arXiv subjects

Akil Narayan

Publications and source records attributed to Akil Narayan.

At least 19 recordsLinked to original sources

arXiv2026

Constructive Tchakaloff results and well-conditioned quadrature through randomized least squares

We consider using randomized least squares to construct quadrature rules exact on a subspace of functions. Using new conditions that we call relative admissibility and reference weight concentration, we establish both that the quadrature weights from such a procedure concentrate close to their asymptotic values with prescribed and arbitrarily large probability, and this in turn provides useful finite-sample probabilistic bounds on the stability of the resulting quadrature rules for very general classes of possibly complex-valued functions. Our analysis both significantly generalizes the existing analysis of randomized least squares quadrature construction, and provides new bounds on stability for these rules. These results specialize to existence results for positive quadrature rules, when such rules can be theoretically expected. Because our procedures are formally algorithmic, our analysis is a substantive advance toward computationally constructive generalized Tchakaloff theorems.

math.NA↗

arXiv2026

Structure-Preserving Discontinuous Galerkin Methods for Stochastic Shallow Water Equations

Shallow water equations (SWE) are fundamental models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. In many practical applications, uncertainties arising from initial conditions and bottom topography must be taken into account, motivating the development of stable and accurate numerical methods for stochastic SWE. Building on the hyperbolicity-preserving stochastic Galerkin formulation for SWE [Dai, Epshteyn, Narayan, 2021 SISC] and a stochastic extension of the entropy stable discontinuous Galerkin methods for skew-symmetric SWE [Fu, 2022 JSC], we develop a structure-preserving, entropy conservative, and entropy stable discontinuous Galerkin--stochastic Galerkin method for the stochastic shallow water system, with the well-balanced property. We demonstrate the accuracy, applicability, and robustness of the proposed structure-preserving algorithms through several numerical experiments.

math.NA↗

arXiv2025

Greedy Rational Approximation for Frequency-Domain Model Reduction of Parametric LTI Systems

We investigate model reduction of parametric linear time-invariant (LTI) dynamical systems. When posed in the frequency domain, this problem can be formulated as seeking a low-order rational function approximation of a high-order rational function. We propose to use a standard reduced basis method (RBM) to construct this low-order rational function. Algorithmically, this procedure is an iterative greedy approach, where the greedy objective is evaluated through an error estimator that exploits the linearity of the frequency domain representation. The greedy framework is motivated through theoretical results of rational approximability of functions. This framework provides a principled approach to rational compression of high-order rational functions, and provides a computational pathway for model reduction of parametric LTI systems.

math.NA↗

arXiv2025

An Optimal Weighted Least-Squares Method for Operator Learning

We consider the problem of learning an unknown, possibly nonlinear operator between separable Hilbert spaces from supervised data. Inputs are drawn from a prescribed probability measure on the input space, and outputs are (possibly noisy) evaluations of the target operator. We regard admissible operators as square-integrable maps with respect to a fixed approximation measure, and we measure reconstruction error in the corresponding Bochner norm. For a finite-dimensional approximation space $V$ of dimension $N$, we study weighted least squares estimators in $V$ and establish probabilistic stability and accuracy bounds in the Bochner norm. We show that there exist sampling measures and weights - defined via an operator-level Christoffel function - that yield uniformly well-conditioned Gram matrices and near-optimal sample complexity, with a number of training samples $M$ on the order of $N \log N$. We complement the analysis by constructing explicit operator approximation spaces in cases of interest: rank-one linear operators that are dense in the class of bounded linear operators, and rank-one polynomial operators that are dense in the Bochner space under mild assumptions on the approximation measure. For both families we describe implementable procedures for sampling from the associated optimal measures. Finally, we demonstrate the effectiveness of this framework on several benchmark problems, including learning solution operators for the Poisson equation, viscous Burgers' equation, and the incompressible Navier-Stokes equations.

math.NA↗

arXiv2025

Integrating Uncertainty Quantification into Computational Fluid Dynamics Models of Coronary Arteries Under Steady Flow

