arXiv subjects

Zakhar Shumaylov

Publications and source records attributed to Zakhar Shumaylov.

At least 19 recordsLinked to original sources

arXiv2026

When is a System Discoverable from Data? Discovery Requires Chaos

The deep learning revolution has spurred a rise in advances of using AI in sciences. Within physical sciences the main focus has been on discovery of dynamical systems from observational data. Yet the reliability of learned surrogates and symbolic models is often undermined by the fundamental problem of non-uniqueness. The resulting models may fit the available data perfectly, but lack genuine predictive power. This raises the question: under what conditions can the systems governing equations be uniquely identified from a finite set of observations? We show, counter-intuitively, that chaos, typically associated with unpredictability, is crucial for ensuring a system is discoverable in the space of continuous or analytic functions. The prevalence of chaotic systems in benchmark datasets may have inadvertently obscured this fundamental limitation. More concretely, we show that systems chaotic on their entire domain are discoverable from a single trajectory within the space of continuous functions, and systems chaotic on a strange attractor are analytically discoverable under a geometric condition on the attractor. As a consequence, we demonstrate for the first time that the classical Lorenz system is analytically discoverable. Moreover, we establish that analytic discoverability is impossible in the presence of first integrals, common in real-world systems. These findings help explain the success of data-driven methods in inherently chaotic domains like weather forecasting, while revealing a significant challenge for engineering applications like digital twins, where stable, predictable behavior is desired. For these non-chaotic systems, we find that while trajectory data alone is insufficient, certain prior physical knowledge can help ensure discoverability. These findings warrant a critical re-evaluation of the fundamental assumptions underpinning purely data-driven discovery.

math.DS↗

arXiv2026

Muon is Not That Special: Random or Inverted Spectra Work Just as Well

The recent empirical success of the Muon optimizer has renewed interest in non-Euclidean optimization, typically justified by similarities with second-order methods, and linear minimization oracle (LMO) theory. In this paper, we challenge this geometric narrative through three contributions, demonstrating that precise geometric structure is not the key factor affecting optimization performance. First, we introduce Freon, a family of optimizers based on Schatten (quasi-)norms, powered by a novel, provably optimal QDWH-based iterative approximation. Freon naturally interpolates between SGD and Muon, while smoothly extrapolating into the quasi-norm regime. Empirically, the best-performing Schatten parameters for GPT-2 lie strictly within the quasi-norm regime, and thus cannot be represented by any unitarily invariant LMO. Second, noting that Freon performs well across a wide range of exponents, we introduce Kaon, an absurd optimizer that replaces singular values with random noise. Despite lacking any coherent geometric structure, Kaon matches Muon's performance and retains classical convergence guarantees, proving that strict adherence to a precise geometry is practically irrelevant. Third, having shown that geometry is not the primary driver of performance, we demonstrate it is instead controlled by two local quantities: alignment and descent potential. Ultimately, each optimizer must tune its step size around these two quantities. While their dynamics are difficult to predict a-priori, evaluating them within a stochastic random feature model yields a precise insight: Muon succeeds not by tracking an ideal global geometry, but by guaranteeing step-size optimality.

cs.LG↗

arXiv2026

Adaptive Coordinate Transforms for Neural Operators

Neural operators have achieved promising performance on partial differential equations (PDEs), but most existing models are built on fixed Eulerian coordinates. This mismatch between evolving physical structures and static coordinates creates spatial misalignment, leading to unnecessarily non-local operator mappings and reinforcing a smoothness preference near sharp transitions. Inspired by adaptive coordinate transformations in classical PDE analysis, we propose the Adaptive Coordinate Transform (ACT) block, a plug-and-play module for data-driven geometric adaptation in neural operators. ACT blocks resolve this structural limitation by learning adaptive coordinate systems within the operator learning pipeline. Specifically, given an input feature, the ACT block learns a coordinate transformation and represents the same feature under the transformed coordinates via differentiable sampling. This operation preserves the underlying signal while changing its spatial representation, equivalent to expressing the same physical quantity in different coordinate systems. By adapting the coordinate system to the data, ACT allows the network to better track evolving structures, reduce operator complexity, and dynamically focus on critical features to improve learning. We evaluate the proposed approach across diverse PDE benchmarks and multiple neural operator architectures. Experimental results demonstrate consistent and significant improvements in predictive accuracy, indicating that learning coordinate systems provides a powerful mechanism for enhancing operator learning.

