arXiv subjects

Akshunna S. Dogra

Publications and source records attributed to Akshunna S. Dogra.

12 recordsLinked to original sources

arXiv2026

Distance by de-correlation: Computing distance with heterogeneous grid cells

Encoding the distance between locations in space is essential for accurate navigation. Grid cells, a functional class of neurons in medial entorhinal cortex, are believed to support this computation. Inspired by recent work finding populations of grid cells to have small, but robust heterogeneity in their grid properties, we hypothesize that distance coding can be achieved by a simple de-correlation of population activity. We develop a mathematical theory for describing this de-correlation in one-dimension, showing that its predictions are consistent with simulations of noisy grid cells. Our simulations highlight a non-intuitive prediction of such a distance by de-correlation framework. Namely, some further distances are better encoded than some nearer distances. We find preliminary evidence of this ``sweet spot'' in previously published rodent behavioral experiments and demonstrate that a decoder which estimates distance from the de-correlation of populations of simulated noisy grid cells leads to a similar pattern of errors. Finally, we extend our theory to two-dimensions and, by simulating noisy grid cells in two-dimensions, find that there exists a trade-off between the range of distances that can be encoded by de-correlation of population activity and the distinguishability between different distances, which is controlled by the amount of variability in grid properties. We show that the previously measured average amount of grid property variability strikes a balance, enabling the encoding of distances up to several meters. Our work provides new insight on how grid cells can underlie the encoding of distance and why grid cells may have small amounts of heterogeneity in their grid properties.

q-bio.NC↗

arXiv2026

Man, Machine, and Mathematics

Nonlinear models and optimization methods have successfully tackled a rapidly growing set of problems in recent years. Indeed, a relatively small toolbox of such models and methods can provide sufficient performance across a large landscape of tasks: deep learning alone has made significant recent contributions in scientific modelling, natural language processing, visual analysis, etc. A similar relationship exists between physical theories and phenomena, where many applications and observations emerge neatly from remarkably minimal foundations. It is natural to wonder if sparse unified frameworks could be built to steer discussion and discovery in the fields concerned with learning, optimization, and modelling. In this work, we posit and examine a possible outline for such a unified theory, interpreting the notion of ''learning'' in a broad sense. In particular, we pursue our goals by viewing learning as an inter-connected process on multiple levels: problem setup, choosing methods, and the analysis of their interplay via imposed optimisation dynamics. We begin by proposing a precise yet versatile definition for ''solvable'' problems. We then define the ''parametrised methods'' by which their solution(s) may be ''learned''. Our goal is to sketch a ''universal convergence theorem'', specifying how and when solvable problems become amenable to the methods chosen for them. We find these constructions reduce the study of learning down to remarkably few ideas and tools - many of which are simply adapted from existing ones in dynamical systems theory, geometry, and fundamental physics.

math.OC↗

arXiv2025

FINDER: Feature Inference on Noisy Datasets using Eigenspace Residuals

''Noisy'' datasets (regimes with low signal to noise ratios, small sample sizes, faulty data collection, etc) remain a key research frontier for classification methods with both theoretical and practical implications. We introduce FINDER, a rigorous framework for analyzing generic classification problems, with tailored algorithms for noisy datasets. FINDER incorporates fundamental stochastic analysis ideas into the feature learning and inference stages to optimally account for the randomness inherent to all empirical datasets. We construct ''stochastic features'' by first viewing empirical datasets as realizations from an underlying random field (without assumptions on its exact distribution) and then mapping them to appropriate Hilbert spaces. The Kosambi-Karhunen-Loéve expansion (KLE) breaks these stochastic features into computable irreducible components, which allow classification over noisy datasets via an eigen-decomposition: data from different classes resides in distinct regions, identified by analyzing the spectrum of the associated operators. We validate FINDER on several challenging, data-deficient scientific domains, producing state of the art breakthroughs in: (i) Alzheimer's Disease stage classification, (ii) Remote sensing detection of deforestation. We end with a discussion on when FINDER is expected to outperform existing methods, its failure modes, and other limitations.

cs.LG↗

arXiv2023

SHAPER: Can You Hear the Shape of a Jet?