Computational models are continuously integrated in the clinical space, where they support clinicians in disease diagnosis, prognosis, and prevention strategies. While assisting in clinical space, these computational models frequently use deterministic approaches, where the inherent (aleatoric) variability of input parameters is ignored. This questions the credibility and often hinders the clinical adoption of these computational models. Therefore, in this study, we introduced uncertainty quantification in the computational fluid dynamics models of the left main coronary artery to analyze the influence of input hemodynamics parameters on wall shear stress (WSS). UncertainSCI was used, where an emulator was built using polynomial chaos expansion between the input parameters and the output quantity of interest, and the output sensitivities and statistics were directly extracted from the emulator. The uncertainty-informed framework was first applied to an analytical solution of the Navier-Stokes equation (Poiseuille flow) and then to a patient-specific model of the left main coronary artery. Different input hemodynamics parameters are considered, such as pressure, viscosity, density, velocity, and radius, whereas wall shear stress was considered as our output quantity of interest. The results suggest that velocity dominated the variability in WSS in the analytical model (~79%), whereas viscosity dominated in the patient-specific model (~59%). The results further suggest that out of all the Sobol indices interactions, unary interactions were the most dominant ones, contributing ~93.2% and ~99% for the analytical and patient-specific model, respectively. This study will enhance confidence in computational models, facilitating their adoption in the clinical space to improve decision-making for coronary artery disease diagnosis, prognosis, and therapeutic strategies.

physics.med-ph↗

arXiv2025

SUPN: Shallow Universal Polynomial Networks

Deep neural networks (DNNs) and Kolmogorov-Arnold networks (KANs) are popular methods for function approximation due to their flexibility and expressivity. However, they typically require a large number of trainable parameters to produce a suitable approximation. Beyond making the resulting network less transparent, overparameterization creates a large optimization space, likely producing local minima in training that have quite different generalization errors. In this case, network initialization can have an outsize impact on the model's out-of-sample accuracy. For these reasons, we propose shallow universal polynomial networks (SUPNs). These networks replace all but the last hidden layer with a single layer of polynomials with learnable coefficients, leveraging the strengths of DNNs and polynomials to achieve sufficient expressivity with far fewer parameters. We prove that SUPNs converge at the same rate as the best polynomial approximation of the same degree, and we derive explicit formulas for quasi-optimal SUPN parameters. We complement theory with an extensive suite of numerical experiments involving SUPNs, DNNs, KANs, and polynomial projection in one, two, and ten dimensions, consisting of over 13,000 trained models. On the target functions we numerically studied, for a given number of trainable parameters, the approximation error and variability are often lower for SUPNs than for DNNs and KANs by an order of magnitude. In our examples, SUPNs even outperform polynomial projection on non-smooth functions.

cs.LG↗

arXiv2025

Efficient and Robust Carath\'{e}odory-Steinitz Pruning of Positive Discrete Measures

In many applications, one seeks to approximate integration against a positive measure of interest by a positive discrete measure: a numerical quadrature rule with positive weights. One common desired discretization property is moment preservation over a finite dimensional function space, e.g., bounded-degree polynomials. Carath\'{e}odory's theorem asserts that if there is any finitely supported quadrature rule with more nodes than the dimension of the given function space, one can form a smaller (and hence more efficient) positive, nested, quadrature rule that preserves the moments of the original rule. We describe an efficient streaming procedure for Carath\'{e}odory-Steinitz pruning, a numerical procedure that implements Carath\'{e}odory's theorem for this measure compression. The new algorithm makes use of Givens rotations and on-demand storage of arrays to successfully prune very large rules whose storage complexity only depends on the dimension of the function space. This approach improves on a naive implementation of Carath\'{e}odory-Steinitz pruning whose runtime and storage complexity are quadratic and linear, respectively, in the size of the original measure. We additionally prove mathematical stability properties of our method with respect to a set of admissible, total-variation perturbations of the original measure. Our method is compared to two alternative approaches with larger storage requirements: non-negative least squares and linear programming, and we demonstrate comparable runtimes, with improved stability and storage robustness. Finally, we demonstrate practical usage of this algorithm to generate quadrature for discontinuous Galerkin finite element simulations on cut-cell meshes.

math.NA↗

arXiv2025

Hybrid least squares for learning functions from highly noisy data

Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal when large noise is present. To address this issue, we propose a hybrid approach that combines Christoffel sampling with optimal experimental design. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convexity-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we further extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance.

stat.ML↗

arXiv2025

Optimally balancing exploration and exploitation to automate multi-fidelity statistical estimation

