional domain knowledge about the problem. We empirically demonstrate on a range of regression problems, including dynamical systems forecasting, that models whose training has been controlled using our method perform better than unconstrained models and are more data-efficient. Moreover, we highlight that interpretations obtained from the former actually align with the user-provided knowledge, whereas those obtained from the latter do not.

A series of results from the NeuroAI over the past fifteen years have raised core questions both about how to compare Deep Neural Network (DNN) models to the brain, and about how much convergent evolution to expect between artificial networks and real brain networks. Here, we show that for any two minimal DNN solutions to a sufficiently hard task: (i) "weak" alignment of network representations based on affine mappings guarantees "strong" alignment of privileged axes, and (ii) alignment "zippers" up the network hierarchy, causing the emergence of privileged axes from end-to-end task optimization. These results formalize the notion of contravariance from Cao and Yamins [2024], and illustrate important consequences for the theory of NeuroAI: with sufficiently strong tasks, choice of metric for inter-network comparison is not all that sensitive, and that convergent evolution is probably inevitable.

Backpropagation (BP) dominates deep learning training, but its reliance on gradients brings inherent troubles -- vanishing and exploding gradients. The pursuit of gradient-free methods has long been a goal in the field of artificial intelligence. This paper shows that indeed the simplest Monte Carlo algorithm implemented on a single GPU -- randomly mutate a parameter, keep it if the loss decreases, otherwise retry -- can practically train deep networks. This gradient-free method does not even need common techniques such as batch normalization or residual connections to directly train sufficiently deep networks. More remarkably, its flexibility extends to several nontrivial scenarios: it enables pure pruning training, supports discrete weights, accommodates unconventional transfer functions such as Gaussian, and reveals the substantial redundancy of deep networks. We have demonstrated its feasibility on deep networks with more than 20 layers, single-hidden-layer wide networks with up to 16,384 hidden neurons, and even a simple Transformer architecture trained on both image classification (MNIST) and character-level language modeling (Tiny Shakespeare). This simple gradient-free method may offer a complementary perspective for understanding the self-organization and learning mechanisms of neural networks, and also provides an alternative route for building physically inspired deep learning systems.

We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.

For seemless control of advanced hand prostheses and augmented reality, accurate and immediate hand gestures recognition is essential. Surface electromyography (sEMG) signals obtained from the forearm are commonly employed for this purpose. In this paper, we present a novel approach for sEMG representation that utilizes graph networks which contain information about muscle activation patterns in the forearm. Based on these graph networks, we have developed a machine learning algorithm capable of real-time hand gesture recognition using a graph neural network. The algorithm's performance was evaluated using sEMG signals acquired from myoband, which has 8 electrodes placed around the forearm, involving 8 healthy subjects. The proposed method demonstrated an average classification accuracy of 99%, surpassing the performance of state-of-the-art techniques. The average time for both graph construction and prediction stood at 48ms utilizing a M1 pro CPU, rendering the approach well-suited for real-time applications.

The Hessian of the training loss governs the local geometry of the loss landscape, yet despite existing explanations for its largest eigenvalues, the origin of the vast multitude of vanishingly small eigenvalues remains elusive. We argue that the bulk consists of the weakly lifted pseudo-Goldstone modes of the continuous symmetries of the network parametrization. In deep linear networks these symmetries are exact: they generate flat directions and hence exact zero modes, whose eigenvectors we construct explicitly. Introducing a ReLU nonlinearity as a perturbation, we show that it breaks these symmetries weakly and explicitly. Resolving the spectrum at the level of eigenvectors, we find that the high-curvature directions are orthogonal to the symmetry subspace, while the bulk lies almost entirely within it. We demonstrate the mechanism in a two-layer ReLU student--teacher model and in a network trained on CIFAR-10. A convolutional example demonstrates that the same diagnostic extends beyond fully connected layers. Together, these results link the Hessian bulk to weakly broken symmetries and clarify the origin of near-zero modes.

The quadratic cost of causal self-attention severely bottlenecks long-context transformer inference. While numerous post hoc linearization pipelines exist, it is difficult to identify which components preserve model quality. This work isolates the effect of state update design in a strict frozen-backbone regime. We show that softmax relies on key-dependent, rank-1 orthogonal projections, elucidating why delta-style networks outperform purely gated accumulation. We identify a potential source of approximation errors and introduce structural interventions, specifically sink tokens, short convolutions, and fixed-budget cache routing, which reduces the remaining gap. We scale this linearization approach across LLaMA and Qwen models up to 32B parameters, outperforming prior post hoc baselines on MMLU and matching the long-context retrieval of complex adaptive-caching frameworks.

Bubeck, Li and Nagaraj conjectured that, for generic data, any two-layer neural network with $m$ neurons that fits $n$ noisy labels must have Lipschitz constant at least of order $\sqrt{n/m}$, with no restriction on the size of the weights. Bubeck and Sellke proved a universal version of this law for Lipschitz-parameterized classes, but under a polynomial bound on the parameters; at depth three that boundedness hypothesis is genuinely necessary. The two-layer unbounded-weight case requires a different argument. We prove the conjectured law, up to one logarithmic factor, for every continuous piecewise-linear activation, in particular for ReLU networks. For data drawn uniformly from $\mathbb{S}^{d-1}$, $d\ge3$, or from $N(0,I_d/d)$, labels in $[-1,1]$ with noise level $σ^2>0$, and any width-$m$ two-layer network with arbitrary real weights, biases and affine skip connection, fitting the data $\varepsilon$ below the noise floor forces $\mathrm{Lip}(f)\ge c\,\varepsilon\sqrt{n/(\bar m\log(C\bar m nd/\varepsilon))}$, $\bar m=(K-1)m+1$, with high probability. A realized-kink-count version holds on the same event: every realized two-layer piecewise-linear function with $k(f)\le n$ distinct kink hyperplanes obeys the bound with $\bar m$ replaced by $k(f)+1$, irrespective of how many redundant hidden units parameterize it. The proof replaces parameter-space covering, impossible for unbounded weights, by a function-space covering. The central deterministic ingredient is a rigidity lemma: on $B_2$, and on $\mathbb{S}^{d-1}$ for $d\ge3$, the coefficient of each canonical kink is controlled by the Lipschitz constant of the realized function, because kinks on distinct hyperplanes cannot cancel at generic points. Rigidity genuinely fails at $d=2$, and an explicit two-layer ReLU interpolant with $O(1)$ Lipschitz constant at width $2n$ matches the law at the overparameterized endpoint.

