Neural Network Generalization

For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.

This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $Γ$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary $Γ$-semiring with $|T|=4$, $|Γ|=1$}, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational $Γ$-Algebra.

In 2020 Domingos introduced an interpolation formula valid for "every model trained by gradient descent". He concluded that such models behave approximately as kernel machines. In this work, we extend the Domingos formula to stochastic training. We introduce a stochastic gradient kernel that extends the deterministic version via a continuous-time diffusion approximation. We prove stochastic Domingos theorems and show that the expected network output admits a kernel-machine representation with optimizer-specific weighting. It reveals that training samples contribute through loss-dependent weights and gradient alignment along the training trajectory. We then link the generalization error to the null space of the integral operator induced by the stochastic gradient kernel. The same path-kernel viewpoint provides a unified interpretation of diffusion models and GANs: diffusion induces stage-wise, noise-localized corrections, whereas GANs induce distribution-guided corrections shaped by discriminator geometry. We visualize the evolution of implicit kernels during optimization and quantify out-of-distribution behaviors through a series of numerical experiments. Our results support a feature-space memory view of learning: training stores data-dependent information in an evolving tangent feature geometry, and predictions at test time arise from kernel-weighted retrieval and aggregation of these stored features, with generalization governed by alignment between test points and the learned feature memory.

The use of Deep Neural Network based systems in the real world is growing. They have achieved state-of-the-art performance on many image, speech and text datasets. They have been shown to be powerful systems that are capable of learning detailed relationships and abstractions from the data. This is a double-edged sword which makes such systems vulnerable to learning the noise in the training set, thereby negatively impacting performance. This is also known as the problem of overfitting or poor generalization. In a practical setting, analysts typically have limited data to build models that must generalize to unseen data. In this work, we explore the use of a differential-privacy based approach to improve generalization in Deep Neural Networks.

Structured pruning is widely applied to compress large language models (LLMs), but its performance depends heavily on how neuron importance is estimated. Most existing methods rely on activation statistics from a single calibration set, which introduces calibration bias and degrades downstream cross-task generalization. We observe that neurons exhibit heterogeneous distribution sensitivity, ranging from maintaining relatively stable rankings across calibration datasets to showing substantially larger cross-dataset variation. Ignoring this heterogeneity, existing methods rank all neurons in shared spaces with a uniform scoring source, so calibration-specific neurons dominate the ranking and weakly-activated neurons are scored unreliably. To address this, we propose GPrune-LLM, a structured pruning framework that reduces calibration bias by measuring and exploiting the cross-distribution behavior of neurons for fair comparison. Specifically, we restructure the neuron ranking space into behavior-consistent local spaces, adapt the scoring source where the calibration signal is unreliable, and learn per-module sparsity allocation under a global budget. Experiments across multiple models and downstream tasks show that GPrune-LLM improves the generalization of its base pruning metrics, with gains most pronounced at high sparsity, and reduces dependence on the choice of importance metric.

Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.

Flatness measures based on the spectrum or the trace of the Hessian of the loss are widely used as proxies for the generalization ability of deep networks. However, most existing definitions are either tailored to fully connected architectures, relying on stochastic estimators of the Hessian trace, or ignore the specific geometric structure of modern Convolutional Neural Networks (CNNs). In this work, we develop a flatness measure that is both exact and architecturally faithful for a broad and practically relevant class of CNNs. We first derive a closed-form expression for the trace of the Hessian of the cross-entropy loss with respect to convolutional kernels in networks that use global average pooling followed by a linear classifier. Building on this result, we then specialize the notion of relative flatness to convolutional layers and obtain a parameterization-aware flatness measure that properly accounts for the scaling symmetries and filter interactions induced by convolution and pooling. Finally, we empirically investigate the proposed measure on families of CNNs trained on standard image-classification benchmarks. The results obtained suggest that the proposed measure can serve as a robust tool to assess and compare the generalization performance of CNN models, and to guide the design of architecture and training choices in practice.

