arXiv subjects

Fehmi Cirak

Publications and source records attributed to Fehmi Cirak.

At least 19 recordsLinked to original sources

arXiv2026

A Quantum Spectral Solver for Periodic Incompressible Stokes Flow

We present a quantum spectral solver for the steady incompressible Stokes equations on a two-dimensional periodic domain. The method uses the Quantum Fourier Transform as a coherent change of basis and exploits the resulting spectral structure of the Stokes operator: the Laplacian becomes diagonal, while incompressibility is enforced mode by mode through a Helmholtz projection. In two dimensions, this projection is realized by a mode-dependent rotation from Cartesian velocity components to longitudinal--transverse coordinates, followed by component-conditioned inverse-Laplacian scaling. The velocity and pressure fields are encoded as quantum states over Fourier modes and physical components, and the corresponding spectral factors are implemented through polynomially encoded amplitude blocks. The construction extends recent quantum spectral methods in computational mechanics to an incompressible flow operator with explicit pressure--velocity splitting and divergence-free projection. The approach is also compatible with multiscale finite-element architectures in which quantum parallelism can simultaneously update all representative volume element (RVE) states. Numerical verification includes a steady vortex, a regularized periodic force-dipole benchmark, and an RVE-inspired Kolmogorov-like fluctuation benchmark. The latter illustrates how the circuit can recover a homogenized kinetic-energy observable without reconstructing the full velocity field, consistent with the role of averaged quantities in multiscale flow calculations. Under the standard assumptions of efficient state preparation and observable estimation, the circuit has polylogarithmic dependence on the grid resolution, with the polynomial degree and tile count appearing as explicit approximation and implementation parameters.

math.NA↗

arXiv2026

Statistical finite elements for sequential data synthesis in solid dynamics

We present an approach for synthesising observational data with elastodynamic finite element models by extending the statistical finite element method (statFEM) framework. The proposed formulation adopts a Bayesian filtering approach to account for uncertainties in the data, the finite element model, and the discrepancies between the model and the physical system. Observational data are assimilated while the state of the spatially discretised finite element problem is advanced in time using the stochastic variant of the explicit Newmark scheme. The prior probability density of the state is obtained by solving an incremental probabilistic forward problem, and the corresponding posterior density is obtained by conditioning on the data available at each time step. In the probabilistic forward problem, spatio-temporal Gaussian random fields representing the forcing, model misspecification, and material parameters are specified via their stochastic PDE formulation. The resulting non-Gaussian prior probability density is approximated using a perturbation approach, yielding a Gaussian posterior with closed-form mean and covariance. The hyperparameters of the random field representing model misspecification are calibrated by maximising the marginal likelihood of the data. The proposed approach is illustrated on one- and two-dimensional elastodynamic examples with synthetic data.

math.NA↗

arXiv2026

Neural parametric representations for thin-shell shape optimisation

Shape optimisation of thin-shell structures requires a flexible, differentiable geometric representation suitable for gradient-based optimisation. We propose a neural parametric geometry representation (NRep) for shells based on a neural network with periodic activation functions. The NRep is defined using a multi-layer perceptron (MLP), which maps the parametric coordinates of mid-surface vertices to their physical coordinates. A structural compliance optimisation problem is posed to optimise the shape of a thin-shell parameterised by the NRep subject to a volume constraint, with the network parameters as design variables. The resulting shape optimisation problem is solved using a gradient-based optimisation algorithm. Benchmark examples with classical solutions and comparisons with the free-form deformation method demonstrate that the proposed NRep is capable of representing shell geometries with local geometric features using a small set of network parameters. The robustness of the approach has been demonstrated with different initial geometries, boundary conditions and neural network hyperparameters. The approach also exhibits potential for complex lattice-skin structures, owing to the compact and expressive geometry representation afforded by the NRep.