This work presents a novel approach for adapting neural network architecture along the depth based on a posteriori error estimation. By formulating neural network training as a continuous-time optimal control problem, we derive rigorous error estimates that quantify how approximation error distributes across network layers. This error decomposition enables a principled depth adaptation strategy: new layers are inserted at locations of maximum estimated error, allowing the network to efficiently capture complex, nonlinear variations in the underlying problem. Our framework introduces a novel network architecture that treats weights and biases as piecewise linear functions varying across layers, with the error estimator bounding the discrepancy between this discrete representation and the true continuous optimal control solution. The approach leverages dual weighted residual methodology from finite element analysis to derive computable upper bounds on the functional error. A key theoretical contribution is the derivation of explicit error bounds that decompose the total approximation error into interval-wise contributions, providing a rigorous basis for targeted architecture refinement. We demonstrate the effectiveness of our method on scientific datasets, including learning the observable-to-parameter map for the Navier-Stokes equation. Numerical results reveal that our approach consistently outperforms existing architecture adaptation methods in terms of generalization performance.

Deformable shape representations have proven to be robust complements to texture features in cardiac image classification, offering geometric priors that are invariant to imaging artifacts and intensity variations. However, existing deep networks perform simple concatenation to combine these distinct feature representations, which neither fully exploits their complementary nature nor learns cross-modal feature dependencies. Furthermore, this results in uniform attention across all timepoints; hence ignoring the varying diagnostic importance across the cardiac phases. In this paper, we propose a novel cardiac video classification model that, for the first time, learns temporal features in an integrated space of deformable shape and image texture representations. In particular, we design a bi-directional cross-attention in the latent space to fuse latent deformable shape and image features, allowing each modality to adaptively weight the other based on spatio-temporal correspondence. In contrast to current methods that apply uniform weighting across all the cardiac phases, our approach learns to dynamically adjust the contributions of shape and texture representations, derived from images, over time. We demonstrate state-of-the-art classification performance on a cine cardiac magnetic resonance (CMR) video dataset, achieving improved interpretability from attention mechanisms that identify diagnostically critical cardiac phases and modality contributions.

Navigation for social organisms rarely is a fully independent activity. Group structure and dynamics, as well as embodied interactions, critically influence useful behavior. Individual neural network controlled agents are trained to navigate in different social contexts, where social dependence and behavioral strategy learned is determined by relative task performance and spatial effect. Increasing high quality social information drives phase transitions from individual to following navigational strategy, and to collision avoidance in response to a crowded foraging patch. Predictable, nonstationary environmental dynamics drive behavioral hybridization between individual and social navigation, far and near the patch. Our findings challenge the approach of only inspecting individual behavior for social organisms and highlight the importance of taking a bottom-up approach in understanding how organisms behave.

Many biological processes are governed by complex dynamical mechanisms that remain incompletely understood despite increasing volumes of experimental data. Biologically-informed neural networks (BINNs) seek to address this challenge by embedding mechanistic differential equations into neural network training, enabling interpretable constitutive operators to be recovered directly from sparse and noisy observations. However, reliable operator recovery depends sensitively on network architecture, optimisation strategy, and data informativeness. Here, we present a systematic empirical study of how these factors influence mechanistic inference using BINNs applied to canonical one-dimensional advection-diffusion-reaction partial differential equation models. Across a suite of benchmark problems, we investigate how network expressivity, learning rate, loss weighting, and batch size influence optimisation behaviour and operator recovery. We show that successful mechanistic inference depends on balancing competing objectives rather than maximising any single aspect of the model or optimisation. Moderately expressive architectures outperform overly complex networks, intermediate learning rates improve optimisation stability, balanced data and PDE losses are essential for accurate operator recovery, and intermediate batch sizes provide the best compromise between computational efficiency and reproducibility. We further identify practical diagnostics for recognising common failure modes, including over-fitting, unstable optimisation, and poor mechanistic recovery when the ground truth is unavailable. Together, these findings provide evidence-based guidelines for deploying BINNs as credible tools for biological model discovery.

This article offers a comprehensive overview of mechanistic interpretability, an emerging field that seeks to reverse-engineer the internal algorithms of modern neural networks. While traditional explainable AI methods often stop at surface-level input-output correlations, this approach directly addresses the opaque "black box" nature of machine learning models, which is essential for ensuring safety and auditability in high-stakes deployments. The paper provides a detailed examination of Transformer circuit analysis, exploring how internal components like the residual stream, attention mechanisms, and induction heads drive complex tasks and in-context learning. It subsequently tackles the core challenge of superposition and polysemanticity, demonstrating how tools like Sparse Autoencoders (SAEs) and transcoders can decompose tangled network activations into distinct, human-interpretable features. Furthermore, the paper explores methods for actively controlling and modifying model behavior through steering vectors and causal interventions. Finally, it connects these mechanistic insights with neurosymbolic AI frameworks designed to translate neural representations into explicit, executable logical rules.

