arXiv subjects

Bamdad Hosseini

Publications and source records attributed to Bamdad Hosseini.

At least 19 recordsLinked to original sources

arXiv2026

Conditional Sampling via Wasserstein Autoencoders and Triangular Transport

We present Conditional Wasserstein Autoencoders (CWAEs), a framework for conditional simulation that exploits low-dimensional structure in both the conditioned and the conditioning variables. The key idea is to modify a Wasserstein autoencoder to use a (block-) triangular decoder and impose an appropriate independence assumption on the latent variables. We show that the resulting model gives an autoencoder that can exploit low-dimensional structure while simultaneously the decoder can be used for conditional simulation. We explore various theoretical properties of CWAEs, including their connections to conditional optimal transport (OT) problems. We also present alternative formulations that lead to three architectural variants forming the foundation of our algorithms. We present a series of numerical experiments that demonstrate that our different CWAE variants achieve substantial reductions in approximation error relative to the low-rank ensemble Kalman filter (LREnKF), particularly in problems where the support of the conditional measures is truly low-dimensional.

cs.LG↗

arXiv2026

Error Analysis of Bayesian Inverse Problems with Generative Priors

Data-driven methods for the solution of inverse problems have become widely popular in recent years thanks to the rise of machine learning techniques. A popular approach concerns the training of a generative model on additional data to learn a bespoke prior for the problem at hand. In this article we present an analysis for such problems by presenting quantitative error bounds for minimum Wasserstein-2 generative models for the prior. We show that under some assumptions, the error in the posterior due to the generative prior will inherit the same rate as the prior with respect to the Wasserstein-1 distance. We further present numerical experiments that verify that aspects of our error analysis manifests in some benchmarks followed by an elliptic PDE inverse problem where a generative prior is used to model a non-stationary field.

stat.ML↗

arXiv2025

Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations

Recent studies have demonstrated the success of deep learning in solving forward and inverse problems in engineering and scientific computing domains, such as physics-informed neural networks (PINNs). Source inversion problems under sparse measurements for parabolic partial differential equations (PDEs) are particularly challenging to solve using PINNs, due to their severe ill-posedness and the multiple unknowns involved including the source function and the PDE parameters. Although the neural tangent kernel (NTK) of PINNs has been widely used in forward problems involving a single neural network, its extension to inverse problems involving multiple neural networks remains less explored. In this work, we propose a weighted adaptive approach based on the NTK of PINNS including multiple separate networks representing the solution, the unknown source, and the PDE parameters. The key idea behind our methodology is to simultaneously solve the joint recovery of the solution, the source function along with the unknown parameters thereby using the underlying partial differential equation as a constraint that couples multiple unknown functional parameters, leading to more efficient use of the limited information in the measurements. We apply our method on the advection-diffusion equation and we present various 2D and 3D numerical experiments using different types of measurements data that reflect practical engineering systems. Our proposed method is successful in estimating the unknown source function, the velocity and diffusion parameters as well as recovering the solution of the equation, while remaining robust to additional noise in the measurements.

stat.ML↗

arXiv2025

Operator Learning at Machine Precision

Neural operator learning methods have garnered significant attention in scientific computing for their ability to approximate infinite-dimensional operators. However, increasing their complexity often fails to substantially improve their accuracy, leaving them on par with much simpler approaches such as kernel methods and more traditional reduced-order models. In this article, we set out to address this shortcoming and introduce CHONKNORIS (Cholesky Newton--Kantorovich Neural Operator Residual Iterative System), an operator learning paradigm that can achieve machine precision. CHONKNORIS draws on numerical analysis: many nonlinear forward and inverse PDE problems are solvable by Newton-type methods. Rather than regressing the solution operator itself, our method regresses the Cholesky factors of the elliptic operator associated with Tikhonov-regularized Newton--Kantorovich updates. The resulting unrolled iteration yields a neural architecture whose machine-precision behavior follows from achieving a contractive map, requiring far lower accuracy than end-to-end approximation of the solution operator. We benchmark CHONKNORIS on a range of nonlinear forward and inverse problems, including a nonlinear elliptic equation, Burgers' equation, a nonlinear Darcy flow problem, the Calder\'{o}n problem, an inverse wave scattering problem, and a problem from seismic imaging. We also present theoretical guarantees for the convergence of CHONKNORIS in terms of the accuracy of the emulated Cholesky factors. Additionally, we introduce a foundation model variant, FONKNORIS (Foundation Newton--Kantorovich Neural Operator Residual Iterative System), which aggregates multiple pre-trained CHONKNORIS experts for diverse PDEs to emulate the solution map of a novel nonlinear PDE. Our FONKNORIS model is able to accurately solve unseen nonlinear PDEs such as the Klein--Gordon and Sine--Gordon equations.

