arXiv subjects

Hessam Babaee

Publications and source records attributed to Hessam Babaee.

At least 19 recordsLinked to original sources

arXiv2026

Implicit Tensor-Train Cross Integration of High-Dimensional Nonlinear PDEs via Fiber-Dependency Elimination

Tensor-train (TT) representations have emerged as an effective framework for mitigating the curse of dimensionality in the numerical solution of high-dimensional tensor differential equations. Among existing approaches, TT-cross methods are particularly attractive because they require only pointwise evaluations of the governing equations, naturally accommodate arbitrary nonlinearities, and avoid tangent-space projections and the numerical difficulties associated with nearly singular low-rank factors. However, existing TT-cross rank-truncation methods have been restricted to explicit time integration. Extending TT-cross methods to implicit schemes presents an obstacle: the collocation equations associated with the cross-selected fibers depend on neighboring fibers that are not part of the unknown set. Consequently, the resulting nonlinear system is not closed, preventing the direct application of standard implicit solvers. In this work, we introduce a principled fiber-dependency elimination framework that resolves this obstacle by expressing neighboring fibers as linear combinations of the cross-selected fibers through cross interpolation identities. The resulting formulation produces a closed collocation system while preserving the principal advantages of TT-cross methods. The proposed framework applies to both linear and nonlinear high-dimensional partial differential equations and is naturally combined with Newton iterations and rank adaptivity. Numerical experiments demonstrate rapid convergence of the dependency-elimination iterations, preservation of the temporal accuracy of implicit multistep schemes, and efficient implicit integration of high-dimensional nonlinear problems with full-order discretizations containing up to $10^{55}$ degrees of freedom.

math.NA↗

arXiv2025

Kinetic-Mamba: Mamba-Assisted Predictions of Stiff Chemical Kinetics

Accurate chemical kinetics modeling is essential for combustion simulations, as it governs the evolution of complex reaction pathways and thermochemical states. In this work, we introduce Kinetic-Mamba, a Mamba-based neural operator framework that integrates the expressive power of neural operators with the efficient temporal modeling capabilities of Mamba architectures. The framework comprises three complementary models: (i) a standalone Mamba model that predicts the time evolution of thermochemical state variables from given initial conditions; (ii) a constrained Mamba model that enforces mass conservation while learning the state dynamics; and (iii) a regime-informed architecture employing two standalone Mamba models to capture dynamics across temperature-dependent regimes. We additionally develop a latent Kinetic-Mamba variant that evolves dynamics in a reduced latent space and reconstructs the full state on the physical manifold. The accuracy and robustness of Kinetic-Mamba was evaluated using both time-decomposition and recursive-prediction strategies. We further assess the extrapolation capabilities of the model on varied out-of-distribution datasets. Computational experiments on Syngas and GRI-Mech 3.0 reaction mechanisms demonstrate that our framework achieves high fidelity in predicting complex kinetic behavior using only the initial conditions of the state variables.

cs.LG↗

arXiv2025

A CUR Krylov Solver for Large-Scale Linear Matrix Equations

Developing efficient solvers for large-scale multi-term linear matrix equations remains a central challenge in numerical linear algebra and is still largely unresolved. This paper introduces a methodology leveraging CUR decomposition for solving large-scale generalized Sylvester as well as non-Sylvester multi-term equations on low-rank matrix manifolds. The approach decomposes the original equation into two smaller subproblems: one involving all columns with a small subset of rows, and the other involving all rows with a small subset of columns. The rows and columns are strategically selected using the discrete empirical interpolation method. We further utilize the CUR properties and propose a novel iterative scheme that removes the dependencies between selected and unselected rows (and likewise for columns), thereby enabling the subset problems to be solved independently. We present a Krylov-based scheme for solving the resulting subproblems, which scales effectively to large problems and does not rely on a Sylvester structure. The method incorporates rank adaptivity, dynamically adjusting computational rank to reach the desired accuracy. The methodology is demonstrated in three representative settings: (i) implicit time integration of matrix differential equations on low-rank manifolds, leading to multi-term linear matrix equations; (ii) large-scale steady-state generalized Lyapunov equations including cases of size up to $10^{13}$ unknown entries; and (iii) non-Sylvester linear matrix equations with Hadamard product terms, such as those arising in nonlinear partial differential equations.

