arXiv subjects

Assad A Oberai

Publications and source records attributed to Assad A Oberai.

7 recordsLinked to original sources

arXiv2025

Time-dependent density estimation using binary classifiers

We propose a data-driven method to learn the time-dependent probability density of a multivariate stochastic process from sample paths, assuming that the initial probability density is known and can be evaluated. Our method uses a novel time-dependent binary classifier trained using a contrastive estimation-based objective that trains the classifier to discriminate between realizations of the stochastic process at two nearby time instants. Significantly, the proposed method explicitly models the time-dependent probability distribution, which means that it is possible to obtain the value of the probability density within the time horizon of interest. Additionally, the input before the final activation in the time-dependent classifier is a second-order approximation to the partial derivative, with respect to time, of the logarithm of the density. We apply the proposed approach to approximate the time-dependent probability density functions for systems driven by stochastic excitations. We also use the proposed approach to synthesize new samples of a random vector from a given set of its realizations. In such applications, we generate sample paths necessary for training using stochastic interpolants. Subsequently, new samples are generated using gradient-based Markov chain Monte Carlo methods because automatic differentiation can efficiently provide the necessary gradient. Further, we demonstrate the utility of an explicit approximation to the time-dependent probability density function through applications in unsupervised outlier detection. Through several numerical experiments, we show that the proposed method accurately reconstructs complex time-dependent, multi-modal, and near-degenerate densities, scales effectively to moderately high-dimensional problems, and reliably detects rare events among real-world data.

stat.ML↗

arXiv2025

Unifying and extending Diffusion Models through PDEs for solving Inverse Problems

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these models have been derived using principles of variational inference, denoising, statistical signal processing, and stochastic differential equations. In contrast to the conventional presentation, in this study we derive diffusion models using ideas from linear partial differential equations and demonstrate that this approach has several benefits that include a constructive derivation of the forward and reverse processes, a unified derivation of multiple formulations and sampling strategies, and the discovery of a new class of variance preserving models. We also apply the conditional version of these models to solve canonical conditional density estimation problems and challenging inverse problems. These problems help establish benchmarks for systematically quantifying the performance of different formulations and sampling strategies in this study and for future studies. Finally, we identify and implement a mechanism through which a single diffusion model can be applied to measurements obtained from multiple measurement operators. Taken together, the contents of this manuscript provide a new understanding of and several new directions in the application of diffusion models to solving physics-based inverse problems.

cs.LG↗

arXiv2023

A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows

We propose a novel modular inference approach combining two different generative models -- generative adversarial networks (GAN) and normalizing flows -- to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows.

cs.CE↗

arXiv2022

Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called "curse of dimensionality") and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data.

stat.ML↗

arXiv2021

The Effect of Super-spreader Events in Epidemics

The spread of infectious epidemics is often accelerated by super-spreader events. Understanding their effect is important, particularly in the context of standard epidemiological models, which require estimates for parameters such as $R_0$. In this letter, we show that the effective value of $R_0$ in super-spreader situations is significantly large, of the order of hundreds, suggesting a delta-function-like behavior during the event. Use of a well-mixed room model supports these findings. They elucidate infection kinetic modeling in enclosed environments, which differ from the standard SIR model, and provide expressions for $R_0$ in terms of physical and operational parameters. The overall impact of super-spreader events can be significant, depending on the state of the epidemic and how the infections generated by the event subsequently spread in the community.

q-bio.PE↗

arXiv2020

A comprehensive spatial-temporal infection model

Motivated by analogies between the spreading of human-to-human infections and of chemical processes, we develop a comprehensive model that accounts both for infection and for transport. In this analogy, the three different populations of infection models correspond to three chemical species. Areal densities emerge as the key variables, thus capturing the effect of spatial density. We derive expressions for the kinetics of the infection rates and for the important parameter R0, that include areal density and its spatial distribution. Coupled with mobility the model allows the study of various effects. We first present results for a batch reactor, the chemical process equivalent of the SIR model. Because density makes R0 a decreasing function of the process extent, the infection curves are different and smaller than for the standard SIR model. We show that the effect of the initial conditions is limited to the onset of the epidemic. We derive effective infection curves for a number of cases, including a back-and-forth commute between regions of low and high R0 environments. We then consider spatially distributed systems. We show that diffusion leads to traveling waves, which in 1-D geometries propagate at a constant speed and with a constant shape, both of which are sole functions of R0. The infection curves are slightly different than for the batch problem, as diffusion mitigates the infection intensity, thus leading to an effective lower R0. The dimensional wave speed is found to be proportional to the product of the square root of the diffusivity and of an increasing function of R0, confirming the importance of restricting mobility in arresting the propagation of infection. We examine the interaction of infection waves under various conditions and scenarios, and extend the wave propagation analysis to 2-D heterogeneous systems.

q-bio.PE↗

arXiv2019

Bayesian Inference with Generative Adversarial Network Priors

Bayesian inference is used extensively to infer and to quantify the uncertainty in a field of interest from a measurement of a related field when the two are linked by a physical model. Despite its many applications, Bayesian inference faces challenges when inferring fields that have discrete representations of large dimension, and/or have prior distributions that are difficult to represent mathematically. In this manuscript we consider the use of Generative Adversarial Networks (GANs) in addressing these challenges. A GAN is a type of deep neural network equipped with the ability to learn the distribution implied by multiple samples of a given field. Once trained on these samples, the generator component of a GAN maps the iid components of a low-dimensional latent vector to an approximation of the distribution of the field of interest. In this work we demonstrate how this approximate distribution may be used as a prior in a Bayesian update, and how it addresses the challenges associated with characterizing complex prior distributions and the large dimension of the inferred field. We demonstrate the efficacy of this approach by applying it to the problem of inferring and quantifying uncertainty in the initial temperature field in a heat conduction problem from a noisy measurement of the temperature at later time.

stat.ML↗