cs.LG↗

arXiv2025

A joint optimization approach to identifying sparse dynamics using least squares kernel collocation

We develop an all-at-once modeling framework for learning systems of ordinary differential equations (ODE) from scarce, partial, and noisy observations of the states. The proposed methodology amounts to a combination of sparse recovery strategies for the ODE over a function library combined with techniques from reproducing kernel Hilbert space (RKHS) theory for estimating the state and discretizing the ODE. Our numerical experiments reveal that the proposed strategy leads to significant gains in terms of accuracy, sample efficiency, and robustness to noise, both in terms of learning the equation and estimating the unknown states. This work demonstrates capabilities well beyond existing and widely used algorithms while extending the modeling flexibility of other recent developments in equation discovery.

stat.ME↗

arXiv2025

Learning Paths for Dynamic Measure Transport: A Control Perspective

We bring a control perspective to the problem of identifying paths of measures for sampling via dynamic measure transport (DMT). We highlight the fact that commonly used paths may be poor choices for DMT and connect existing methods for learning alternate paths to mean-field games. Based on these connections we pose a flexible family of optimization problems for identifying tilted paths of measures for DMT and advocate for the use of objective terms which encourage smoothness of the corresponding velocities. We present a numerical algorithm for solving these problems based on recent Gaussian process methods for solution of partial differential equations and demonstrate the ability of our method to recover more efficient and smooth transport models compared to those which use an untilted reference path.

stat.ML↗

arXiv2025

Error Analysis of Triangular Optimal Transport Maps for Filtering

We present a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation. Along the way, we extend previous error analyses of Brenier maps to the case of conditional Brenier maps that arise in the context of simulation based inference. We then apply these results in a filtering scenario to analyze the optimal transport filtering algorithm of Al-Jarrah et al. (2024, ICML). An extension of that algorithm along with numerical benchmarks on various non-Gaussian and high-dimensional examples are provided to demonstrate its effectiveness and practical potential.

math.ST↗

arXiv2025

Nonlinear Filtering with Brenier Optimal Transport Maps

This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance resampling (SIR) particle filters suffer from fundamental limitations, in scenarios involving degenerate likelihoods or high-dimensional states, due to the weight degeneracy issue. In this paper, we explore an alternative method, which is based on estimating the Brenier optimal transport (OT) map from the current prior distribution of the state to the posterior distribution at the next time step. Unlike SIR particle filters, the OT formulation does not require the analytical form of the likelihood. Moreover, it allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare the OT method to the SIR particle filter and the ensemble Kalman filter, evaluating the performance in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.

cs.LG↗

arXiv2025

Fast filtering of non-Gaussian models using Amortized Optimal Transport Maps

In this paper, we present the amortized optimal transport filter (A-OTF) designed to mitigate the computational burden associated with the real-time training of optimal transport filters (OTFs). OTFs can perform accurate non-Gaussian Bayesian updates in the filtering procedure, but they require training at every time step, which makes them expensive. The proposed A-OTF framework exploits the similarity between OTF maps during an initial/offline training stage in order to reduce the cost of inference during online calculations. More precisely, we use clustering algorithms to select relevant subsets of pre-trained maps whose weighted average is used to compute the A-OTF model akin to a mixture of experts. A series of numerical experiments validate that A-OTF achieves substantial computational savings during online inference while preserving the inherent flexibility and accuracy of OTF.

math.OC↗

arXiv2025

Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis

We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational cost, in terms of training procedures. Our approach is mathematically interpretable and backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation. Numerical benchmarks demonstrate significant improvements in computational complexity and robustness while achieving one to two orders of magnitude improvements in terms of accuracy compared to state-of-the-art algorithms.

stat.ML↗

arXiv2024

An Approximation Theory Framework for Measure-Transport Sampling Algorithms

This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical inference and generative modeling. We provide a priori error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps and on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the distance between two maps to the distance~(or divergence) between the pushforward measures they define. We present a series of applications of our framework, where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback--Leibler divergence. Specialized rates for approximations of the popular triangular Kn{ö}the-Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory.

math.NA↗

arXiv2024

Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation

The article presents a systematic study of the problem of conditioning a Gaussian random variable $ξ$ on nonlinear observations of the form $F \circ ϕ(ξ)$ where $ϕ: \mathcal{X} \to \mathbb{R}^N$ is a bounded linear operator and $F$ is nonlinear. Such problems arise in the context of Bayesian inference and recent machine learning-inspired PDE solvers. We give a representer theorem for the conditioned random variable $ξ\mid F\circ ϕ(ξ)$, stating that it decomposes as the sum of an infinite-dimensional Gaussian (which is identified analytically) as well as a finite-dimensional non-Gaussian measure. We also introduce a novel notion of the mode of a conditional measure by taking the limit of the natural relaxation of the problem, to which we can apply the existing notion of maximum a posteriori estimators of posterior measures. Finally, we introduce a variant of the Laplace approximation for the efficient simulation of the aforementioned conditioned Gaussian random variables towards uncertainty quantification.