In the training of neural networks, adaptive moment estimation (Adam) typically converges fast but exhibits suboptimal generalization performance. A widely accepted explanation for its defect in generalization is that it often tends to converge to sharp minima. To enhance its ability to find flat minima, we propose its new variant named inverse Adam (InvAdam). The key improvement of InvAdam lies in its parameter update mechanism, which is opposite to that of Adam. Specifically, it computes element-wise multiplication of the first-order and second-order moments, while Adam computes the element-wise division of these two moments. This modification aims to increase the step size of the parameter update when the elements in the second-order moments are large and vice versa, which helps the parameter escape sharp minima and stay at flat ones. However, InvAdam's update mechanism may face challenges in convergence. To address this challenge, we propose dual Adam (DualAdam), which integrates the update mechanisms of both Adam and InvAdam, ensuring convergence while enhancing generalization performance. Additionally, we introduce the diffusion theory to mathematically demonstrate InvAdam's ability to escape sharp minima. Extensive experiments are conducted on image classification tasks and large language model (LLM) fine-tuning. The results validate that DualAdam outperforms Adam and its state-of-the-art variants in terms of generalization performance. The code is publicly available at https://github.com/LongJin-lab/DualAdam.

Grokking -- the sudden generalisation that appears long after a model has perfectly memorised its training data -- has been widely observed but lacks a quantitative theory explaining the length of the delay. We show that grokking is a norm-driven representational phase transition in regularised training dynamics, and establish the Norm-Separation Delay Law: $T_{\mathrm{grok}} - T_{\mathrm{mem}} = Θ(γ_{\mathrm{eff}}^{-1} \log(\|θ_{\mathrm{mem}}\|^2 / \|θ_{\mathrm{post}}\|^2))$, where $γ_{\mathrm{eff}}$ is the optimiser's effective contraction rate ($γ_{\mathrm{eff}} = ηλ$ for SGD, $γ_{\mathrm{eff}} \ge ηλ$ for AdamW). The upper bound follows from a discrete Lyapunov contraction argument; the matching lower bound from dynamical constraints of regularised first-order optimisation. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity, we confirm three falsifiable predictions: inverse scaling with weight decay ($R^2 = 0.97$), inverse scaling with learning rate ($R^2 = 0.92$), and logarithmic dependence on the norm ratio (Pearson $r = 0.91$). A fourth finding reveals that grokking requires an optimiser capable of decoupling memorisation from contraction: SGD fails entirely at the same hyperparameters where AdamW reliably groks. These results reframe grokking not as a mysterious optimisation artefact but as a predictable consequence of norm separation between competing interpolating representations. We further derive a practical three-input algorithm that predicts grokking delay at memorisation time with 34.6% mean absolute error (bootstrap 95% CI [30.0%, 39.4%], $N=60$ seeds), enabling principled early stopping.

Multi--task learning seeks to improve the generalization error by leveraging the common information shared by multiple related tasks. One challenge in multi--task learning is identifying formulations capable of uncovering the common information shared between different but related tasks. This paper provides a precise asymptotic analysis of a popular multi--task formulation associated with misspecified perceptron learning models. The main contribution of this paper is to precisely determine the reasons behind the benefits gained from combining multiple related tasks. Specifically, we show that combining multiple tasks is asymptotically equivalent to a traditional formulation with additional regularization terms that help improve the generalization performance. Another contribution is to empirically study the impact of combining tasks on the generalization error. In particular, we empirically show that the combination of multiple tasks postpones the double descent phenomenon and can mitigate it asymptotically.