math.NA↗

arXiv2025

AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics

Time integration of stiff systems is a primary source of computational cost in combustion, hypersonics, and other reactive transport systems. This stiffness can introduce time scales significantly smaller than those associated with other physical processes, requiring extremely small time steps in explicit schemes or computationally intensive implicit methods. Consequently, strategies to alleviate challenges posed by stiffness are important. While neural operators (DeepONets) can act as surrogates for stiff kinetics, a reliable operator learning strategy is required to appropriately account for differences in error between output variables and samples. Here, we develop AMORE, Adaptive Multi-Output Operator Network, a framework comprising an operator capable of predicting multiple outputs and adaptive loss functions ensuring reliable operator learning. The operator predicts all thermochemical states from given initial conditions. We propose two adaptive loss functions within the framework, considering each state variable's and sample's error to penalize the loss function. We designed the trunk to automatically satisfy Partition of Unity. To enforce unity mass-fraction constraint exactly, we propose an invertible analytical map that transforms the $n$-dimensional species mass-fraction vector into an ($n-1$)-dimensional space. We extend the proposed adaptive loss functions to trunk and branch training in two-step training of DeepONet with multiple outputs. We implemented another unity mass fraction constraint exactly using a softmax function on the predicted mass fraction. We demonstrate efficacy and applicability of our models through two examples: syngas (12 states), GRI-Mech 3.0 (24 active states out of 54). The proposed DeepONet will be a backbone for future CFD studies to accelerate turbulent combustion simulations. AMORE is a general framework, and here, we also demonstrate it for FNO.

cs.LG↗

arXiv2025

An Adaptive CUR Algorithm and its Application to Reduced-Order Modeling of Random PDEs

Certain classes of CUR algorithms, also referred to as cross or pseudoskeleton algorithms, are widely used for low-rank matrix approximation when direct access to all matrix entries is costly. Their key advantage lies in constructing a rank-r approximation by sampling only r columns and r rows of the target matrix. This property makes them particularly attractive for reduced-order modeling of nonlinear matrix differential equations, where nonlinear operations on low-rank matrices can otherwise produce high-rank or even full-rank intermediates that must subsequently be truncated to rank $r$. CUR cross algorithms bypass the intermediate step and directly form the rank-$r$ matrix. However, standard cross algorithms may suffer from loss of accuracy in some settings, limiting their robustness and broad applicability. In this work, we propose a cross oversampling algorithm that augments the intersection with additional sampled columns and rows. We provide an error analysis demonstrating that the proposed oversampling improves robustness. We also present an algorithm that adaptively selects the number of oversampling entries based on efficiently computable indicators. We demonstrate the performance of the proposed CUR algorithm for time integration of several nonlinear stochastic PDEs on low-rank matrix manifolds.

math.NA↗

arXiv2025

Skeletal Reaction Models for Gasoline Surrogate Combustion

Skeletal reaction models are derived for a four-component gasoline surrogate model via an instantaneous local sensitivity analysis technique. The sensitivities of the species mass fractions and the temperature with respect to the reaction rates are estimated by a reduced-order modeling (ROM) methodology. Termed "implicit time-dependent basis CUR (implicit TDB-CUR)," this methodology is based on the CUR matrix decomposition and incorporates implicit time integration for evolving the bases. The estimated sensitivities are subsequently analyzed to develop skeletal reaction models with a fully automated procedure. The 1389-species gasoline surrogate model developed at Lawrence Livermore National Laboratory (LLNL) is selected as the detailed kinetics model. The skeletal reduction procedure is applied to this model in a zero-dimensional constant-pressure reactor over a wide range of initial conditions. The performances of the resulting skeletal models are appraised by comparison against the results via the LLNL detailed model, and also predictions via other skeletal models. Two new skeletal models are developed consisting of 679 and 494 species, respectively. The first is an alternative to an existing model with the same number of species. The predictions with this model reproduces the detailed models vital flame results with less than 1% errors. The errors via the second model are less than 10%.

cs.CE↗

arXiv2025

On-the-fly Reduced-Order Modeling of the Filter Density Function with Time-Dependent Subspaces