Multi-fidelity methods that use an ensemble of models to compute a Monte Carlo estimator of the expectation of a high-fidelity model can significantly reduce computational costs compared to single-model approaches. These methods use oracle statistics, specifically the covariance between models, to optimally allocate samples to each model in the ensemble. However, in practice, the oracle statistics are estimated using additional model evaluations, whose computational cost and induced error are typically ignored. To address this issue, this paper proposes an adaptive algorithm to optimally balance the resources between oracle statistics estimation and final multi-fidelity estimator construction, leveraging ideas from multilevel best linear unbiased estimators in Schaden and Ullmann (2020) and a bandit-learning procedure in Xu et al. (2022). Under mild assumptions, we demonstrate that the multi-fidelity estimator produced by the proposed algorithm exhibits mean-squared error commensurate with that of the best linear unbiased estimator under the optimal allocation computed with oracle statistics. Our theoretical findings are supported by detailed numerical experiments, including a parametric elliptic PDE and an ice-sheet mass-change modeling problem.

stat.CO↗

arXiv2025

Entropy stable reduced order modeling of nonlinear conservation laws using discontinuous Galerkin methods

Reduced order models (ROMs) are inexpensive surrogate models that reduce costs associated with many-query scenarios. Current methods for constructing entropy stable ROMs for nonlinear conservation laws utilize full order models (FOMs) based on finite volume methods (FVMs). This work describes how to generalize the construction of entropy stable ROMs from FVM FOMs to high order discontinuous Galerkin (DG) FOMs. Significant innovations of our work include the introduction of a new "test basis" which significantly improves accuracy for DG FOMs, a dimension-by-dimension hyper-reduction strategy, and a simplification of the boundary hyper-reduction step based on "Carath\'eodory pruning".

math.NA↗

arXiv2024

Energy Stable and Structure-Preserving Algorithms for the Stochastic Galerkin System of 2D Shallow Water Equations

Shallow water equations (SWE) are fundamental nonlinear hyperbolic PDE-based models in fluid dynamics that are essential for studying a wide range of geophysical and engineering phenomena. Therefore, stable and accurate numerical methods for SWE are needed. Although some algorithms are well studied for deterministic SWE, more effort should be devoted to handling the SWE with uncertainty. In this paper, we incorporate uncertainty through a stochastic Galerkin (SG) framework, and building on an existing hyperbolicity-preserving SG formulation for 2D SWE, we construct the corresponding entropy flux pair, and develop structure-preserving, well-balanced, second-order energy conservative and energy stable finite volume schemes for the SG formulation of the two-dimensional shallow water system. We demonstrate the efficacy, applicability, and robustness of these structure-preserving algorithms through several challenging numerical experiments.

math.NA↗

arXiv2024

Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation

Modern physics simulation often involves multiple functions of interests, and traditional numerical approaches are known to be complex and computationally costly. While machine learning-based surrogate models can offer significant cost reductions, most focus on a single task, such as forward prediction, and typically lack uncertainty quantification -- an essential component in many applications. To overcome these limitations, we propose Arbitrarily-Conditioned Multi-Functional Diffusion (ACM-FD), a versatile probabilistic surrogate model for multi-physics emulation. ACM-FD can perform a wide range of tasks within a single framework, including forward prediction, various inverse problems, and simulating data for entire systems or subsets of quantities conditioned on others. Specifically, we extend the standard Denoising Diffusion Probabilistic Model (DDPM) for multi-functional generation by modeling noise as Gaussian processes (GP). We propose a random-mask based, zero-regularized denoising loss to achieve flexible and robust conditional generation. We induce a Kronecker product structure in the GP covariance matrix, substantially reducing the computational cost and enabling efficient training and sampling. We demonstrate the effectiveness of ACM-FD across several fundamental multi-physics systems.

cs.LG↗

arXiv2024

Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions). The KNO decouples the choice of kernel from the numerical integration scheme (quadrature), thereby naturally allowing for operator learning with explicitly-chosen trainable kernels on irregular geometries. On irregular domains, this allows the KNO to utilize domain-specific quadrature rules. To help ameliorate the curse of dimensionality, we also leverage an efficient dimension-wise factorization algorithm on regular domains. More importantly, the ability to explicitly specify kernels also allows the use of highly expressive, non-stationary, neural anisotropic kernels whose parameters are computed by training neural networks. We present universal approximation theorems showing that both the continuous and fully discretized KNO are universal approximators on operator learning problems. Numerical results demonstrate that on existing benchmarks the training and test accuracy of KNOs is closely comparable to or higher than that of popular neural operators while typically using an order of magnitude fewer trainable parameters, with the more expressive kernels proving important to attaining high accuracy. KNOs thus facilitate low-memory, geometrically-flexible, deep operator learning, while retaining the implementation simplicity and transparency of traditional kernel methods from both scientific computing and machine learning.