stat.ML↗

arXiv2024

Data-Driven Approximation of Stationary Nonlinear Filters with Optimal Transport Maps

The nonlinear filtering problem is concerned with finding the conditional probability distribution (posterior) of the state of a stochastic dynamical system, given a history of partial and noisy observations. This paper presents a data-driven nonlinear filtering algorithm for the case when the state and observation processes are stationary. The posterior is approximated as the push-forward of an optimal transport (OT) map from a given distribution, that is easy to sample from, to the posterior conditioned on a truncated observation window. The OT map is obtained as the solution to a stochastic optimization problem that is solved offline using recorded trajectory data from the state and observations. An error analysis of the algorithm is presented under the stationarity and filter stability assumptions, which decomposes the error into two parts related to the truncation window during training and the error due to the optimization procedure. The performance of the proposed method, referred to as optimal transport data-driven filter (OT-DDF), is evaluated for several numerical examples, highlighting its significant computational efficiency during the online stage while maintaining the flexibility and accuracy of OT methods in nonlinear filtering.

math.OC↗

arXiv2024

Diffeomorphic Measure Matching with Kernels for Generative Modeling

This article presents a general framework for the transport of probability measures towards minimum divergence generative modeling and sampling using ordinary differential equations (ODEs) and Reproducing Kernel Hilbert Spaces (RKHSs), inspired by ideas from diffeomorphic matching and image registration. A theoretical analysis of the proposed method is presented, giving a priori error bounds in terms of the complexity of the model, the number of samples in the training set, and model misspecification. An extensive suite of numerical experiments further highlights the properties, strengths, and weaknesses of the method and extends its applicability to other tasks, such as conditional simulation and inference.

stat.ML↗

arXiv2024

Conditional Optimal Transport on Function Spaces

We present a systematic study of conditional triangular transport maps in function spaces from the perspective of optimal transportation and with a view towards amortized Bayesian inference. More specifically, we develop a theory of constrained optimal transport problems that describe block-triangular Monge maps that characterize conditional measures along with their Kantorovich relaxations. This generalizes the theory of optimal triangular transport to separable infinite-dimensional function spaces with general cost functions. We further tailor our results to the case of Bayesian inference problems and obtain regularity estimates on the conditioning maps from the prior to the posterior. Finally, we present numerical experiments that demonstrate the computational applicability of our theoretical results for amortized and likelihood-free inference of functional parameters.

math.OC↗

arXiv2023

Kernel Methods are Competitive for Operator Learning

We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator $\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V}$ are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations $ϕ(u_i), φ(v_i)$ of input/output functions $v_i=\mathcal{G}^\dagger(u_i)$ ($i=1,\ldots,N$), and the measurement operators $ϕ\,:\, \mathcal{U}\to \mathbb{R}^n$ and $φ\,:\, \mathcal{V} \to \mathbb{R}^m$ are linear. Writing $ψ\,:\, \mathbb{R}^n \to \mathcal{U}$ and $χ\,:\, \mathbb{R}^m \to \mathcal{V}$ for the optimal recovery maps associated with $ϕ$ and $φ$, we approximate $\mathcal{G}^\dagger$ with $\bar{\mathcal{G}}=χ\circ \bar{f} \circ ϕ$ where $\bar{f}$ is an optimal recovery approximation of $f^\dagger:=φ\circ \mathcal{G}^\dagger \circ ψ\,:\,\mathbb{R}^n \to \mathbb{R}^m$. We show that, even when using vanilla kernels (e.g., linear or Matérn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.

stat.ML↗

arXiv2023

Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference

We present a novel framework for conditional sampling of probability measures, using block triangular transport maps. We develop the theoretical foundations of block triangular transport in a Banach space setting, establishing general conditions under which conditional sampling can be achieved and drawing connections between monotone block triangular maps and optimal transport. Based on this theory, we then introduce a computational approach, called monotone generative adversarial networks (M-GANs), to learn suitable block triangular maps. Our algorithm uses only samples from the underlying joint probability measure and is hence likelihood-free. Numerical experiments with M-GAN demonstrate accurate sampling of conditional measures in synthetic examples, Bayesian inverse problems involving ordinary and partial differential equations, and probabilistic image in-painting.

stat.ML↗

arXiv2023

Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

math.NA↗