math.NA↗

arXiv2026

QAFE$^2$: Quantum accelerated multiscale finite element analysis

The computational cost of concurrent multiscale finite element methods is dominated by the repeated solution of microscopic representative volume element (RVE) problems at macroscopic quadrature points. In this work, we introduce a quantum-classical framework for multiscale finite element analysis (QAFE$^2$) that leverages quantum parallelism to fundamentally alter the scaling of RVE-based homogenisation. At the single-RVE level, the proposed quantum solver attains polylogarithmic complexity with respect to the microscopic discretisation size, yielding an exponential asymptotic speedup over the best available classical solvers. More importantly, QAFE$^2$ exploits quantum superposition and entanglement to evaluate, in a single quantum execution, the entire ensemble of RVE problems associated with all macroscopic quadrature points. This capability is a form of intrinsic quantum concurrency with no classical analogue. Numerical experiments on one- and two-dimensional model problems with known analytical solutions confirm the accuracy of the proposed formulation and verify the theoretical computational scaling and parallel performance.

math.NA↗

arXiv2025

A Quantum Spectral Method for Non-Periodic Boundary Value Problems

Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as $\mathscr{O}((\log N)^c)$, where $N$ is the problem size and $c$ a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.

math.NA↗

arXiv2025

Unsupervised Constitutive Model Discovery from Sparse and Noisy Data

Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations. In this work, we integrate statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM-EUCLID. The framework is demonstrated for isotropic hyperelastic materials. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws.

cs.CE↗

arXiv2025

Bayesian buckling load optimisation for structures with geometric uncertainties

Optimised lightweight structures, such as shallow domes and slender towers, are prone to sudden buckling failure because geometric uncertainties/imperfections can lead to a drastic reduction in their buckling loads. We introduce a framework for the robust optimisation of buckling loads, considering geometric nonlinearities and random geometric imperfections. The mean and standard deviation of buckling loads are estimated by Monte Carlo sampling of random imperfections and performing a nonlinear finite element computation for each sample. The extended system method is employed to compute the buckling load directly, avoiding costly path-following procedures. Furthermore, the quasi-Monte Carlo sampling using the Sobol sequence is implemented to generate more uniformly distributed samples, which significantly reduces the number of finite element computations. The objective function consisting of the weighted sum of the mean and standard deviation of the buckling load is optimised using Bayesian optimisation. The accuracy and efficiency of the proposed framework are demonstrated through robust sizing optimisation of several geometrically nonlinear truss examples.

math.NA↗

arXiv2025

Multi-view Bayesian optimisation in an input-output reduced space for engineering design

Bayesian optimisation is an adaptive sampling strategy for constructing a Gaussian process surrogate to efficiently search for the global minimum of a black-box computational model. Gaussian processes have limited applicability in engineering design problems, which usually have many design variables but typically a low intrinsic dimensionality. Their scalability can be significantly improved by identifying a low-dimensional space of latent variables that serve as inputs to the Gaussian process. In this paper, we introduce a multi-view learning strategy that considers both the input design variables and output data representing the objective or constraint functions, to identify a low-dimensional latent subspace. Adopting a fully probabilistic viewpoint, we use probabilistic partial least squares (PPLS) to learn an orthogonal mapping from the design variables to the latent variables using training data consisting of inputs and outputs of the black-box computational model. The latent variables and posterior probability densities of the PPLS and Gaussian process models are determined sequentially and iteratively, with retraining occurring at each adaptive sampling iteration. We compare the proposed probabilistic partial least squares Bayesian optimisation (PPLS-BO) strategy with its deterministic counterpart, partial least squares Bayesian optimisation (PLS-BO), and classical Bayesian optimisation, demonstrating significant improvements in convergence to the global minimum.

stat.AP↗

arXiv2024

Mechanical State Estimation with a Polynomial-Chaos-Based Statistical Finite Element Method