Gradient descent on overparameterized neural networks typically operates at the Edge of Stability (EoS), where the largest Hessian eigenvalue hovers around a step-size-dependent threshold. We study how sparse connectivity changes generalization below this threshold in two-layer ReLU networks. Prior results have shown that for fully-connected networks (FCNs), generalization guarantees in this regime degrade and become vacuous on high-dimensional spherical inputs. Our analysis reveals that sparse connectivity fundamentally alters this picture. Under sparse connectivity, the network processes a collection of low-dimensional patches rather than the full input vector, so the effective constraint imposed by the stability condition is governed by the geometry of the training patch collection. We prove that when the receptive fields are small relative to the ambient dimension, the effective constraint yields non-vacuous generalization bounds in precisely the spherical regime where FCNs provably fail. The same framework also reveals a contrasting failure mode: if the patch collection lacks geometric structure, the constraint becomes unable to prevent overfitting. We corroborate this theory by analyzing the patch geometry of natural images, showing that standard convolutional designs produce patch multiset with low-dimensional structure that facilitates generalization. This provides a principled explanation for the generalization advantage of convolutional networks. Thus, our analysis yields a unified framework that identifies how architecture, data geometry, and gradient descent jointly govern generalization performance.

Standard accounts of memory consolidation emphasise the stabilisation of stored representations, but struggle to explain representational drift, semanticisation, or the necessity of offline replay. Here we propose that high-capacity neocortical networks optimise stored representations for generalisation by reducing complexity via predictive forgetting, i.e. the selective retention of experienced information that predicts future outcomes or experience. We show that predictive forgetting formally improves information-theoretic generalisation bounds on stored representations. Under high-fidelity encoding constraints, such compression is generally unattainable in a single pass; high-capacity networks therefore benefit from temporally separated, iterative refinement of stored traces without re-accessing sensory input. We demonstrate this capacity dependence with simulations in autoencoder-based neocortical models, biologically plausible predictive coding circuits, and Transformer-based language models, and derive quantitative predictions for consolidation-dependent changes in neural representational geometry. These results identify a computational role for off-line consolidation beyond stabilisation, showing that outcome-conditioned compression optimises the retention-generalisation trade-off.

Generalization in generative modeling is defined as the ability to learn an underlying distribution from a finite dataset and produce novel samples, with evaluation largely driven by held-out performance and perceived sample quality. In practice, training is often stopped at the minimum of the test loss, taken as an operational indicator of generalization. We challenge this viewpoint by identifying a phase of biased generalization during training, in which the model continues to decrease the test loss while favoring samples with anomalously high proximity to training data. By training the same network on two disjoint datasets and comparing the mutual distances of generated samples and their similarity to training data, we introduce a quantitative measure of bias and demonstrate its presence on real images. We then study the mechanism of bias, using a controlled hierarchical data model where access to exact scores and ground-truth statistics allows us to precisely characterize its onset. We attribute this phenomenon to the sequential nature of feature learning in deep networks, where coarse structure is learned early in a data-independent manner, while finer features are resolved later in a way that increasingly depends on individual training samples. Our results show that early stopping at the test loss minimum, while optimal under standard generalization criteria, may be insufficient for privacy-critical applications.

While deep neural networks (DNNs) have achieved remarkable performance in tasks such as image recognition, they often struggle with generalization, learning from few examples, and continuous adaptation - abilities inherent in biological neural systems. These challenges arise due to DNNs' failure to emulate the efficient, adaptive learning mechanisms of biological networks. To address these issues, we explore the integration of neurobiologically inspired assumptions in neural network learning. This study introduces a biologically inspired learning rule that naturally integrates neurobiological principles, including sparsity, lognormal weight distributions, and adherence to Dale's law, without requiring explicit enforcement. By aligning with these core neurobiological principles, our model enhances robustness against adversarial attacks and demonstrates superior generalization, particularly in few-shot learning scenarios. Notably, integrating these constraints leads to the emergence of biologically plausible neural representations, underscoring the efficacy of incorporating neurobiological assumptions into neural network design. Preliminary results suggest that this approach could extend from feature-specific to task-specific encoding, potentially offering insights into neural resource allocation for complex tasks.