cs.CE↗

arXiv2026

Symplectic Neural Flows for Modeling and Discovery

Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose, but neural network-based methods that incorporate these principles remain underexplored. This work introduces SympFlow, a time-dependent symplectic neural network designed using parameterized Hamiltonian flow maps. This design allows for backward error analysis and ensures the preservation of the symplectic structure. SympFlow allows for two key applications: (i) providing a time-continuous symplectic approximation of the exact flow of a Hamiltonian system purely based on the differential equations it satisfies, and (ii) approximating the flow map of an unknown Hamiltonian system relying on trajectory data. We demonstrate the effectiveness of SympFlow on diverse problems, including chaotic and dissipative systems, showing improved energy conservation compared to general-purpose numerical methods and accurate approximations from sparse irregular data. We also provide a thorough theoretical analysis of SympFlow, showing it can approximate the flow of any time-dependent Hamiltonian system, and providing an a-posteriori error estimate in terms of energy conservation.

cs.LG↗

arXiv2026

Diffeomorphism-Equivariant Neural Networks

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for overcoming the efficiency and data demands of modern deep learning. While most existing approaches, such as group convolutions and averaging-based methods, focus on compact, finite, or low-dimensional groups with linear actions, this work explores how equivariance can be extended to infinite-dimensional groups. We propose a strategy designed to induce diffeomorphism equivariance in pre-trained neural networks via energy-based canonicalisation. Formulating equivariance as an optimisation problem allows us to access the rich toolbox of already established differentiable image registration methods. Empirical results on segmentation and classification tasks confirm that our approach achieves approximate equivariance and generalises to unseen transformations without relying on extensive data augmentation or retraining.

cs.LG↗

arXiv2026

Learning Regularization Functionals for Inverse Problems: A Comparative Study

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

cs.LG↗

arXiv2025

On Information Geometry and Iterative Optimization in Model Compression: Operator Factorization

The ever-increasing parameter counts of deep learning models necessitate effective compression techniques for deployment on resource-constrained devices. This paper explores the application of information geometry, the study of density-induced metrics on parameter spaces, to analyze existing methods within the space of model compression, primarily focusing on operator factorization. Adopting this perspective highlights the core challenge: defining an optimal low-compute submanifold (or subset) and projecting onto it. We argue that many successful model compression approaches can be understood as implicitly approximating information divergences for this projection. We highlight that when compressing a pre-trained model, using information divergences is paramount for achieving improved zero-shot accuracy, yet this may no longer be the case when the model is fine-tuned. In such scenarios, trainability of bottlenecked models turns out to be far more important for achieving high compression ratios with minimal performance degradation, necessitating adoption of iterative methods. In this context, we prove convergence of iterative singular value thresholding for training neural networks subject to a soft rank constraint. To further illustrate the utility of this perspective, we showcase how simple modifications to existing methods through softer rank reduction result in improved performance under fixed compression rates.

cs.LG↗

arXiv2025

Data-driven approaches to inverse problems

Inverse problems are concerned with the reconstruction of unknown physical quantities using indirect measurements and are fundamental across diverse fields such as medical imaging, remote sensing, and material sciences. These problems serve as critical tools for visualizing internal structures beyond what is visible to the naked eye, enabling quantification, diagnosis, prediction, and discovery. However, most inverse problems are ill-posed, necessitating robust mathematical treatment to yield meaningful solutions. While classical approaches provide mathematically rigorous and computationally stable solutions, they are constrained by the ability to accurately model solution properties and implement them efficiently. A more recent paradigm considers deriving solutions to inverse problems in a data-driven manner. Instead of relying on classical mathematical modeling, this approach utilizes highly over-parameterized models, typically deep neural networks, which are adapted to specific inverse problems using carefully selected training data. Current approaches that follow this new paradigm distinguish themselves through solution accuracy paired with computational efficiency that was previously inconceivable. These notes offer an introduction to this data-driven paradigm for inverse problems. The first part of these notes will provide an introduction to inverse problems, discuss classical solution strategies, and present some applications. The second part will delve into modern data-driven approaches, with a particular focus on adversarial regularization and provably convergent linear plug-and-play denoisers. Throughout the presentation of these methodologies, their theoretical properties will be discussed, and numerical examples will be provided. The lecture series will conclude with a discussion of open problems and future perspectives in the field.