The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover's Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (SHAPER), which is a general framework for defining and computing shape-based observables. SHAPER generalizes N-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the SHAPER framework by performing empirical jet substructure studies using several examples of new shape-based observables.

hep-ph↗

arXiv2022

Universality of Winning Tickets: A Renormalization Group Perspective

Foundational work on the Lottery Ticket Hypothesis has suggested an exciting corollary: winning tickets found in the context of one task can be transferred to similar tasks, possibly even across different architectures. This has generated broad interest, but methods to study this universality are lacking. We make use of renormalization group theory, a powerful tool from theoretical physics, to address this need. We find that iterative magnitude pruning, the principal algorithm used for discovering winning tickets, is a renormalization group scheme, and can be viewed as inducing a flow in parameter space. We demonstrate that ResNet-50 models with transferable winning tickets have flows with common properties, as would be expected from the theory. Similar observations are made for BERT models, with evidence that their flows are near fixed points. Additionally, we leverage our framework to study winning tickets transferred across ResNet architectures, observing that smaller models have flows with more uniform properties than larger models, complicating transfer between them.

cs.LG↗

arXiv2022

Hamiltonian neural networks for solving equations of motion

There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine learning method where the optimization process of the network depends solely on the predicted functions without using any ground truth data. The model learns solutions that satisfy, up to an arbitrarily small error, Hamilton's equations and, therefore, conserve the Hamiltonian invariants. The choice of an appropriate activation function drastically improves the predictability of the network. Moreover, an error analysis is derived and states that the numerical errors depend on the overall network performance. The Hamiltonian network is then employed to solve the equations for the nonlinear oscillator and the chaotic Henon-Heiles dynamical system. In both systems, a symplectic Euler integrator requires two orders more evaluation points than the Hamiltonian network in order to achieve the same order of the numerical error in the predicted phase space trajectories.

physics.comp-ph↗

arXiv2021

Error Estimation and Correction from within Neural Network Differential Equation Solvers

Neural Network Differential Equation (NN DE) solvers have surged in popularity due to a combination of factors: computational advances making their optimization more tractable, their capacity to handle high dimensional problems, easy interpret-ability of their models, etc. However, almost all NN DE solvers suffer from a fundamental limitation: they are trained using loss functions that depend only implicitly on the error associated with the estimate. As such, validation and error analysis of solution estimates requires knowledge of the true solution. Indeed, if the true solution is unknown, we are often reduced to simply hoping that a "low enough" loss implies "small enough" errors, since explicit relationships between the two are not available/well defined. In this work, we describe a general strategy for efficiently constructing error estimates and corrections for Neural Network Differential Equation solvers. Our methods do not require advance knowledge of the true solutions and obtain explicit relationships between loss functions and the error associated with solution estimates. In turn, these explicit relationships directly allow us to estimate and correct for the errors.

math.NA↗

arXiv2021

Local error quantification for Neural Network Differential Equation solvers

Neural networks have been identified as powerful tools for the study of complex systems. A noteworthy example is the neural network differential equation (NN DE) solver, which can provide functional approximations to the solutions of a wide variety of differential equations. Such solvers produce robust functional expressions, are well suited for further manipulations on the quantities of interest (for example, taking derivatives), and capable of leveraging the modern advances in parallelization and computing power. However, there is a lack of work on the role precise error quantification can play in their predictions: usually, the focus is on ambiguous and/or global measures of performance like the loss function and/or obtaining global bounds on the errors associated with the predictions. Precise, local error quantification is seldom possible without external means or outright knowledge of the true solution. We address these concerns in the context of dynamical system NN DE solvers, leveraging learnt information within the NN DE solvers to develop methods that allow them to be more accurate and efficient, while still pursuing an unsupervised approach that does not rely on external tools or data. We achieve this via methods that can precisely estimate NN DE solver prediction errors point-wise, thus allowing the user the capacity for efficient and targeted error correction. We exemplify the utility of our methods by testing them on a nonlinear and a chaotic system each.

