arXiv subjects

Nisha Chandramoorthy

Publications and source records attributed to Nisha Chandramoorthy.

At least 19 recordsLinked to original sources

arXiv2026

Diffusion models recover accurate mixture weights despite score function insensitivity

Score-based generative models exhibit a puzzling behavior: they often appear to cover all modes of a target multimodal distribution and yet may fail to learn the correct relative mode amplitudes, which can be interpreted as mixture weights. We resolve this apparent paradox by relating the diffusion score matching (DSM) loss to the error in estimating mixture weights from generated samples. We show that, even when the target score is insensitive to mixture weights, generated samples can recover the weights accurately if the scores at intermediate noise levels are informative about the weights. Accordingly, we define the diffusion score sensitivity index (DSSI) as the variation in the DSM loss relative to changes in a parameter. We then show that the DSSI governs the accuracy with which the parameter of the target distribution can be estimated from generated samples. For Gaussian mixtures in arbitrary dimensions, we prove that the mixture weight estimation errors are on the same order as the DSM loss under mild conditions. Empirically, we show the emergence of sensitivity during the noising process of benchmark data distributions under typical noise schedules, and that these sensitivity values predict how well a well-trained model recovers mixture weights. Furthermore, we show that the choice of noise schedule can reduce diffusion sensitivity, leading to mode amplification. Although we focus on mixture weights, the proposed sensitivity framework governs the recovery of any qualitative parameter of the target distribution.

cs.LG↗

arXiv2026

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

stat.ML↗

arXiv2026

On the Robustness of Distribution Support under Diffusion Guidance

Diffusion guidance is a powerful technique that enables controllable and high-fidelity sample generation with diffusion models. At a high level, it modifies the score function by incorporating a guidance term that steers the generative process toward a desired condition. Despite its empirical success, the theoretical properties of diffusion guidance remain largely unexplored, and it is not well understood why it consistently produces high-quality samples. In this work, we explain the effectiveness of diffusion guidance by establishing a robustness of support property. Specifically, we show that, given exact access to the score functions, guided diffusion processes almost always generate samples that remain close to the target support. This property is particularly desirable, as samples that lie off the support are often structurally implausible and may adversely affect downstream tasks. Our analysis covers both Denoising Diffusion Implicit Models (DDIM) and Denoising Diffusion Probabilistic Models (DDPM), and applies to a wide range of discretization schemes induced by exponential integrators. Our results provide a rigorous foundation for understanding why diffusion guidance produces physically meaningful and structurally plausible samples.

cs.LG↗

arXiv2026

Flow Matching from Viewpoint of Proximal Operators

We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), a class of dynamical generative models, showing that it admits an exact proximal formulation via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is exactly given by a proximal operator, which yields an explicit proximal expression of the vector field. We also discuss the convergence of minibatch OT-CFM to the population formulation as the batch size increases. Finally, using second epi-derivatives of convex potentials, we prove that, for manifold-supported targets, OT-CFM is terminally normally hyperbolic: after time rescaling, the dynamics contracts exponentially in directions normal to the data manifold while remaining neutral along tangential directions.

cs.LG↗

arXiv2025

When and how can inexact generative models still sample from the data manifold?

A curious phenomenon observed in some dynamical generative models is the following: despite learning errors in the score function or the drift vector field, the generated samples appear to shift \emph{along} the support of the data distribution but not \emph{away} from it. In this work, we investigate this phenomenon of \emph{robustness of the support} by taking a dynamical systems approach on the generating stochastic/deterministic process. Our perturbation analysis of the probability flow reveals that infinitesimal learning errors cause the predicted density to be different from the target density only on the data manifold for a wide class of generative models. Further, what is the dynamical mechanism that leads to the robustness of the support? We show that the alignment of the top Lyapunov vectors (most sensitive infinitesimal perturbation directions) with the tangent spaces along the boundary of the data manifold leads to robustness and prove a sufficient condition on the dynamics of the generating process to achieve this alignment. Moreover, the alignment condition is efficient to compute and, in practice, for robust generative models, automatically leads to accurate estimates of the tangent bundle of the data manifold. Using a finite-time linear perturbation analysis on samples paths as well as probability flows, our work complements and extends existing works on obtaining theoretical guarantees for generative models from a stochastic analysis, statistical learning and uncertainty quantification points of view. Our results apply across different dynamical generative models, such as conditional flow-matching and score-based generative models, and for different target distributions that may or may not satisfy the manifold hypothesis.

cs.LG↗

arXiv2025

When are dynamical systems learned from time series data statistically accurate?