While accuracy, robustness, and calibration are all essential for reliable neural networks, they are often studied separately; developing models that satisfy all three simultaneously remains a central challenge. Lipschitz-constrained models guarantee robustness by design, yet the manual selection of the Lipschitz constraint L governs the resulting accuracy-robustness trade-off, and their calibration properties remain largely underexplored. In this work, we highlight a theoretical and empirical link between the enforced Lipschitz constraint and Temperature Scaling, a state-of-the-art calibration method. Specifically, we find that for a given training scheme, there exists a non-trivial value L* that yields an out-of-the-box calibrated network, and that calibration acts as a principled criterion to select a well-defined operating point on the accuracy-robustness Pareto front. Leveraging these insights, we introduce Lipschitz Scaling Training (LiST), a novel training paradigm that iteratively adjusts the global Lipschitz constant to reach this operating point. Through a margin parameter in the training loss, LiST further enables the construction of a fully calibrated Pareto front, allowing users to navigate the accuracy-robustness trade-off while remaining calibrated throughout. At convergence, LiST also enables the reintegration of calibration data into training, improving sample efficiency without sacrificing calibration. We validate LiST on CIFAR-10/100 and Tiny-ImageNet, demonstrating competitive accuracy and robustness against constrained and unconstrained baselines, while remaining calibrated out of the box. Code is available at GitHub.

The architecture of deep feedforward neural networks is ubiquitous in deep learning, either as a whole system or as a subnetwork of other architectures, and thus its mechanism is a key ingredient of the black box of neural networks. On the basis of the simplest two-layer ReLU network, this paper systematically studies the mechanism of deep feedforward ReLU networks with multiple hidden layers and successfully explains the training solution obtained by the back-propagation algorithm. The concept of a path, especially in terms of the relationships between paths, plays a central role in uncovering the mystery of the black box. It is shown that a unit of a deep ReLU network can form a piecewise linear manifold to divide the input space, instead of a hyperplane of the two-layer case. How to efficiently use the hidden-layer units to produce both linear functions and partitions of the input space is also a central problem. The principles of a two-layer ReLU network can be generalized to the deeper case to a large extent, such as multiple strict partial orders and continuity restriction. The combination of the basic and simple principles proposed can yield complicated instantiations including the training solutions, and in this sense the black box of deep feedforward ReLU networks is revealed.

Deep learning applications have been widely adopted on edge devices, to mitigate the privacy and latency issues of accessing cloud servers. Deciding the number of neurons during the design of a deep neural network to maximize performance is not intuitive. Particularly, many application scenarios are real-time and have a strict latency constraint, while conventional neural network optimization methods do not directly change the temporal cost of model inference for latency-critical edge systems. In this work, we propose a latency-oriented neural network learning method to optimize models for high accuracy while fulfilling the latency constraint. For efficiency, we also introduce a universal hardware-customized latency predictor to optimize this procedure to learn a model that satisfies the latency constraint by only a one-shot training process. The experiment results reveal that, compared to state-of-the-art methods, our approach can well-fit the 'hard' latency constraint and achieve high accuracy. Under the same training settings as the original model and satisfying a 34 ms latency constraint on the ImageNet-100 dataset, we reduce GoogLeNet's latency from 40.32 ms to 34 ms with a 0.14% accuracy reduction on the NVIDIA Jetson Nano. When coupled with quantization, our method can be further improved to only 0.04% drop for GoogLeNet. On the NVIDIA Jetson TX2, we compress VGG-19 from 119.98 ms to 34 ms and even improve its accuracy by 0.5%, and we scale GoogLeNet up from 20.27 ms to 34 ms and achieve higher accuracy by 0.78%. We also open source this framework at https://github.com/ntuliuteam/ZeroBN

Existing NAS benchmarks (e.g., NAS-Bench, NATS-Bench) cover only narrow, task-specific regions of the architectural design space and lack cross-domain or deployment-aware evaluation. LEMUR 2 introduces a large-scale, extensible framework unifying generative, evaluative, and deployment pipelines to unlock neural-network diversity. It comprises over 14,000 distinct architectures and more than 750,000 structured training records documenting model performance, hyperparameters, and task outcomes. These models were produced through AST-based code mutation, genetic and reinforcement-learning evolution, generation of fractal architectures, and synthesis guided by a Large Language Model (LLM). This includes deep models generated with the retrieval-augmented system NN-RAG, which derived and used architectural motifs from over 900 PyTorch modules extracted from public repositories. LEMUR 2 further employs NN-VR and NN-Lite pipelines for automated deployment and latency benchmarking on heterogeneous mobile and Unity-based VR platforms, providing real-device performance metadata. It spans multimodal tasks, image captioning, text-to-image synthesis, and language modeling, supporting cross-domain analysis of architectural transferability. By linking diverse architectures, tasks, and deployment data, LEMUR 2 provides the data foundation for LLM fine-tuning and coupling diverse architectural origins with large-scale, cross-platform empirical validation. This dataset defines a new basis for reproducible and data-driven AI design, advancing the emerging paradigm of LLM-driven AutoML and architectural generalization across modalities and hardware.

In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error. We resolve this issue by introducing the Chinese Remainder Theorem as a constructive encoding mechanism. For Lipschitz continuous functions on $[0,1]^D$, we construct a width-$\max\{D,4\}$, depth-$5$ network with explicit parameter-error trade-offs. For Hölder-smooth functions in $C^{r,γ}_A\left([0,1]^D\right)$, our fixed network of width $\max\{2D,\ D+5N+1\}$ and depth $r + 9$ achieves the parameter magnitude $\mathcal{P}$ bounded by $\log_2 \mathcal{P}=\mathcal{O}\bigl(\varepsilon^{-2D/(r+γ)}\log(1/\varepsilon)\bigr)$. This is the dual result compared to those in the parameter-bounded and architecture-unbounded paradigm.