Generalization in deep neural networks remains only partially understood. Inspired by the stronger generalization tendency of biological systems, we explore the hypothesis that robust internal representations should remain effective across both dense and sparse activation regimes. To test this idea, we introduce a simple training strategy that applies global top-k constraints to hidden activations and repeatedly cycles a single model through multiple activation budgets via progressive compression and periodic reset. Using CIFAR-10 without data augmentation and a WRN-28-4 backbone, we find in single-run experiments that two adaptive keep-ratio control strategies both outperform dense baseline training. These preliminary results suggest that joint training across multiple activation sparsity regimes may provide a simple and effective route to improved generalization.

Adam and AdamW are a class of default optimizers for training deep learning models in machine learning. These adaptive algorithms converge faster but generalize worse compared to SGD. In fact, their proved generalization error $O(\frac{1}{\sqrt{N}})$ also is larger than $O(\frac{1}{N})$ of SGD, where $N$ denotes training sample size. Recently, although some variants of Adam have been proposed to improve its generalization, their improved generalizations are still unexplored in theory. To fill this gap, in the paper, we restudy generalization of Adam and AdamW via algorithmic stability, and first prove that Adam and AdamW without square-root (i.e., Adam(W)-srf) have a generalization error $O(\frac{\hatρ^{-2T}}{N})$, where $T$ denotes iteration number and $\hatρ>0$ denotes the smallest element of second-order momentum plus a small positive number. To improve generalization, we propose a class of efficient clever Adam (i.e., HomeAdam(W)) algorithms via sometimes returning momentum-based SGD. Moreover, we prove that our HomeAdam(W) have a smaller generalization error $O(\frac{1}{N})$ than $O(\frac{\hatρ^{-2T}}{N})$ of Adam(W)-srf, since $\hatρ$ is generally very small. In particular, it is also smaller than the existing $O(\frac{1}{\sqrt{N}})$ of Adam(W). Meanwhile, we prove our HomeAdam(W) have a faster convergence rate of $O(\frac{1}{T^{1/4}})$ than $O(\frac{\breveρ^{-1}}{T^{1/4}})$ of the Adam(W)-srf, where $\breveρ\leq\hatρ$ also is very small. Extensive numerical experiments demonstrate efficiency of our HomeAdam(W) algorithms.

Grokking, the sudden transition from memorization to generalization, is characterized by the emergence of low-dimensional representations, yet the mechanism underlying this organization remains elusive. We propose that intrinsic task symmetries primarily drive grokking and shape the geometry of the model's representation space. We identify a consistent three-stage training dynamic underlying grokking: (i) memorization, (ii) symmetry acquisition, and (iii) geometric organization. We show that generalization emerges during the symmetry acquisition phase, after which representations reorganize into a structured, task-aligned geometry. We validate this symmetry-driven account across diverse algorithmic domains, including algebraic, structural, and relational reasoning tasks. Building on these findings, we introduce a symmetry-based diagnostic that anticipates the onset of generalization and propose strategies to accelerate it. Together, our results establish intrinsic symmetry as the key factor enabling neural networks to move beyond memorization and achieve robust algorithmic reasoning.

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a bottom-up'' mechanistic route by reverse-engineering interpretable features or circuits to build explanatory models. While insightful, these methods often struggle to provide the high-level, predictive signals for anticipating failure in real-world deployment. Here, we propose using a top-down'' approach to studying generalization failures inspired by medical biomarkers: identifying system-level measurements that serve as robust indicators of a model's future performance. Rather than mapping out detailed internal mechanisms, we systematically design and test network markers to probe structure, function links, identify prognostic indicators, and validate predictions in real-world settings. In image classification, we find that task-relevant geometric properties of in-distribution (ID) object manifolds consistently forecast poor out-of-distribution (OOD) generalization. In particular, reductions in two geometric measures, effective manifold dimensionality and utility, predict weaker OOD performance across diverse architectures, optimizers, and datasets. We apply this finding to transfer learning with ImageNet-pretrained models. We consistently find that the same geometric patterns predict OOD transfer performance more reliably than ID accuracy. This work demonstrates that representational geometry can expose hidden vulnerabilities, offering more robust guidance for model selection and AI interpretability.