math.NA↗

arXiv2025

Generalized Lie Symmetries in Physics-Informed Neural Operators

Physics-informed neural operators (PINOs) have emerged as powerful tools for learning solution operators of partial differential equations (PDEs). Recent research has demonstrated that incorporating Lie point symmetry information can significantly enhance the training efficiency of PINOs, primarily through techniques like data, architecture, and loss augmentation. In this work, we focus on the latter, highlighting that point symmetries oftentimes result in no training signal, limiting their effectiveness in many problems. To address this, we propose a novel loss augmentation strategy that leverages evolutionary representatives of point symmetries, a specific class of generalized symmetries of the underlying PDE. These generalized symmetries provide a richer set of generators compared to standard symmetries, leading to a more informative training signal. We demonstrate that leveraging evolutionary representatives enhances the performance of neural operators, resulting in improved data efficiency and accuracy during training.

cs.LG↗

arXiv2025

Score-based Pullback Riemannian Geometry: Extracting the Data Manifold Geometry using Anisotropic Flows

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on diverse datasets, including image data, we demonstrate that the proposed framework produces high-quality geodesics passing through the data support, reliably estimates the intrinsic dimension of the data manifold, and provides a global chart of the manifold. To the best of our knowledge, this is the first scalable framework for extracting the complete geometry of the data manifold.

cs.LG↗

arXiv2025

Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups

The quest for robust and generalizable machine learning models has driven recent interest in exploiting symmetries through equivariant neural networks. In the context of PDE solvers, recent works have shown that Lie point symmetries can be a useful inductive bias for Physics-Informed Neural Networks (PINNs) through data and loss augmentation. Despite this, directly enforcing equivariance within the model architecture for these problems remains elusive. This is because many PDEs admit non-compact symmetry groups, oftentimes not studied beyond their infinitesimal generators, making them incompatible with most existing equivariant architectures. In this work, we propose Lie aLgebrA Canonicalization (LieLAC), a novel approach that exploits only the action of infinitesimal generators of the symmetry group, circumventing the need for knowledge of the full group structure. To achieve this, we address existing theoretical issues in the canonicalization literature, establishing connections with frame averaging in the case of continuous non-compact groups. Operating within the framework of canonicalization, LieLAC can easily be integrated with unconstrained pre-trained models, transforming inputs to a canonical form before feeding them into the existing model, effectively aligning the input for model inference according to allowed symmetries. LieLAC utilizes standard Lie group descent schemes, achieving equivariance in pre-trained models. Finally, we showcase LieLAC's efficacy on tasks of invariant image classification and Lie point symmetry equivariant neural PDE solvers using pre-trained models.

cs.LG↗

arXiv2024

Benchmarking learned algorithms for computed tomography image reconstruction tasks

Computed tomography (CT) is a widely used non-invasive diagnostic method in various fields, and recent advances in deep learning have led to significant progress in CT image reconstruction. However, the lack of large-scale, open-access datasets has hindered the comparison of different types of learned methods. To address this gap, we use the 2DeteCT dataset, a real-world experimental computed tomography dataset, for benchmarking machine learning based CT image reconstruction algorithms. We categorize these methods into post-processing networks, learned/unrolled iterative methods, learned regularizer methods, and plug-and-play methods, and provide a pipeline for easy implementation and evaluation. Using key performance metrics, including SSIM and PSNR, our benchmarking results showcase the effectiveness of various algorithms on tasks such as full data reconstruction, limited-angle reconstruction, sparse-angle reconstruction, low-dose reconstruction, and beam-hardening corrected reconstruction. With this benchmarking study, we provide an evaluation of a range of algorithms representative for different categories of learned reconstruction methods on a recently published dataset of real-world experimental CT measurements. The reproducible setup of methods and CT image reconstruction tasks in an open-source toolbox enables straightforward addition and comparison of new methods later on. The toolbox also provides the option to load the 2DeteCT dataset differently for extensions to other problems and different CT reconstruction tasks.

eess.IV↗

arXiv2024

Hamiltonian Matching for Symplectic Neural Integrators

Hamilton's equations of motion form a fundamental framework in various branches of physics, including astronomy, quantum mechanics, particle physics, and climate science. Classical numerical solvers are typically employed to compute the time evolution of these systems. However, when the system spans multiple spatial and temporal scales numerical errors can accumulate, leading to reduced accuracy. To address the challenges of evolving such systems over long timescales, we propose SympFlow, a novel neural network-based symplectic integrator, which is the composition of a sequence of exact flow maps of parametrised time-dependent Hamiltonian functions. This architecture allows for a backward error analysis: we can identify an underlying Hamiltonian function of the architecture and use it to define a Hamiltonian matching objective function, which we use for training. In numerical experiments, we show that SympFlow exhibits promising results, with qualitative energy conservation behaviour similar to that of time-stepping symplectic integrators.