Event-based cameras are gaining popularity as the sensor of choice for mobile robotics, due to their high performance in dynamic environments. However, these applications require efficient real-time data processing with low latency and power consumption. One strategy to meet these stringent requirements is hardware acceleration of efficient algorithms that preserve the temporal sparsity of event data. In this work, we propose an optimization strategy for Graph Convolutional Neural Networks models aimed at adapting their architecture to the limited resources of embedded heterogeneous FPGA platforms. Our method incorporates hardware-aware pruning and quantization, taking into account the trade-off between on-chip memory savings and inference accuracy. Strategic exploration of the design space with Fine Grid Search and Greedy layer-wise Iterative Deepening Search methods enables flexible adaptation of the model architecture to the target platform. Our approach was evaluated across various network configurations and multiple datasets, resulting in BRAM memory reductions of 28.8% for CIFAR-10 (with a 1.65% decrease in accuracy), 31.4% for MNIST-DVS (accuracy drop of 3.55%), and 26.5% for N-Caltech101 (with a 5.18% accuracy reduction).

Energy-efficient neuromorphic computing at the edge requires simulation tools that can capture the non-ideal behavior of mixed-signal spiking neural network (SNN) hardware while supporting system-level design exploration. This work presents an open-source hardware-aware simulation framework for mixed-signal SNNs that enables comparative analysis across neuron, synapse and architecture choices. The framework supports multiple neuron models, including Leaky Integrate-and-Fire (LIF), Hodgkin-Huxley (HH) and Axon-Hillock (AH), together with non-volatile analog synapses based on floating-gate transistors and ReRAM devices. By incorporating device-level nonlinearities directly into PyTorch-based training and inference, the tool enables optimization of physical synaptic parameters rather than idealized abstract weights. The framework is evaluated on standard neuromorphic benchmarks, including N-MNIST, DVS Gesture and Spiking Heidelberg Digits (SHD). For each model dataset configuration, it reports classification accuracy together with hardware-oriented metrics such as silicon area, power consumption and quantization sensitivity. These capabilities enable cross-layer design space exploration and help identify neuron-synapse configurations that best satisfy application-specific constraints on accuracy, energy efficiency, area and hardware fidelity.

A persistent empirical observation is that trained neural networks outperform their neural tangent kernel (NTK) limit on tasks with compositional structure, yet a quantitative account of $\textbf{when}$ and $\textbf{by how much}$ has been lacking. Working on the unit circle, we give such an account through a dichotomy between two complexity measures of the target: its $\textbf{Fourier complexity}$, which controls NTK kernel regression, and its $\textbf{architectural complexity}$, which controls learning over depth-$L$, width-$w$ ReLU networks with the variation norm of the weights bounded by $R$. We first characterize the minimax rate of the architecture class $\mathcal{C}_{L,w,R}$, pinning it down up to a single factor of $L$: between $Ω(Lw^2R^2/n)$ and $\tilde{O}(L^2w^2R^2/n)$. We then show the NTK estimator sits $\textbf{exponentially}$ above this floor whenever the two complexities decouple: for the depth-$L$ iterated sawtooth, NTK regression needs $Ω(4^L)$ samples while the minimax floor is polynomial in $L$. Numerical experiments confirm the theoretical claims: on bandlimited smooth targets, the NTK is competitive or better, while on the hypercube sparse-parity model, a standard two-layer network beats the NTK by four to six orders of magnitude in test error. The gap is thus a function-space property, a mismatch between the kernel's smoothness bias and the target's compositional structure, rather than a generic kernel-versus-network phenomenon.

Enhancing videos under extreme low-light conditions remains challenging due to the difficulty of balancing restoration quality and computational efficiency in resource-constrained settings. This paper introduces EeveeDark, a low-light video enhancement framework that combines the spatial richness of sensor-level RAW data with the temporal precision of event streams. Central to our model is a Binary Neural Network (BNN) architecture that reduces computational overhead by quantizing weights and activations while preserving detail. EeveeDark incorporates (i) modality-specific binary encoders for processing RAW frames and event data, (ii) a lightweight fusion block for integrating spatial and temporal cues, and (iii) an event-guided skip gating mechanism for dynamic spatiotemporal refinement. Experiments on synthetic and real-world datasets show that EeveeDark outperforms prior BNN-based methods and offers a favorable performance-efficiency trade-off compared to full-precision models. The project page is available at https://cyberiada.github.io/EeveeDark.

Real-world data distributions evolve over time, inducing temporal distribution shift that can substantially degrade the reliability of deployed machine learning systems. However, the extent to which architectural choices and their associated inductive biases affect temporal robustness remains insufficiently understood. We present a systematic empirical comparison of temporal robustness across three heterogeneous, time-indexed domains encompassing image classification, multi-label text classification, and text regression tasks. Using a unified evaluation framework based on temporal drift matrices, we train models on cumulative historical data and evaluate their performance on both earlier and later time periods, thereby quantifying cross-temporal generalization. Our study spans model families ranging from simple multilayer perceptrons and convolutional networks to recurrent networks and pretrained Transformer-based encoders. Collectively, the results show that architectural inductive biases systematically shape temporal robustness: models whose inductive biases lead them to exploit localized, highly discriminative features attain the highest in-distribution accuracy, yet those features are often the ones that change most over time, so these models degrade fastest, while pretrained encoders that draw on coarser, more stable representations drift more gradually. These observations offer practical guidance for selecting architectures for real-world systems subject to temporal drift.