Conventional notions of generalization often fail to describe the ability of learned models to capture meaningful information from dynamical data. A neural network that learns complex dynamics with a small test error may still fail to reproduce its \emph{physical} behavior, including associated statistical moments and Lyapunov exponents. To address this gap, we propose an ergodic theoretic approach to generalization of complex dynamical models learned from time series data. Our main contribution is to define and analyze generalization of a broad suite of neural representations of classes of ergodic systems, including chaotic systems, in a way that captures emulating underlying invariant, physical measures. Our results provide theoretical justification for why regression methods for generators of dynamical systems (Neural ODEs) fail to generalize, and why their statistical accuracy improves upon adding Jacobian information during training. We verify our results on a number of ergodic chaotic systems and neural network parameterizations, including MLPs, ResNets, Fourier Neural layers, and RNNs.

cs.LG↗

arXiv2024

Score Operator Newton transport

We propose a new approach for sampling and Bayesian computation that uses the score of the target distribution to construct a transport from a given reference distribution to the target. Our approach is an infinite-dimensional Newton method, involving a linear PDE, for finding a zero of a ``score-residual'' operator. We prove sufficient conditions for convergence to a valid transport map. Our Newton iterates can be computed by exploiting fast solvers for elliptic PDEs, resulting in new algorithms for Bayesian inference and other sampling tasks. We identify elementary settings where score-operator Newton transport achieves fast convergence while avoiding mode collapse.

math.ST↗

arXiv2023

Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems

Ruelle gave a formula for linear response of transitive Anosov diffeomorphisms. Recently, practically computable realizations of Ruelle's formula have emerged that potentially enable sensitivity analysis of certain high-dimensional chaotic numerical simulations encountered in the applied sciences. In this paper, we provide full mathematical justification for the convergence of one such efficient computation, the space-split sensitivity, or S3, algorithm. In S3, Ruelle's formula is computed as a sum of two terms obtained by decomposing the perturbation vector field into a coboundary and a remainder that is parallel to the unstable direction. Such a decomposition results in a splitting of Ruelle's formula that is amenable to efficient computation. We prove the existence of the S3 decomposition and the convergence of the computations of both resulting components of Ruelle's formula.

math.DS↗

arXiv2022

On the generalization of learning algorithms that do not converge

Generalization analyses of deep learning typically assume that the training converges to a fixed point. But, recent results indicate that in practice, the weights of deep neural networks optimized with stochastic gradient descent often oscillate indefinitely. To reduce this discrepancy between theory and practice, this paper focuses on the generalization of neural networks whose training dynamics do not necessarily converge to fixed points. Our main contribution is to propose a notion of statistical algorithmic stability (SAS) that extends classical algorithmic stability to non-convergent algorithms and to study its connection to generalization. This ergodic-theoretic approach leads to new insights when compared to the traditional optimization and learning theory perspectives. We prove that the stability of the time-asymptotic behavior of a learning algorithm relates to its generalization and empirically demonstrate how loss dynamics can provide clues to generalization performance. Our findings provide evidence that networks that "train stably generalize better" even when the training continues indefinitely and the weights do not converge.

cs.LG↗

arXiv2022

Efficient computation of linear response of chaotic attractors with one-dimensional unstable manifolds

This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similar to in a non-chaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficiently computable ergodic average. In this process, we develop new algorithms, which may be useful beyond linear response, to compute i) a fundamental statistical quantity we introduce called the density gradient, and ii) the unstable derivatives of the regularized tangent vector field and the unstable direction. We prove that the S3 algorithm, which combines these computational ingredients that enter the stable and unstable contribution, converges like a Monte Carlo approximation of Ruelle's formula. The algorithm presented here is hence a first step toward full-fledged applications of sensitivity analysis in chaotic systems, wherever such applications have been limited due to lack of availability of long-term sensitivities.

math.DS↗

arXiv2021

An ergodic averaging method to differentiate covariant Lyapunov vectors

Covariant Lyapunov vectors or CLVs span the expanding and contracting directions of perturbations along trajectories in a chaotic dynamical system. Due to efficient algorithms to compute them that only utilize trajectory information, they have been widely applied across scientific disciplines, principally for sensitivity analysis and predictions under uncertainty. In this paper, we develop a numerical method to compute the directional derivatives of CLVs along their own directions. Similar to the computation of CLVs, the present method for their derivatives is iterative and analogously uses the second-order derivative of the chaotic map along trajectories, in addition to the Jacobian. We validate the new method on a super-contracting Smale-Williams Solenoid attractor. We also demonstrate the algorithm on several other examples including smoothly perturbed Arnold Cat maps, and the Lorenz attractor, obtaining visualizations of the curvature of each attractor. Furthermore, we reveal a fundamental connection of the CLV self-derivatives with a statistical linear response formula.

