Efficiently decomposing finite element stiffness matrices into the Pauli basis is challenging due to the exponential growth of Pauli strings with problem size. A naive Pauli expansion requires Θ(8⌈log2N⌉) operations, where N denotes the number of degrees of freedom, rendering direct decomposition infeasible for large systems. Existing approaches exploit algebraic sparsity or operator structure but do not incorporate the geometric organization intrinsic to finite element discretizations, and consequently exhibit poor scaling for stiffness matrices. To address this problem, we introduce PHASE, a hierarchical, geometry-aware Pauli decomposition algorithm that leverages recursive mesh partitioning to organize element contributions across multiple spatial scales. PHASE employs a hybrid strategy that combines full- and reduced-space Tensorized Pauli Decomposition with Fast Walsh–Hadamard Transform-based aggregation to assemble global Pauli coefficients efficiently. We show that this approach yields a dimension-dependent reduction in the exponential scaling exponent of Pauli assembly asymptotic complexity relative to existing methods, reducing the cost from O(N2logN) to O(NγdlogN) with γd<2 under standard mesh regularity and balanced partition assumptions. These results substantially improve the feasibility of quantum-compatible operator synthesis for large-scale finite element models.
This manuscript explores a variational quantum formulation for nonlinear elasticity problems arising from hyperelastic material models. The approach leverages the potential energy structure of hyperelasticity and employs a hybrid quantum–classical framework in which the energy functional is evaluated using parameterized quantum circuits and optimized through classical routines. To enable a hybrid (classical–quantum implementation), polynomial approximations of the nonlinear terms in strain energy density are introduced, yielding a representation compatible with variational quantum algorithms. The methodology is demonstrated on a special case of the Neo-Hookean material model in a one-dimensional setting using finite element discretizations with first- and second-order shape functions and nonhomogeneous boundary conditions. Numerical experiments investigate the influence of the polynomial approximation order on the accuracy and efficiency of the proposed approach, illustrating its feasibility for near-term quantum devices.
Medical computer vision presents unique challenges compared to natural images, largely due to the high dimensionality of the data (e.g., 3D and 4D imaging) and limited training samples. Certain medical imaging modalities, such as Diffusion-weighted Magnetic Resonance Imaging (dMRI) — a critical tool for examining brain microstructure and neural connectivity — poses even greater complexity with its data dimensionality, encompassing three spatial dimensions and at least three degrees of freedom for the diffusion weighting. Therefore, current deep learning methods for dMRI are often limited by a “case-by-case” design. In this work, we introduce a Unified Polyhedral Representation (UPER) to model multi-dimensional dMRI data across over 10 unique shells, sourced from 2800 scans across 17 datasets. The key innovation involves revisiting the dMRI data representation by transforming traditional model-based or signal-based methods into a polyhedral representation. We utilized a masked auto encoder with both self-supervised and supervised learning within a single unified, end-to-end deep learning framework. This approach effectively encodes complex spatial, radial, and angular information within a simple polyhedron. Furthermore, the transformer-based UPER framework models heterogeneous shell configurations through a unified model. To our knowledge, UPER is the first deep learning model capable of (1) modeling varying radial and angular spaces, and (2) accommodating diverse shell configurations. Note that signal-processing approaches such as SHORE and SHARD can also handle multi-shell configurations through analytical formulations; UPER’s novelty lies in its unified, end-to-end deep learning framework. Extensive experiments demonstrate that our pretrained unified model achieves superior performance in various downstream tasks as compared with traditional deep learning framework, offering a holistic model for analyzing diverse dMRI datasets. The code for this paper is publicly available at https://github.com/hrlblab/UPER
This paper addresses the classic problem of parameter estimation (PE) in multimachine power system models. Such models are typically described by a set of nonlinear differential-algebraic equations (DAE), where generator physics and network power flow equations are coupled. DAE models are well established in classic power system textbooks, but parameter identification and estimation of generator inertia and damping together with network branch resistances and reactances for these models remain relatively underexplored. In contrast to prior approaches that rely on ODE approximations, this paper develops a joint Bayesian inference framework to perform PE of generator and network parameters while exploiting grid DAE models. It further combines physics-aware statistical modeling with computationally efficient posterior sampling to make joint Bayesian calibration practical. Results on the IEEE 9-bus system show accurate parameter recovery with well-behaved posterior uncertainty, while a short 39-bus study provides evidence that the framework remains effective on a materially larger joint-estimation problem. These results are obtained without requiring overly conservative priors.
We propose a novel framework for efficiently solving the AC optimal power flow (OPF) problem under varying grid topologies. Traditional learning-based approaches to OPF often retrain entire models when the network topology changes, leading to high computational cost and limited generalization. The proposed approach exploits the fact that different components of the OPF problem respond to topology changes to varying degrees: device-level operating limits remain largely unaffected, the network equations are modified locally around a tripped line, whereas the resulting optimal operating point may change throughout the network. It consists of two complementary training phases. In Phase I (modular pretraining), the OPF learning task under the intact grid topology is decomposed into three sequential subtasks (voltage setpoint prediction, power generation estimation, and generation cost computation) representing distinct functional mappings. In Phase II (adaptive learning), these pretrained modules are jointly fine-tuned for new topologies using a composite loss that combines supervised objectives with physics-informed regularization for branch-flow accuracy, nodal power balance, and operational constraint satisfaction. Extensive evaluations on benchmark grids demonstrate accurate predictions for the intact grid as well as for grids with seen and unseen contingencies. In our numerical experiments, at most 60 N−1 contingency topologies are used for adaptation, and the largest mean prediction error among the reported variables under unseen topologies is approximately 3.1%.