Neural PDE solvers are often described as learning solution operators that map problem data to PDE solutions. In this work, we argue that this interpretation is generally incorrect when boundary conditions vary. We show that standard neural operator training implicitly learns a boundary-indexed family of operators, rather than a single boundary-agnostic operator, with the learned mapping fundamentally conditioned on the boundary-condition distribution seen during training. We formalize this perspective by framing operator learning as conditional risk minimization over boundary conditions, which leads to a non-identifiability result outside the support of the training boundary distribution. As a consequence, generalization in forcing terms or resolution does not imply generalization across boundary conditions. We support our theoretical analysis with controlled experiments on the Poisson equation, demonstrating sharp degradation under boundary-condition shifts, cross-distribution failures between distinct boundary ensembles, and convergence to conditional expectations when boundary information is removed. Our results clarify a core limitation of current neural PDE solvers and highlight the need for explicit boundary-aware modeling in the pursuit of foundation models for PDEs.

In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.

While Adam has long been the ubiquitous default optimizer for deep neural networks, Muon has recently seen rapid adoption due to its superior training speed. Although much of the literature focuses on validating the benefits of Muon, our work investigates the potential downsides of the mechanism driving this speedup. On the theoretical front, we analyze the learning dynamics of simplified Muon on deep linear networks and linear attention. Our analysis reveals that Muon gains speed by avoiding saddle points, but does so at the expense of the simplicity bias characteristic of Gradient Descent (GD), where the complexity of the functional solution learned grows sequentially. Experiments demonstrate the consequences of losing the simplicity bias, showing that Muon struggles to uncover common underlying structure across tasks and may be prone to fitting spurious features. More broadly, this paper serves as a reminder that faster optimization is rarely a free lunch; improvements in optimization can come at the cost of changes in the inductive biases that shape generalization.

Full Waveform Inversion (FWI) is a highly nonlinear and ill-posed problem that aims to recover subsurface velocity maps from surface-recorded seismic waveforms data. Existing data-driven FWI typically uses small models, as available datasets have limited volume, geological diversity, and spatial extent, leading to substantial concerns about overfitting. Although they perform well on synthetic datasets, current methods fail to generalize to more realistic geological structures. In this work, we show that a model trained entirely on simulated and relatively simple data can generalize remarkably well to challenging and unseen geological benchmarks. We provide a working recipe that tames a billion-parameter model for FWI through coordinated scaling across three axes: model capacity, data diversity, and training strategy. Our model achieves state-of-the-art performance on OpenFWI and significantly narrows the generalization gap in data-driven FWI. Across six challenging geophysical benchmarks, including Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP, Sigsbee, and SEAM Phase I, it infers complex structures absent from the training set and delivers significant performance improvements (SSIM from 0.5844 to 0.7669). Overall, our results demonstrate that with an appropriate scaling strategy, large models trained on simple synthetic data can achieve substantial generalization to more complex and realistic geological structures.

Inspired by measurement incompatibility and Bell-family inequalities in quantum mechanics, we propose the Non-Classical Network (NCnet), a simple classical neural architecture that stably exhibits non-classical statistical behaviors under typical and interpretable experimental setups. We find non-classicality, measured by the $S$ statistic of CHSH inequality, arises from gradient competitions of hidden-layer neurons shared by multi-tasks. Remarkably, even without physical links supporting explicit communication, one task head can implicitly sense the training task of other task heads via local loss oscillations, leading to non-local correlations in their training outcomes. Specifically, in the low-resource regime, the value of $S$ increases gradually with increasing resources and approaches toward its classical upper-bound 2, which implies that underfitting is alleviated with resources increase. As the model nears the critical scale required for adequate performance, $S$ may temporarily exceed 2. As resources continue to grow, $S$ then asymptotically decays down to and fluctuates around 2. Empirically, when model capacity is insufficient, $S$ is positively correlated with generalization performance, and the regime where $S$ first approaches $2$ often corresponding to good generalization. Overall, our results suggest that non-classical statistics can provide a novel perspective for understanding internal interactions and training dynamics of deep networks.