Infinite-width limits are a standard way to reason about neural networks, but it is not automatic that the limiting learner has the same complexity-theoretic inductive bias as large finite networks. We study this question for Bayesian neural networks at the mean-field, or critical feature-learning, scaling. The central quantity is the \emph{reduced entropy} $$s_\infty(y,\varepsilon)=\limsup_N -\frac{1}{N}\log π_N^0(L\le \varepsilon),$$ the intensive prior cost of representing a target function $y$ to population mean-squared error $\varepsilon$. Our main result is a width-robust learnability theorem. At fixed depth, a family of Boolean-cube targets is learnable from polynomially many samples at infinite width if and only if it is learnable at polynomial width, if and only if its reduced entropy is polynomially bounded. Equivalently, up to polynomial slack in accuracy, the Bayesian mean-field learner generalizes exactly on the targets that can be represented by polynomial-size networks. The forward direction is proved by a form of subsampling: from the infinitely many hidden neurons in the mean-field solution, one can select polynomially many representatives and still preserve the learned function on every input simultaneously. At the critical scaling this subsampling has both an active'' component, which keeps the data-dependent low-dimensional statistics, and a lazy'' component, which resamples the entropy-dominated directions from the prior. Thus the infinite-width mean-field limit gives a clean analytic description of learning without introducing spurious width-dependent generalization power.

Implantable brain-computer interfaces require on-node spike sorting to reduce telemetry bandwidth and power while maintaining reliable neural decoding. This paper presents a hardware-oriented deep binarized neural network (DBNN) spike-sorting system with two binarized hidden layers with 256 neurons and a fixed-point output layer to enable multiplier-free inference dominated by sign-controlled accumulation and bit-wise logic. The proposed classifier operates on compact 16-sample spike waveforms to reduce the implementation cost (16-256-256-3) and achieves a median classification accuracy of 98.7% on both synthetic and in-vivo datasets. An FPGA prototype on a Cyclone V device operates at 50 MHz and requires 528 cycles per spike, corresponding to a 0.01 ms compute latency, while consuming 828 ALMs and 1023 registers with zero DSP blocks. For ASIC feasibility, the DBNN is implemented using FreePDK45-based flow; synthesis in Synopsys Design Compiler indicates an estimated silicon area of 0.014 mm2 and an operating power of 122 nW at 20 kHz under a 1.1 V supply. These results demonstrate that the proposed DBNN spike sorter offers a favorable trade-off between accuracy and implementation cost, supporting low-power, implantable neural interfaces. Overall, the proposed DBNN spike sorter achieves high accuracy (98.7%) with extremely low hardware cost (0.014 mm2, 122 nW at 20 kHz) and multiplier-free operation, making it suitable for low-power, implantable neural interfaces. This paper introduces the first DBNN designed for real-time neural spike sorting, striking an excellent balance between input data size and network complexity.

We develop a unified function space theory of deep fully connected neural networks. Functions in our spaces are defined recursively as $\ell^1$-bounded linear combinations of activated functions from preceding layers, with a dictionary of affine functions at the first layer. Unlike existing theories that are largely specialized to homogeneous activations such as the ReLU, our framework provides a meaningful notion of functional complexity for deep networks with a broad range of homogeneous and non-homogeneous activation functions commonly used in practice. This simple construction unites several seemingly disparate ideas from the literature, including norm-based complexity bounds and variational characterizations of depth, and facilitates novel analyses of what kinds of functions deep norm-constrained networks can represent. To this end, we prove a novel representer theorem for our spaces and establish novel function-space complexity bounds showing that the associated function classes remain qualitatively small at arbitrary depth. In the univariate ReLU case, we prove a "depth saturation" result: depth in this setting yields only a small constant rescaling of the function class, with no added functional diversity. As a consequence, we show that deep norm-controlled ReLU functions in any dimension cannot exhibit high frequencies along any direction. This finding reveals that some commonly cited expressivity benefits of depth disappear once network complexity is controlled by an appropriate function space norm, rather than parameter count or other representational costs that permit compounded rescaling across layers. Overall, our results illustrate how a function space perspective yields new structural insights into the relationship between depth and complexity.

The AInstein architecture introduced an unsupervised neural method for solving the Riemannian Einstein equations on arbitrary manifolds. This Physics Informed Neural Network approach (PINN) is extended here to Lorentzian signature, validated by recovering the maximally extended Schwarzschild geometry, and tested as novel search method for arbitrary black hole solutions. The topology is built into the architecture by treating $S^{2}$ globally through its standard embedding, such that the network learns an ambient metric on the manifold $\mathbb{R}^{2} \times \mathbb{R}^{3}$, where Penrose coordinates are chosen for $\mathbb{R}^2$ and the metric on $S^{2}$ is obtained by pullback. The architecture is first trained with the objective of recovering the Schwarzschild metric via losses encoding the vacuum Einstein equation, a quadratic Weyl scalar constraint, and the $SO(3)$ symmetry of the resultant metric; directly motivated by the Birkhoff--Jebsen theorem. Following this, the objective is generalised to use the Petrov speciality index, a horizon curvature anchor, and a trapped-surface constraint, to allow search for algebraically general Petrov type I solutions, finding potentially novel general-type Lorentzian Einstein metrics with a genuinely trapped interior.

Integrating complex, multi-omics data presents significant challenges. Existing approaches often face a trade-off between model interpretability and representational capacity, with most either relying on post-hoc interpretation or use linear models that may overlook complex interactions. We report Pathway Activity Autoencoders for the multi-omics setting, which embed prior knowledge via pathway-informed architectural constraints, fostering interpretability, while preserving representational power. Our multi-omic framework is applied in the context of breast cancer and is evaluated in survival prediction and subtype classification with results indicating a positive effect of integration. We conduct analysis of individual omics layer impact on end-task performance, revealing that gene, protein, and microRNA expression layers provide the strongest contribution. Repeatability studies indicate that, while dropout improves model robustness and consistency, excessive regularisation can reduce predictive performance. Finally, visualizations of the learned feature space illustrate the framework's intrinsic transparency and clinical relevance. The results underscore the value of multi-omic integration and delineate the impact of individual omics layers, establishing practical guidelines for integration within our framework. Overall, our pathway activity autoencoder frameworks yield superior latent representations that are biologically meaningful and are directly translatable into clinically relevant insights.