A dynamical low-rank approximation is developed for reduced-order modeling (ROM) of the filtered density function (FDF) transport equation, which is utilized for large eddy simulation (LES) of turbulent reacting flows. In this methodology, the evolution of the composition matrix describing the FDF transport via a set of Langevin equations is constrained to a low-rank matrix manifold. The composition matrix is approximated using a low-rank factorization, which consists of two thin, time-dependent matrices representing spatial and composition bases, along with a small time-dependent coefficient matrix. The evolution equations for spatial and composition subspaces are derived by projecting the composition transport equation onto the tangent space of the low-rank matrix manifold. Unlike conventional ROMs, such as those based on principal component analysis, both subspaces are time-dependent and the ROM does not require any prior data to extract the low-dimensional subspaces. As a result, the constructed ROM adapts on the fly to changes in the dynamics. For demonstration, LES via the time-dependent bases (TDB) is conducted of the canonical configuration of a temporally developing planar CO/H2 jet flame. The flame is rich with strong flame-turbulence interactions resulting in local extinction followed by re-ignition. The combustion chemistry is modeled via the skeletal kinetics, containing 11 species with 21 reaction steps. It is shown that the FDF-TDB yields excellent predictions of various statistics of the thermo-chemistry variables, as compared to the full-order model (FOM).

physics.flu-dyn↗

arXiv2025

Optimally time-dependent modes of vortex gust-airfoil interactions

We find the optimally time-dependent (OTD) orthogonal modes about a time-varying flow generated by a strong gust vortex impacting a NACA 0012 airfoil. This OTD analysis reveals the amplification characteristics of perturbations about the unsteady base flow and their amplified spatiotemporal structures that evolve over time. We consider four time-varying laminar base flows in which a vortex with a strength corresponding to the gust ratio $G$ of $\{-1,-0.5,0.5,1\}$ impinges on the leading edge of the airfoil at an angle of attack of $12^\circ$. In these cases, the impingement of the strong gust vortex causes massive separation and the generation of large-scale vortices around the airfoil within two convective time units. The highly unsteady nature of these vortex-airfoil interactions necessitates an advanced analytical technique capable of capturing the transient perturbation dynamics. For each of the considered gust ratios, the OTD analysis identifies the most amplified region to perturbations, the location of which changes as the wake evolves differently. For interactions between a moderate positive vortex gust ($G=0.5$) and the airfoil, the area where perturbations are amplified transitions from the leading-edge vortex sheet to the forming leading-edge vortex. Later, this most amplified structure becomes supported in the airfoil wake directly behind the trailing edge. In contrast, a strong vortex gust ($G=\pm 1$) encountered by the airfoil shows the most amplified OTD mode to appear around the core of the shed vortices. This study provides an analysis technique and fundamental insights into the broader family of unsteady aerodynamic problems.

physics.flu-dyn↗

arXiv2024

CUR for Implicit Time Integration of Random Partial Differential Equations on Low-Rank Matrix Manifolds

Dynamical low-rank approximation allows for solving large-scale matrix differential equations (MDEs) with significantly fewer degrees of freedom and has been applied to a growing number of applications. However, most existing techniques rely on explicit time integration schemes. In this work, we introduce a cost-effective Newton's method for the implicit time integration of stiff, nonlinear MDEs on low-rank matrix manifolds. Our methodology is focused on MDEs resulting from the discretization of random partial differential equations (PDEs). Cost-effectiveness is achieved by solving the MDE at the minimum number of entries required for a rank-$r$ approximation. We present a novel CUR low-rank approximation that requires solving the parametric PDE at $r$ strategically selected parameters and $\mathcal{O}(r)$ grid points using Newton's method. The selected random samples and grid points adaptively vary over time and are chosen using the discrete empirical interpolation method or similar techniques. The proposed methodology is developed for high-order implicit multistep and Runge-Kutta schemes and incorporates rank adaptivity, allowing for dynamic rank adjustment over time to control error. Several analytical and PDE examples, including the stochastic Burgers' and Gray-Scott equations, demonstrate the accuracy and efficiency of the presented methodology.

math.NA↗

arXiv2024

Cross Interpolation for Solving High-Dimensional Dynamical Systems on Low-Rank Tucker and Tensor Train Manifolds