cs.LG↗

arXiv2024

Quantum initial conditions for curved inflating universes

We discuss the challenges of motivating, constructing, and quantizing a canonically normalized inflationary perturbation in spatially curved universes. We show that this has historically proved challenging due to the interaction of nonadiabaticity with spatial curvature. We construct a novel curvature perturbation that is canonically normalized in the sense of its equation of motion and is unique up to a single scalar parameter. With this construction it becomes possible to set initial conditions invariant under canonical transformations, overcoming known ambiguities in the literature. This corrected quantization has potentially observational consequences via modifications to the primordial power spectrum at large angular scales, as well as theoretical implications for quantization procedures in curved cosmologies filled with a scalar field.

gr-qc↗

arXiv2024

Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation

Variational regularisation is the primary method for solving inverse problems, and recently there has been considerable work leveraging deeply learned regularisation for enhanced performance. However, few results exist addressing the convergence of such regularisation, particularly within the context of critical points as opposed to global minimisers. In this paper, we present a generalised formulation of convergent regularisation in terms of critical points, and show that this is achieved by a class of weakly convex regularisers. We prove convergence of the primal-dual hybrid gradient method for the associated variational problem, and, given a Kurdyka-Lojasiewicz condition, an $\mathcal{O}(\log{k}/k)$ ergodic convergence rate. Finally, applying this theory to learned regularisation, we prove universal approximation for input weakly convex neural networks (IWCNN), and show empirically that IWCNNs can lead to improved performance of learned adversarial regularisers for computed tomography (CT) reconstruction.

math.OC↗

arXiv2024

The Curse of Recursion: Training on Generated Data Makes Models Forget

Stable Diffusion revolutionised image creation from descriptive text. GPT-2, GPT-3(.5) and GPT-4 demonstrated astonishing performance across a variety of language tasks. ChatGPT introduced such language models to the general public. It is now clear that large language models (LLMs) are here to stay, and will bring about drastic change in the whole ecosystem of online text and images. In this paper we consider what the future might hold. What will happen to GPT-{n} once LLMs contribute much of the language found online? We find that use of model-generated content in training causes irreversible defects in the resulting models, where tails of the original content distribution disappear. We refer to this effect as Model Collapse and show that it can occur in Variational Autoencoders, Gaussian Mixture Models and LLMs. We build theoretical intuition behind the phenomenon and portray its ubiquity amongst all learned generative models. We demonstrate that it has to be taken seriously if we are to sustain the benefits of training from large-scale data scraped from the web. Indeed, the value of data collected about genuine human interactions with systems will be increasingly valuable in the presence of content generated by LLMs in data crawled from the Internet.

cs.LG↗

arXiv2023

Provably Convergent Data-Driven Convex-Nonconvex Regularization

An emerging new paradigm for solving inverse problems is via the use of deep learning to learn a regularizer from data. This leads to high-quality results, but often at the cost of provable guarantees. In this work, we show how well-posedness and convergent regularization arises within the convex-nonconvex (CNC) framework for inverse problems. We introduce a novel input weakly convex neural network (IWCNN) construction to adapt the method of learned adversarial regularization to the CNC framework. Empirically we show that our method overcomes numerical issues of previous adversarial methods.

cs.LG↗

arXiv2022

Primordial power spectra from $k$-inflation with curvature

We investigate the primordial power spectra for general kinetic inflation models that support a period of kinetic dominance in the case of curved universes. We present derivations of the Mukhanov-Sasaki equations with a nonstandard scalar kinetic Lagrangian which manifests itself through the inflationary sound speed $c_s^2$. We extend the analytical approximations exploited in Contaldi et al [1] and Thavanesan et al [2] to general kinetic Lagrangians and show the effect of $k$-inflation on the primordial power spectra for models with curvature. In particular, the interplay between sound speed and curvature results in a natural low wavenumber cutoff for the power spectra in the case of closed universes. Using the analytical approximation, we further show that a change in the inflationary sound speed between different epochs of inflation results in non-decaying oscillations in the resultant power spectra for the comoving curvature perturbation.

astro-ph.CO↗