Physics-informed neural networks (PINNs) often face ill-posed optimization, competing losses, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse source-task representations, but direct fine-tuning may induce negative transfer when source and target physics differ, leading to low field error but inaccurate parameter recovery. To address this issue, we propose Target-Guided Selective Reweighting PINN (TGSR-PINN), a target-evidence-driven representation correction method for PINN inverse transfer learning. TGSR-PINN transfers source network weights and biases but initializes target physical parameters independently. After short target adaptation, it scores neurons using first-order Taylor sensitivity and pre-activation variance on fixed batches. These scores are converted into continuous weak-adaptation signals using a Gaussian mixture model with rank fallback. TGSR-PINN then applies bounded selective soft decay to the corresponding input weight rows and biases without pruning or resetting them. Experiments on a zero-source high-Péclet inflow--outflow problem with nonzero Dirichlet data and an outflow boundary layer, Allen--Cahn to Burgers cross-PDE transfer, and 5%-noise reaction--diffusion inverse problems show that TGSR-PINN improves parameter recovery while maintaining low field error. Ablation studies indicate that neuron target scoring, weak-adaptation estimation, layer protection, and selective soft decay jointly contribute to the observed benefits.

Neural network verification and data privacy are inherently in tension: verification demands full access to model parameters and input data, yet both are increasingly restricted by privacy regulations and intellectual property constraints. This tension has left robustness verification impractical in privacy-sensitive domains. In this work, we address this gap with SecureCROWN, the first framework for privacy-preserving neural network robustness verification. Built upon secure two-party computation (2PC), our framework enables a model owner and a data owner to jointly compute certified robustness bounds -- revealing only the final result while provably protecting both parties' private data under the semi-honest security model. A key challenge is securely computing the conditional operations in Linear Bound Propagation, where the data-dependent branching is incompatible with standard secure computation protocols. We eliminate branching by formulating conditional logic as continuous arithmetic operations. Additionally, we introduce a Newton--Raphson refinement method to improve numerical stability. Extensive analysis and experiments show that SecureCROWN strictly matches plaintext verification results, while completing in 0.1--200s across varied model sizes and communication settings (LAN/WAN), demonstrating the feasibility of privacy-preserving neural network verification.

Many phenomena of deep learning are dynamical: they concern not only which minima exist, but how gradient descent reaches, avoids, or selects among them. Edge-of-stability behavior, sharpness oscillations, catapult phases, balancing, and movement toward flatter representations are effects of the training map itself, and are poorly captured by the small-step gradient-flow limit. This paper studies fixed-step gradient descent as a discrete dynamical system in a hierarchy of exactly solvable models retaining basic structures of deep learning: depth, factorization, width, data coupling, activation, and stochasticity. The starting point is the balanced scalar reduction of a deep linear chain, giving a quartic loss and a cubic gradient map whose post-edge behavior is explicit. Under the natural large-depth scaling, this dynamics converges to a universal Ricker-type map. The edge of stability is therefore not a breakdown of optimization, but the first bifurcation of the training map. Embedding the scalar dynamics back into factored models turns these regimes into learning phenomena. Finite steps break conservation laws of gradient flow and contract factorization imbalance; residual oscillations move parameters toward flatter, more balanced representations. Wider linear networks produce a ladder of spectral edges, so the optimal learning rate can lie beyond the first edge. Data coupling, nonlinear activations, and stochastic targets preserve the same organizing principle: finite-step oscillations drive alignment, balancing, and representation selection. Thus the learning rate is not merely a numerical stability parameter. It is a structural parameter of the training dynamics, determining its attractors and shaping the representations gradient descent selects.

Score-based Generative Models (SGMs) have achieved impressive performance in data generation across a wide range of applications. While the statistical properties of their sampling procedures are increasingly well understood, the optimization dynamics underlying their training remain less explored. SGMs are typically trained by minimizing a weighted denoising scorematching objective, yet optimization guarantees with stochastic gradients remain limited. In this work, we study Stochastic Gradient Descent (SGD) for SGMs, contributing results in two complementary regimes. First, for general score parameterizations, we establish a non-convex convergence rate for SGD on the weighted denoising score-matching objective, with explicit dependence on the schedule-dependent weighting factors. Second, for overparameterized two-layer ReLU networks, we develop a Neural Tangent Kernel analysis tailored to diffusion training with stochastic gradients, yielding score-approximation error bounds along the SGD trajectory. Finally, our analysis quantifies the role of the reweighting factor in the score approximation error, providing theoretical guidance for weighting choices used in practice.

Low-precision neural networks are attractive for resource-constrained hardware, but fixed-point arithmetic introduces failure modes that are often hidden by idealised quantisation models. In particular, two's-complement overflow wrapping can corrupt hidden activations by changing both their magnitude and sign, leading to unstable numerical error propagation and severe accuracy degradation. This paper proposes a Lyapunov-stabilised quantisation framework for low-precision neural networks operating under hardware-style wrapping arithmetic. The hidden-state energy is monitored through a layerwise Lyapunov function, and a monotone projection is applied to enforce bounded and non-increasing state evolution across depth. The method is evaluated on MNIST using a compact patch-based transformer under post-training quantisation and quantisation-aware training with fixed-point bit-widths from 4 to 16 bits. Monte Carlo results show that unconstrained wrapped quantisation-aware training collapses to near-chance accuracy across 6-16 bits, with activation overflow rates exceeding 11%. In contrast, the proposed monotone Lyapunov projection suppresses activation overflow to below 0.012% and restores stable low-precision learning, achieving 86.55% accuracy at 12 bits. These results demonstrate that Lyapunov-based state control can act as a hardware-aware stabilisation mechanism for reliable fixed-point neural inference and training.