cs.LG↗

arXiv2024

TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs

We introduce the Transformed Generative Pre-Trained Physics-Informed Neural Networks (TGPT-PINN) for accomplishing nonlinear model order reduction (MOR) of transport-dominated partial differential equations in an MOR-integrating PINNs framework. Building on the recent development of the GPT-PINN that is a network-of-networks design achieving snapshot-based model reduction, we design and test a novel paradigm for nonlinear model reduction that can effectively tackle problems with parameter-dependent discontinuities. Through incorporation of a shock-capturing loss function component as well as a parameter-dependent transform layer, the TGPT-PINN overcomes the limitations of linear model reduction in the transport-dominated regime. We demonstrate this new capability for nonlinear model reduction in the PINNs framework by several nontrivial parametric partial differential equations.

math.NA↗

arXiv2024

Conformal Finite Element Methods for Nonlinear Rosenau-Burgers-Biharmonic Models

We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence, and uniqueness statements of the corresponding variational methods. We also obtain optimal error estimates of the semidiscrete scheme in corresponding B\^ochner spaces. Finally, we construct a fully discrete scheme through a backward Euler discretization of the time derivative, and prove well-posedness statements for this fully discrete scheme. Our findings show that the mixed approach removes some theoretical impediments to analysis and is numerically easier to implement. We provide numerical simulations for the mixed formulation approach using $C^0$ Taylor-Hood finite elements on several domains. Our numerical results confirm that the algorithm has optimal convergence in accordance with the observed theoretical results.

math.NA↗

arXiv2024

Influence of Material Parameter Variability on the Predicted Coronary Artery Biomechanical Environment via Uncertainty Quantification

Central to the clinical adoption of patient-specific modeling strategies is demonstrating that simulation results are reliable and safe. Simulation frameworks must be robust to uncertainty in model input(s), and levels of confidence should accompany results. In this study we applied a coupled uncertainty quantification-finite element (FE) framework to understand the impact of uncertainty in vascular material properties on variability in predicted stresses. Univariate probability distributions were fit to material parameters derived from layer-specific mechanical behavior testing of human coronary tissue. Parameters were assumed to be probabilistically independent, allowing for efficient parameter ensemble sampling. In an idealized coronary artery geometry, a forward FE model for each parameter ensemble was created to predict tissue stresses under physiologic loading. An emulator was constructed within the UncertainSCI software using polynomial chaos techniques, and statistics and sensitivities were directly computed. Results demonstrated that material parameter uncertainty propagates to variability in predicted stresses across the vessel wall, with the largest dispersions in stress within the adventitial layer. Variability in stress was most sensitive to uncertainties in the anisotropic component of the strain energy function. Unary and binary interactions within the adventitial layer were the main contributors to stress variance, and the leading factor in stress variability was uncertainty in the stress-like material parameter summarizing contribution of the embedded fibers to the overall artery stiffness. Results from a patient-specific coronary model confirmed many of these findings. Collectively, this highlights the impact of material property variation on predicted artery stresses and presents a pipeline to explore and characterize uncertainty in computational biomechanics.

q-bio.TO↗

arXiv2023

Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

math.NA↗

arXiv2023

Multi-Resolution Active Learning of Fourier Neural Operators

Fourier Neural Operator (FNO) is a popular operator learning framework. It not only achieves the state-of-the-art performance in many tasks, but also is efficient in training and prediction. However, collecting training data for the FNO can be a costly bottleneck in practice, because it often demands expensive physical simulations. To overcome this problem, we propose Multi-Resolution Active learning of FNO (MRA-FNO), which can dynamically select the input functions and resolutions to lower the data cost as much as possible while optimizing the learning efficiency. Specifically, we propose a probabilistic multi-resolution FNO and use ensemble Monte-Carlo to develop an effective posterior inference algorithm. To conduct active learning, we maximize a utility-cost ratio as the acquisition function to acquire new examples and resolutions at each step. We use moment matching and the matrix determinant lemma to enable tractable, efficient utility computation. Furthermore, we develop a cost annealing framework to avoid over-penalizing high-resolution queries at the early stage. The over-penalization is severe when the cost difference is significant between the resolutions, which renders active learning often stuck at low-resolution queries and inferior performance. Our method overcomes this problem and applies to general multi-fidelity active learning and optimization problems. We have shown the advantage of our method in several benchmark operator learning tasks. The code is available at https://github.com/shib0li/MRA-FNO.

cs.LG↗