The Statistical Finite Element Method (statFEM) offers a Bayesian framework for integrating computational models with observational data, thus providing improved predictions for structural health monitoring and digital twinning. This paper presents an efficient sampling-free statFEM tailored for non-conjugate, non-Gaussian prior probability densities. We assume that constitutive parameters, modeled as weakly stationary random fields, are the primary source of uncertainty and approximate them using Karhunen-Lo\`eve (KL) expansion. The resulting stochastic solution field, i.e., the displacement field, is a non-stationary, non-Gaussian random field, which we approximate via Polynomial Chaos (PC) expansion. The PC coefficients are determined through projection using Smolyak sparse grids. Additionally, we model the measurement noise as a stationary Gaussian random field and the model misspecification as a mean-free, non-stationary Gaussian random field, which is also approximated using KL expansion. The coefficients of the KL expansion are treated as hyperparameters. The PC coefficients of the stochastic posterior displacement field are computed using the Gauss-Markov-K\'alm\'an filter, while the hyperparameters are determined by maximizing the marginal likelihood. We demonstrate the efficiency and convergence of the proposed method through one- and two-dimensional elastostatic problems.

cs.CE↗

arXiv2024

Variational Bayesian surrogate modelling with application to robust design optimisation

Surrogate models provide a quick-to-evaluate approximation to complex computational models and are essential for multi-query problems like design optimisation. The inputs of current deterministic computational models are usually high-dimensional and uncertain. We consider Bayesian inference for constructing statistical surrogates with input uncertainties and intrinsic dimensionality reduction. The surrogate is trained by fitting to data obtained from a deterministic computational model. The assumed prior probability density of the surrogate is a Gaussian process. We determine the respective posterior probability density and parameters of the posited statistical model using variational Bayes. The non-Gaussian posterior is approximated by a Gaussian trial density with free variational parameters and the discrepancy between them is measured using the Kullback-Leibler (KL) divergence. We employ the stochastic gradient method to compute the variational parameters and other statistical model parameters by minimising the KL divergence. We demonstrate the accuracy and versatility of the proposed reduced dimension variational Gaussian process (RDVGP) surrogate on illustrative and robust structural optimisation problems where cost functions depend on a weighted sum of the mean and standard deviation of model outputs.

stat.AP↗

arXiv2024

Towards Quantum Computational Mechanics

The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of $\mathcal{O}((\log N)^c)$, compared to $\mathcal{O}(N^c)$ in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated $\mathcal{O} \left ((\log N)^c \right)$ complexity of the proposed solver.

quant-ph↗

arXiv2024

Point collocation with mollified piecewise polynomial approximants for high-order partial differential equations

The solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell-wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order of the piecewise functions and the smoothness of the mollifier. In this work, we exploit the high-order and high-smoothness properties of the mollified basis functions for solving PDEs through the point collocation method. The basis functions are evaluated at a set of collocation points in the domain. In addition, boundary conditions are imposed at a set of boundary collocation points distributed over the domain boundaries. To ensure the stability of the resulting linear system of equations, the number of collocation points is set larger than the total number of basis functions. The resulting linear system is overdetermined and is solved using the least square technique. The presented numerical examples confirm the convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. We study in particular the influence of the mollifier and the spatial distribution of the collocation points.

math.NA↗

arXiv2024

A three-grid high-order immersed finite element method for the analysis of CAD models

The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. We discretise physical fields on the finite element grid using high-order Lagrange basis functions. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains given by complex CAD models and discretised with tens of millions of degrees of freedom.

math.NA↗

arXiv2023

Robust topology optimisation of lattice structures with spatially correlated uncertainties

The uncertainties in material and other properties of structures are usually spatially correlated. We introduce an efficient technique for representing and processing spatially correlated random fields in robust topology optimisation of lattice structures. Robust optimisation considers the statistics of the structural response to obtain a design whose performance is less sensitive to the specific realisation of the random field. We represent Gaussian random fields on lattices by leveraging the established link between random fields and stochastic partial differential equations (SPDEs). It is known that the precision matrix, i.e. the inverse of the covariance matrix, of a random field with Matérn covariance is equal to the finite element stiffness matrix of a possibly fractional PDE with a second-order elliptic operator. We consider the discretisation of the PDE on the lattice to obtain a random field which, by design, considers its geometry and connectivity. The so-obtained random field can be interpreted as a physics-informed prior by the hypothesis that the elliptic SPDE models the physical processes occurring during manufacturing, like heat and mass diffusion. Although the proposed approach is general, we demonstrate its application to lattices modelled as pin-jointed trusses with uncertainties in member Young's moduli. We consider as a cost function the weighted sum of the expectation and standard deviation of the structural compliance. To compute the expectation and standard deviation and their gradients with respect to member cross-sections we use a first-order Taylor series approximation. The cost function and its gradient are computed using only sparse matrix operations. We demonstrate the efficiency of the proposed approach using several lattice examples with isotropic, anisotropic and non-stationary random fields and up to eighty thousand random and optimisation variables.