Deep neural networks often contain substantial hidden-state redundancy, but most compression methods operate directly on weights, neurons, or quantised representations without explicitly characterising the dynamical role of internal states. This paper proposes a controllability-observability framework for empirical state-order reduction of deep neural networks. By viewing a trained network as a depth-indexed nonlinear dynamical system, we construct data-driven reachability, observability, and balanced Gramians from hidden-state snapshots and output Jacobians. The resulting A/B/C tests estimate layer-wise reachable, observable, and jointly reachable--observable ranks. These ranks are then used not only as diagnostic measures of hidden-state redundancy, but also as actual compressed layer widths for realised reduced networks. Experiments on MNIST and CIFAR-10 compare the proposed balanced realisation against projection-based reduction, unstructured pruning, structured pruning, low-rank SVD, dynamic INT8 quantisation, and linear baselines. On MNIST, a four-layer SiLU DNN is reduced from state order 1024 to 277, giving 72.95% state compression and 73.48% parameter compression, while maintaining 95.45% accuracy compared with 96.60% for the full model. On CIFAR-10, a larger SiLU DNN is reduced from state order 4608 to 1339, giving 70.94% state compression and 83.09% parameter compression, while preserving accuracy from 54.45% to 54.44% and reducing CUDA inference latency by approximately 3X. The results show that balanced reachable-observable ranks provide a principled empirical minimal-realisation criterion for designing compact neural architectures with little or no loss in accuracy.

We present CoFINN (Conservation Flux Informed Neural Networks), a physics-informed deep learning framework for predicting compressible flow fields governed by conservation laws. Unlike conventional data-driven convolutional neural networks (CNNs), which optimize only pixel-wise similarity metrics, CoFINN embeds finite-volume conservation physics directly into the training process. Unlike classical physics-informed methods which enforce differential-equation residuals at collocation points through automatic differentiation, CoFINN adopts a finite-volume perspective consistent with modern CFD methodology. CoFINN interprets CNN output fields as structured computational grids, where each pixel represents a finite-volume cell, and enforces conservation consistency through sophisticated numerical flux calculations. The framework is evaluated on transonic flow prediction around airfoils at (M=0.7, Re=6 * 10^6), including challenging conditions involving shock waves and high angles of attack. Results show that CoFINN improves aerodynamic force prediction accuracy, reducing drag prediction error by up to 34% at extreme angles of attack and by approximately 15% on average across the test set. Improvements are particularly significant in limited-data regimes, demonstrating that the conservation-based loss acts as an effective physical regularizer. The proposed approach maintains the computational efficiency advantages of CNN surrogates while significantly improving physical consistency and conservation behavior. The framework is architecture-agnostic and extensible to broader classes of conservation-law-governed physical systems.

A central goal of current Spiking Neural Network (SNN) research is to improve their accuracy toward becoming low-power alternatives to Artificial Neural Networks (ANNs). This work further argues that realizing this ambition requires improving not only accuracy but also robustness, defined as the ability to maintain correct predictions under input perturbations. We identify two key issues in existing SNN methods that undermine robustness. First, binary spiking activations can produce large activation-state changes under small perturbations. Second, the lack of effective weight constraints makes network outputs more sensitive to input variations. To this end, we propose Burst Spiking Neural Networks (BuSNNs), built upon Burst-enhanced Spiking Neurons (BSNs) and a Dynamic Weight Constraint (DWC) mechanism. BSNs incorporate burst firing to provide a graded spiking pattern. This spiking mechanism mitigates perturbation-induced transitions in activation states and thereby enhances robustness. DWC penalizes connection weights based on activation states, effectively reducing weight magnitudes and improving robustness while preserving accuracy. We provide theoretical analyses to support these robustness effects. Experimental results further show that, on smaller-scale benchmarks such as CIFAR-10, BuSNNs outperform both SNN and ANN counterparts in accuracy and robustness. On large-scale ImageNet, BuSNN with the MS ResNet-34 backbone further improves top-1 accuracy and corruption robustness over the corresponding SNN baseline by 3.18% and 2.66%, respectively. Despite using spike-based activations, BuSNNs surpass 4-bit activation-quantized ANN baselines and approach 8-bit ANN baselines on ImageNet. They also preserve SNNs' low-power advantage. This work studies the accuracy-robustness problem in SNNs, advancing their practical viability in robust and energy-efficient applications.

The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies $d/\|w\|$ as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than $30\%$ of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

We study the implicit bias of noisy stochastic gradient descent in training wide two-layer ReLU networks for multivariate regression. In a mean-field regime, the training dynamics are approximated by a Wasserstein gradient flow that converges to a unique stationary measure. We characterize the structure of this stationary measure and the predictor it represents. We show that, despite the network being infinitely overparameterized, the learned predictor admits an effectively finite representation: the input weights and biases align along finitely many directions, leading to an effective width collapse. In particular, the solution function is continuous piecewise affine, with affine regions determined by the cells of a finite hyperplane arrangement. The number of learned directions, and hence hyperplanes, is bounded above by $2\mathcal{P}-1$, where $\mathcal{P}$ denotes the number of linear dichotomies realizable on the training inputs. We further establish a non-redundancy property of the learned representation by proving that each learned direction induces a unique ternary activation pattern on the training data. Consequently, the complexity of the learned predictor is governed by the combinatorial geometry of the training data.