Generative modeling of neural network parameters is often tied to architectures because standard parameter representations rely on known weight-matrix dimensions. Generation is further complicated by permutation symmetries that allow networks to model similar input-output functions while having widely different, unaligned parameterizations. In this work, we introduce Neural Network Diffusion Transformers (NNiTs), which generate weights in a width-agnostic manner by tokenizing weight matrices into patches and modeling them as locally structured fields. We establish that Graph HyperNetworks (GHNs) with a convolutional neural network (CNN) decoder structurally align the weight space, creating the local correlation necessary for patch-based processing. Focusing on Multilayer Perceptrons (MLPs), where permutation symmetry is especially apparent, NNiTs generate fully functional networks across a range of architectures. Our approach jointly models discrete architecture tokens and continuous weight patches within a single sequence model. On ManiSkill3 robotics tasks, NNiT achieves $>85\%$ success on architecture topologies unseen during training, while baseline approaches fail to generalize; the same pipeline also generalizes to MNIST classification beyond the robotic control setting.

Generalization measures have been studied extensively in the machine learning community to better characterize generalization gaps. However, establishing a reliable generalization measure for statistically singular models such as deep neural networks (DNNs) is difficult due to their complex nature. This study focuses on Takeuchi's information criterion (TIC) to investigate the conditions under which this classical measure can effectively explain the generalization gaps of DNNs. Importantly, the developed theory indicates the applicability of TIC near the neural tangent kernel (NTK) regime. In a series of experiments, we trained more than 5,000 DNN models with 12 architectures, including large models (e.g., VGG-16), on four datasets, and estimated the corresponding TIC values to examine the relationship between the generalization gap and the TIC estimates. We applied several TIC approximation methods with feasible computational costs and assessed the accuracy trade-off. Our experimental results indicate that the estimated TIC values correlate well with the generalization gap under conditions close to the NTK regime. However, we show both theoretically and empirically that outside the NTK regime such correlation disappears. Finally, we demonstrate that TIC provides better trial pruning ability than existing methods for hyperparameter optimization.

Operator learning has the potential to strongly impact scientific computing by learning solution operators for differential equations, potentially accelerating multi-query tasks such as design optimization and uncertainty quantification by orders of magnitude. Despite proven universal approximation properties, deep operator networks (DeepONets) often exhibit limited accuracy and generalization in practice, which hinders their adoption. Understanding these limitations is therefore crucial for further advancing the approach. This work analyzes performance limitations of the classical DeepONet architecture. It is shown that the approximation error is dominated by the branch network when the internal dimension is sufficiently large, and that the learned trunk basis can often be replaced by classical basis functions without a significant impact on performance. To investigate this further, a modified DeepONet is constructed in which the trunk network is replaced by the left singular vectors of the training solution matrix. This modification yields several key insights. First, for examples involving the KdV and Burgers equations, a spectral bias in the branch network is observed, with coefficients of dominant, low-frequency modes learned more effectively. Second, through the interplay of the singular-value weighting and the optimizer's neglect of small modes, the branch error is dominated by modes with large and intermediate singular values. Third, using a shared branch network for all mode coefficients, as in the standard architecture, improves generalization of small modes compared to a stacked architecture in which coefficients are computed separately. Finally, detrimental coupling between modes in parameter space is identified, which weakens with increasing network width.

