arXiv subjects

Sahil Bhola

Publications and source records attributed to Sahil Bhola.

5 recordsLinked to original sources

arXiv2025

Residual-augmented flow matching operators for probabilistic partial differential equations

Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural operators fail to characterize uncertainty, while generative approaches require large ensembles of high-fidelity solution operator simulations and often sacrifice resolution generalizability. In this work, we propose a residual-augmented probabilistic operator learning framework that casts flow-matching-based generative modeling in infinite-dimensional function spaces while leveraging inexpensive low-fidelity solution operators as an inductive bias. Rather than learning the full high-fidelity stochastic solution operator directly, the proposed framework learns probabilistic residual operators that characterize the discrepancy between low- and high-fidelity solutions. By parameterizing the vector field in flow matching using neural operators conditioned on both the known system input and low-fidelity solution, the framework amortizes probabilistic inference across input conditions while enabling uncertainty-aware and resolution-generalizable predictions across spatial discretizations. Numerical experiments on stochastic advection, Burgers', and Darcy flow systems demonstrate that the residual-augmented formulation improves predictive accuracy under the same high-fidelity data budget, while the probabilistic operator learning formulation enables accurate characterization of uncertainty in low-data regimes compared to learning high-fidelity stochastic operators directly from data.

stat.CO↗

arXiv2024

Deterministic and Probabilistic Rounding Error Analysis for Mixed-Precision Arithmetic on Modern Computing Units

Modern computer architectures support low-precision arithmetic, which present opportunities for the adoption of mixed-precision algorithms to achieve high computational throughput and reduce energy consumption. As a growing number of scientific computations leverage specialized hardware accelerators, the risk of rounding errors increases, potentially compromising the reliability of models. This shift towards hardware-optimized, low-precision computations highlights the importance of rounding error analysis to ensure that performance gains do not come at the expense of accuracy, especially in high-stakes scientific applications. In this work, we conduct rounding error analysis on widely used operations such as fused multiply-add (FMA), mixed-precision FMA (MPFMA), and NVIDIA Tensor cores. We present a deterministic and probabilistic approach to quantifying the accumulated rounding errors. Numerical experiments are presented to perform the multiply and accumulate operation (MAC) and matrix-matrix multiplication using Tensor cores with random data. We show that probabilistic bounds produce tighter estimates by nearly an order of magnitude compared to deterministic ones for matrix-matrix multiplication.

stat.CO↗

arXiv2024

Bias- and Variance-Aware Probabilistic Rounding Error Analysis for Floating-Point Arithmetic

Probabilistic rounding error analysis can yield much sharper bounds than classical worst-case theory, but existing results typically rely on zero-mean rounding errors and often leave the confidence parameter implicit. This work revisits probabilistic rounding error analysis in a moment-aware setting. We first derive a confidence-calibrated reformulation of the Higham and Mary [16] bound that makes its confidence parameter explicit. We then introduce a variance-informed probabilistic backward error bound based on the first two moments of $\log(1+\delta)$, where $\delta$ is the relative rounding error. This allows the analysis to accommodate biased rounding error models rather than relying on a zero-mean assumption. To illustrate this framework, we study both a uniform model and a log-space $\operatorname{Beta}$ model for rounding errors, the latter of which provides a simple way to represent bias. This perspective shows that the growth of probabilistic rounding error bounds is not universal: near-zero-mean regimes recover $\sqrt{n}$-like behavior, while biased models can exhibit faster accumulation. $\texttt{CUDA}$ experiments in single and half precision on dot products, sparse matrix-vector products, and a stochastic boundary-value problem show that the proposed framework is especially useful in low-precision regimes where deterministic bounds are overly conservative and where bias-aware modeling better matches observed error growth.

stat.CO↗

arXiv2023

Estimating Global Identifiability Using Conditional Mutual Information in a Bayesian Framework

A novel information-theoretic approach is proposed to assess the global practical identifiability of Bayesian statistical models. Based on the concept of conditional mutual information, an estimate of information gained for each model parameter is used to quantify the identifiability with practical considerations. No assumptions are made about the structure of the statistical model or the prior distribution while constructing the estimator. The estimator has the following notable advantages: first, no controlled experiment or data is required to conduct the practical identifiability analysis; second, unlike popular variance-based global sensitivity analysis methods, different forms of uncertainties, such as model-form, parameter, or measurement can be taken into account; third, the identifiability analysis is global, and therefore independent of a realization of the parameters. If an individual parameter has low identifiability, it can belong to an identifiable subset such that parameters within the subset have a functional relationship and thus have a combined effect on the statistical model. The practical identifiability framework is extended to highlight the dependencies between parameter pairs that emerge a posteriori to find identifiable parameter subsets. The applicability of the proposed approach is demonstrated using a linear Gaussian model and a non-linear methane-air reduced kinetics model. It is shown that by examining the information gained for each model parameter along with its dependencies with other parameters, a subset of parameters that can be estimated with high posterior certainty can be found.

stat.ME↗

arXiv2022

Multi-fidelity reinforcement learning framework for shape optimization

Deep reinforcement learning (DRL) is a promising outer-loop intelligence paradigm which can deploy problem solving strategies for complex tasks. Consequently, DRL has been utilized for several scientific applications, specifically in cases where classical optimization or control methods are limited. One key limitation of conventional DRL methods is their episode-hungry nature which proves to be a bottleneck for tasks which involve costly evaluations of a numerical forward model. In this article, we address this limitation of DRL by introducing a controlled transfer learning framework that leverages a multi-fidelity simulation setting. Our strategy is deployed for an airfoil shape optimization problem at high Reynolds numbers, where our framework can learn an optimal policy for generating efficient airfoil shapes by gathering knowledge from multi-fidelity environments and reduces computational costs by over 30\%. Furthermore, our formulation promotes policy exploration and generalization to new environments, thereby preventing over-fitting to data from solely one fidelity. Our results demonstrate this framework's applicability to other scientific DRL scenarios where multi-fidelity environments can be used for policy learning.

cs.LG↗