cs.LG↗

arXiv2020

Optimizing Neural Networks via Koopman Operator Theory

Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in making use of this connection. As Koopman operator theory is a linear theory, a successful implementation of it in evolving network weights and biases offers the promise of accelerated training, especially in the context of deep networks, where optimization is inherently a non-convex problem. We show that Koopman operator theoretic methods allow for accurate predictions of weights and biases of feedforward, fully connected deep networks over a non-trivial range of training time. During this window, we find that our approach is >10x faster than various gradient descent based methods (e.g. Adam, Adadelta, Adagrad), in line with our complexity analysis. We end by highlighting open questions in this exciting intersection between dynamical systems and neural network theory. We highlight additional methods by which our results could be expanded to broader classes of networks and larger training intervals, which shall be the focus of future work.

cs.NE↗

arXiv2020

Impact of ionizing radiation on superconducting qubit coherence

The practical viability of any qubit technology stands on long coherence times and high-fidelity operations, with the superconducting qubit modality being a leading example. However, superconducting qubit coherence is impacted by broken Cooper pairs, referred to as quasiparticles, with a density that is empirically observed to be orders of magnitude greater than the value predicted for thermal equilibrium by the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. Previous work has shown that infrared photons significantly increase the quasiparticle density, yet even in the best isolated systems, it still remains higher than expected, suggesting that another generation mechanism exists. In this Letter, we provide evidence that ionizing radiation from environmental radioactive materials and cosmic rays contributes to this observed difference, leading to an elevated quasiparticle density that would ultimately limit superconducting qubits of the type measured here to coherence times in the millisecond regime. We further demonstrate that introducing radiation shielding reduces the flux of ionizing radiation and positively correlates with increased coherence time. Albeit a small effect for today's qubits, reducing or otherwise mitigating the impact of ionizing radiation will be critical for realizing fault-tolerant superconducting quantum computers.

quant-ph↗

arXiv2020

Dynamical Systems and Neural Networks

Neural Networks (NNs) have been identified as a potentially powerful tool in the study of complex dynamical systems. A good example is the NN differential equation (DE) solver, which provides closed form, differentiable, functional approximations for the evolution of a wide variety of dynamical systems. A major disadvantage of such NN solvers can be the amount of computational resources needed to achieve accuracy comparable to existing numerical solvers. We present new strategies for existing dynamical system NN DE solvers, making efficient use of the \textit{learnt} information, to speed up their training process, while still pursuing a completely unsupervised approach. We establish a fundamental connection between NN theory and dynamical systems theory via Koopman Operator Theory (KOT), by showing that the usual training processes for Neural Nets are fertile ground for identifying multiple Koopman operators of interest. We end by illuminating certain applications that KOT might have for NNs in general.

math.DS↗

arXiv2019

Geometry of transient chaos in streamwise-localized pipe flow turbulence

In pipes and channels, the onset of turbulence is initially dominated by localized transients, which lead to sustained turbulence through their collective dynamics. In the present work, we study the localized turbulence in pipe flow numerically and elucidate a state space structure that gives rise to transient chaos. Starting from the basin boundary separating laminar and turbulent flow, we identify transverse homoclinic orbits, the presence of which necessitates a homoclinic tangle and chaos. A direct consequence of the homoclinic tangle is the fractal nature of the laminar-turbulent boundary, which was conjectured in various earlier studies. By mapping the transverse intersections between the stable and unstable manifold of a periodic orbit, we identify the 'gateways' that promote an escape from turbulence.

physics.flu-dyn↗