math.NA↗

arXiv2023

Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are either intended for unbounded domains or are too restrictive in terms of possible field properties. Because of these limitations, techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields have been gaining interest. The SPDE representation is especially appealing for engineering applications which already have a finite element discretisation for solving the physical conservation equations. In contrast to the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of an elliptic SPDE. We use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis and Gaussian process (GP) regression on complex geometries. The statistical finite element method (statFEM) introduced by Girolami et al. (2022) is a novel approach for synthesising measurement data and finite element models. In both statFEM and GP regression, we use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the SPDE so that we can model on bounded domains and manifolds anisotropic, non-stationary random fields with arbitrary smoothness. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The respective mean vector and precision matrix and can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with Poisson and thin-shell examples.

math.NA↗

arXiv2023

Fully probabilistic deep models for forward and inverse problems in parametric PDEs

We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational inference to assemble a fully probabilistic coherent framework. In the posited probabilistic model, both the forward and inverse maps are approximated as Gaussian distributions with a mean and covariance parameterized by deep neural networks. The PDE residual is assumed to be an observed random vector of value zero, hence we model it as a random vector with a zero mean and a user-prescribed covariance. The model is trained by maximizing the probability, that is the evidence or marginal likelihood, of observing a residual of zero by maximizing the evidence lower bound (ELBO). Consequently, the proposed methodology does not require any independent PDE solves and is physics-informed at training time, allowing the real-time solution of PDE forward and inverse problems after training. The proposed framework can be easily extended to seamlessly integrate observed data to solve inverse problems and to build generative models. We demonstrate the efficiency and robustness of our method on finite element discretized parametric PDE problems such as linear and nonlinear Poisson problems, elastic shells with complex 3D geometries, and time-dependent nonlinear and inhomogeneous PDEs using a physics-informed neural network (PINN) discretization. We achieve up to three orders of magnitude speed-up after training compared to traditional finite element method (FEM), while outputting coherent uncertainty estimates.

stat.ML↗

arXiv2023

Random Grid Neural Processes for Parametric Partial Differential Equations

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

cs.LG↗

arXiv2023

Quaternion variational integration for inertial maneuvering in a biomimetic UAV

Biological flying, gliding, and falling creatures are capable of extraordinary forms of inertial maneuvering: free-space maneuvering based on fine control of their multibody dynamics, as typified by the self-righting reflexes of cats. However, designing inertial maneuvering capability into biomimetic robots, such as biomimetic unmanned aerial vehicles (UAVs) is challenging. Accurately simulating this maneuvering requires numerical integrators that can ensure both singularity-free integration, and momentum and energy conservation, in a strongly coupled system - properties unavailable in existing conventional integrators. In this work, we develop a pair of novel quaternion variational integrators (QVIs) showing these properties, and demonstrate their capability for simulating inertial maneuvering in a biomimetic UAV showing complex multibody-dynamics coupling. Being quaternion-valued, these QVIs are innately singularity-free; and being variational, they can show excellent energy and momentum conservation properties. We explore the effect of variational integration order (left-rectangle vs. midpoint) on the conservation properties of integrator, and conclude that, in complex coupled systems in which canonical momenta may be time-varying, the midpoint integrator is required. The resulting midpoint QVI is well-suited to the analysis of inertial maneuvering in a biomimetic UAV - a feature that we demonstrate in simulation - and of other complex dynamical systems.

cs.RO↗