Neural networks are trained by optimizing multi-dimensional sets of fitting parameters on non-convex loss landscapes. Low-loss regions of the landscapes correspond to the parameter sets that perform well on the training data. A key issue in machine learning is the performance of trained neural networks on previously unseen test data. Here, we investigate neural network training by stochastic gradient descent (SGD) - a non-convex global optimization algorithm which relies only on the gradient of the objective function. We contrast SGD solutions with those obtained via a non-stochastic quasi-Newton method, which utilizes curvature information to determine step direction and Golden Section Search to choose step size. We use several computational tools to investigate neural network parameters obtained by these two optimization methods, including kernel Principal Component Analysis and a novel, general-purpose algorithm for finding low-height paths between pairs of points on loss or energy landscapes, FourierPathFinder. We find that the choice of the optimizer profoundly affects the nature of the resulting solutions. SGD solutions tend to be separated by lower barriers than quasi-Newton solutions, even if both sets of solutions are regularized by early stopping to ensure adequate performance on test data. When allowed to fit extensively on the training data, quasi-Newton solutions occupy deeper minima on the loss landscapes that are not reached by SGD. These solutions are less generalizable to the test data however. Overall, SGD explores smooth basins of attraction, while quasi-Newton optimization is capable of finding deeper, more isolated minima that are more spread out in the parameter space. Our findings help understand both the topography of the loss landscapes and the fundamental role of landscape exploration strategies in creating robust, transferrable neural network models.

Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. The dynamical system modeling approach has recently emerged as a powerful framework, offering new mathematical insights into the structure and learning behavior of DNNs. In this work, we establish generalization error bounds for both discrete- and continuous-time residual networks (ResNets) by combining Rademacher complexity, flow maps of dynamical systems, and the convergence behavior of ResNets in the deep-layer limit. The resulting bounds are of order $O(1/\sqrt{S})$ with respect to the number of training samples $S$, and include a structure-dependent negative term, yielding depth-uniform and asymptotic generalization bounds under milder assumptions. These findings provide a unified understanding of generalization across both discrete- and continuous-time ResNets, helping to close the gap in both the order of sample complexity and assumptions between the discrete- and continuous-time settings.

Rehearsal is one of the key techniques for mitigating catastrophic forgetting and has been widely adopted in continual learning algorithms due to its simplicity and practicality. However, the theoretical understanding of how rehearsal scale influences learning dynamics remains limited. To address this gap, we formulate rehearsal-based continual learning as a multidimensional effectiveness-driven iterative optimization problem, providing a unified characterization across diverse performance metrics. Within this framework, we derive a closed-form analysis of adaptability, memorability, and generalization from the perspective of rehearsal scale. Our results uncover several intriguing and counterintuitive findings. First, rehearsal can impair model's adaptability, in sharp contrast to its traditionally recognized benefits. Second, increasing the rehearsal scale does not necessarily improve memory retention. When tasks are similar and noise levels are low, the memory error exhibits a diminishing lower bound. Finally, we validate these insights through numerical simulations and extended analyses on deep neural networks across multiple real-world datasets, revealing statistical patterns of rehearsal mechanisms in continual learning.

Pretraining and fine-tuning are central stages in modern machine learning systems. In practice, feature learning plays an important role across both stages: deep neural networks learn a broad range of useful features during pretraining and further refine those features during fine-tuning. However, an end-to-end theoretical understanding of how choices of initialization impact the ability to reuse and refine features during fine-tuning has remained elusive. Here we develop an analytical theory of the pretraining fine-tuning pipeline in diagonal linear networks, deriving exact expressions for the generalization error as a function of initialization parameters and task statistics. We find that different initialization choices place the network into four distinct fine-tuning regimes that are distinguished by their ability to support feature learning and reuse and therefore by the task statistics for which they are beneficial. In particular, a smaller initialization scale in earlier layers enables the network to both reuse and refine its features, leading to superior generalization on fine-tuning tasks that rely on a subset of pretraining features. We demonstrate empirically that the same initialization parameters impact generalization in ResNets trained on CIFAR-100 and SVHN as well as Transformers trained on modular arithmetic tasks. Overall, our results demonstrate an alytically how data and network initialization interact to shape fine-tuning generalization, highlighting an important role for the relative scale of initialization across different layers in enabling continued feature learning during fine-tuning.