Weight space learning aims to learn representations of neural network (NN) weights, enabling different downstream tasks. Existing approaches show promising performance, but lacking a way to shape these weight-space representations using information about the datasets the models were trained on, thus limiting downstream applications. We propose WeightCLIP, a method for learning a dataset-aligned latent space for neural networks, where datasets information is induced during training. The NNs are encoded as latent representations using an autoencoder, while dataset samples are encoded using a dataset encoder. The two representations are aligned using a contrastive objective, effectively reshaping the weight-space representations according to the datasets. We demonstrate that such representations can be used for different downstream tasks, including mapping dataset information to a weight-space representation that decode to strong models. In addition, we introduce a latent refinement process for generating models that outperforms standard fine-tuning. Overall, our results demonstrate that explicitly incorporating dataset information improves what can be achieved with weight-space representations across retrieval, generation, and refinement. Code will be available at https://github.com/HSG-AIML/WeightCLIP.

We investigate whether structured reasoning interventions improve the strategic economic reasoning of large language models, and whether their effects depend on model architecture. Using Hotelling's linear city model as a diagnostic vehicle, we evaluate GPT-4.1-mini (a standard instruction-following model) and GPT-5-mini (a reasoning-optimized model) under five conditions - an unscaffolded baseline and four reasoning interventions - across eight questions spanning deductive and abductive reasoning, three prompt framings, and three repetitions per condition, yielding 720 individually judged responses. We find a statistically significant crossover interaction between scaffolding type and model architecture ($t(7) = 4.79$, $p = 0.002$, $d = 1.69$): commitment scaffolding improves the standard model ($+0.21$) while degrading the reasoning model ($-0.63$), and principled separation shows the opposite pattern ($-0.40$ vs. $+0.31$). Both crossovers are individually significant (commitment: $p = 0.040$; separation: $p = 0.002$) and hold across all eight questions with 7/8 directional consistency. Adversarial stress-testing harms both models, with $2.6\times$ greater degradation for the reasoning model ($-1.47$ vs. $-0.57$; $p = 0.038$), and the damage correlates negatively with baseline difficulty ($R^2 = 0.36$, $p = 0.014$). We further document a persistent declarative-procedural gap in which both models identify correct strategies at rates far exceeding their ability to execute them; separation fully closes this gap for the reasoning model while no intervention helps the standard model.

Activation functions are considered an essential primitive for neural nonlinearity, i.e., they enable neural networks to serve as universal approximators. In this paper, we show that this nonlinearity can also be achieved by input-conditioned threshold gating through branches as a universal primitive. We demonstrate that standard activations -- whether piecewise-linear (ReLU, PReLU, Hardtanh) or smooth (SiLU, Sigmoid, Tanh, GELU) -- are in fact instances of a single Threshold Gating (TG) primitive. For softmax, we show that it admits an exact TG conversion via its equivalent per-element Sigmoid form. We then validate these equivalences by converting pretrained networks across CNNs, transformer-based models, and recurrent architectures, preserving model performance without requiring retraining. Threshold Gating also enables training from scratch that goes beyond replacing existing activations, enabling gains in model compression, performance, and shorter training. We also propose a 'Minimal Branch Theorem' which relates the minimum number of required branches in our primitive to the trainability of general deep neural networks. In terms of hardware implementation, TG maps to a unified implementation in the case of analog in-memory systems, addressing the bottleneck of analog-to-digital and digital-to-analog converters (ADC/DAC) that is known to significantly impact power consumption and on-chip area.

Deep reinforcement learning (DRL) offers powerful control for industrial cyber-physical systems (ICPSs), but its "black-box" exploration risks violating strict hardware safety limits. Typically, these constraints are managed through complex reward shaping. In this work-in-progress paper, we embed a differentiable physics model directly into the proximal policy optimization (PPO) actor loss function. By simulating short-horizon future trajectories during training, the policy is penalized for anticipated safety violations independent of the task-reward signal. Evaluated on a simulated 1-degree-of-freedom helicopter testbed with strict pitch constraints, our physics-informed soft regularizations substantially reduce constraint violations while maintaining reliable target tracking.

We study a graph classification problem involving over 20 million graphs, arising from high-order perturbative computations of correlators in planar $\mathcal{N}=4$ super-Yang--Mills, a model closely related to the theory of the strong nuclear force. We benchmark graph neural networks, including graph transformers, achieving robust generalization to larger graphs with up to $99.996\%$ ROC AUC. Then, we analyze how the models can be used to gain a computational speedup compared to the traditional graphical bootstrap algorithm, through shrinking the redundant data by up to $85.5\%$ at the level of denominator graphs. Finally, we study the embeddings of the models to investigate their interpretability.

Neuromorphic controllers for size, weight, and power-constrained systems require neural architectures that are both energy-efficient and interpretable at the level of system dynamics. However, existing approaches either rely on end-to-end trained spiking networks with limited interpretability, or on converted classical controllers that fail to fully exploit neuromorphic dynamics. We present a spiking neural network (SNN) architecture for generating polar trajectories, using a winner-take-all (WTA) architecture with accessory populations that induce controlled transitions in neural activity. We demonstrate tuning rules for these population dynamics, and utilize a form of shunting inhibition to enable independent control of direction, speed, and radius of the resulting polar trajectories. We implement the network on the SpiNNaker2 neuromorphic processor, and demonstrate a two to three orders of magnitude reduction in wall-clock step time and three to four orders of magnitude reduction in energy expenditure when compared to conventional computing platforms.

Machine learning has shown speedups for numerical methods in physics applications, but integrating Python-based libraries into high-performance C++ solvers creates performance bottlenecks. We present CodeJeNN, which bridges this gap by auto-generating self-contained C++ code from trained Keras models for inference. This eliminates external dependencies through minimal inlined functions, allowing seamless integration into existing frameworks. We describe the Keras-to-C++ workflow, supported architectures, and limitations. CodeJeNN is demonstrated through inference benchmarks against Keras in eager and JIT modes and a CFD test case modeling viscosity in a hydrogen-air mixing layer, showing speedups without sacrificing accuracy.