nlin.CD↗

arXiv2021

Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems

An assumption of smooth response to small parameter changes, of statistics or long-time averages of a chaotic system, is generally made in the field of sensitivity analysis, and the parametric derivatives of statistical quantities are critically used in science and engineering. In this paper, we propose a numerical procedure to assess the differentiability of statistics with respect to parameters in chaotic systems. We numerically show that the existence of the derivative depends on the Lebesgue-integrability of a certain density gradient function, which we define as the derivative of logarithmic SRB density along the unstable manifold. We develop a recursive formula for the density gradient that can be efficiently computed along trajectories, and demonstrate its use in determining the differentiability of statistics. Our numerical procedure is illustrated on low-dimensional chaotic systems whose statistics exhibit both smooth and rough regions in parameter space.

nlin.CD↗

arXiv2020

On the probability of finding a nonphysical solution through shadowing

This paper proves that shadowing solutions can be almost surely nonphysical. This finding invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior. This theoretical finding has implications for many applications in which chaotic mechanics plays an important role. It suggests, for example, that we can control the climate through subtle perturbations. It also suggests that numerical simulations of chaotic dynamics, such as turbulent flows, may fail to predict the true long-term or statistical behavior.

nlin.CD↗

arXiv2020

Variational optimization and data assimilation in chaotic time-delayed systems with automatic-differentiated shadowing sensitivity

In this computational paper, we perform sensitivity analysis of long-time (or ensemble) averages in the chaotic regime using the shadowing algorithm. We introduce automatic differentiation to eliminate the tangent/adjoint equation solvers used in the shadowing algorithm. In a gradient-based optimization, we use the computed shadowing sensitivity to minimize different long-time averaged functionals of a chaotic time-delayed system by optimal parameter selection. In combined state and parameter estimation for data assimilation, we use the computed sensitivity to predict the optimal trajectory given information from a model and data from measurements beyond the predictability time. The algorithms are applied to a thermoacoustic model. Because the computational framework is rather general, the techniques presented in this paper may be used for sensitivity analysis of ensemble averages, parameter optimization and data assimilation of other chaotic problems, where shadowing methods are applicable.

math.DS↗

arXiv2020

Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive explanation of the key components of the S3 algorithm, including the density gradient function.

nlin.CD↗

arXiv2020

A computable realization of Ruelle's formula for linear response of statistics in chaotic systems

We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points. The reformulation is based on splitting the overall sensitivity into that to stable and unstable components of the perturbation. The unstable contribution to the sensitivity is regularized using ergodic properties and the hyperbolic structure of the dynamics. Numerical examples of uniformly hyperbolic attractors are used to validate the S3 formula against a naïve finite-difference calculation; sensitivities match closely, with far fewer sample points required by S3.

math.DS↗

arXiv2019

Feasibility analysis of ensemble sensitivity computation in turbulent flows

In chaotic systems, such as turbulent flows, the solutions to tangent and adjoint equations exhibit an unbounded growth in their norms. This behavior renders the instantaneous tangent and adjoint solutions unusable for sensitivity analysis. The Lea-Allen-Haine ensemble sensitivity (ES) estimates provide a way of computing meaningful sensitivities in chaotic systems by utilizing tangent/adjoint solutions over short trajectories. In this paper, we analyze the feasibility of ES computations under optimistic mathematical assumptions on the flow dynamics. Furthermore, we estimate upper bounds on the rate of convergence of the ES method in numerical simulations of turbulent flow. Even at the optimistic upper bound, the ES method is computationally intractable in each of the numerical examples considered.

physics.comp-ph↗

arXiv2019

Sensitivity computation of statistically stationary quantities in turbulent flows

It is well-known that linearized perturbation methods for sensitivity analysis, such as tangent or adjoint equation-based, finite difference and automatic differentiation are not suitable for turbulent flows. The reason is that turbulent flows exhibit chaotic dynamics, leading to the norm of an infinitesimal perturbation to the state growing exponentially in time. As a result, these conventional methods cannot be used to compute the derivatives of long-time averaged quantities to control or design inputs. The ensemble-based approaches and shadowing-based approaches to circumvent the problems of the conventional methods in chaotic systems, also suffer from computational impracticality and lack of consistency guarantees, respectively. We introduce the space-split sensitivity, or the S3 algorithm, which is a Monte-Carlo approach to the chaotic sensitivity computation problem. In this work, we derive the S3 algorithm under simplifying assumptions on the dynamics and present a numerical validation on a low-dimensional example of chaos.

nlin.CD↗