We present a novel tensor interpolation algorithm for the time integration of nonlinear tensor differential equations (TDEs) on the tensor train and Tucker tensor low-rank manifolds, which are the building blocks of many tensor network decompositions. This paper builds upon our previous work (Donello et al., Proceedings of the Royal Society A, Vol. 479, 2023) on solving nonlinear matrix differential equations on low-rank matrix manifolds using CUR decompositions. The methodology we present offers multiple advantages: (i) It delivers near-optimal computational savings both in terms of memory and floating-point operations by leveraging cross algorithms based on the discrete empirical interpolation method to strategically sample sparse entries of the time-discrete TDEs to advance the solution in low-rank form. (ii) Numerical demonstrations show that the time integration is robust in the presence of small singular values. (iii) High-order explicit Runge-Kutta time integration schemes are developed. (iv) The algorithm is easy to implement, as it requires the evaluation of the full-order model at strategically selected entries and does not use tangent space projections, whose efficient implementation is intrusive. We demonstrate the efficiency of the presented algorithm for several test cases, including a nonlinear 100-dimensional TDE for the evolution of a tensor of size $70^{100} \approx 3.2 \times 10^{184}$ and a stochastic advection-diffusion-reaction equation with a tensor of size $4.7 \times 10^9$.

math.NA↗

arXiv2024

Reconstructing Blood Flow in Data-Poor Regimes: A Vasculature Network Kernel for Gaussian Process Regression

Blood flow reconstruction in the vasculature is important for many clinical applications. However, in clinical settings, the available data are often quite limited. For instance, Transcranial Doppler ultrasound (TCD) is a noninvasive clinical tool that is commonly used in the clinical settings to measure blood velocity waveform at several locations on brain's vasculature. This amount of data is grossly insufficient for training machine learning surrogate models, such as deep neural networks or Gaussian process regression. In this work, we propose a Gaussian process regression approach based on physics-informed kernels, enabling near-real-time reconstruction of blood flow in data-poor regimes. We introduce a novel methodology to reconstruct the kernel within the vascular network, which is a non-Euclidean space. The proposed kernel encodes both spatiotemporal and vessel-to-vessel correlations, thus enabling blood flow reconstruction in vessels that lack direct measurements. We demonstrate that any prediction made with the proposed kernel satisfies the conservation of mass principle. The kernel is constructed by running stochastic one-dimensional blood flow simulations, where the stochasticity captures the epistemic uncertainties, such as lack of knowledge about boundary conditions and uncertainties in vasculature geometries. We demonstrate the performance of the model on three test cases, namely, a simple Y-shaped bifurcation, abdominal aorta, and the Circle of Willis in the brain.

eess.IV↗

arXiv2024

A DEIM Tucker Tensor Cross Algorithm and its Application to Dynamical Low-Rank Approximation

We introduce a Tucker tensor cross approximation method that constructs a low-rank representation of a $d$-dimensional tensor by sparsely sampling its fibers. These fibers are selected using the discrete empirical interpolation method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling (DEIM-FS). For a rank-$r$ approximation of an $\mathcal{O}(N^d)$ tensor, DEIM-FS requires access to only $dNr^{d-1}$ tensor entries, a requirement that scales linearly with the tensor size along each mode. We demonstrate that DEIM-FS achieves an approximation accuracy close to the Tucker-tensor approximation obtained via higher-order singular value decomposition at a significantly reduced cost. We also present DEIM-FS (iterative) that does not require access to singular vectors of the target tensor unfolding and can be viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the computational cost associated with solving nonlinear tensor differential equations (TDEs) using dynamical low-rank approximation (DLRA). The computational cost of solving DLRA equations can become prohibitive when the exact rank of the right-hand side tensor is large. This issue arises in many TDEs, especially in cases involving non-polynomial nonlinearities, where the right-hand side tensor has full rank. This necessitates the storage and computation of tensors of size $\mathcal{O}(N^d)$. We show that DEIM-FS results in significant computational savings for DLRA by constructing a low-rank Tucker approximation of the right-hand side tensor on the fly. Another advantage of using DEIM-FS is to significantly simplify the implementation of DLRA equations, irrespective of the type of TDEs. We demonstrate the efficiency of the algorithm through several examples including solving high-dimensional partial differential equations.

math.NA↗

arXiv2023

Time-Dependent Low-Rank Input-Output Operator for Forced Linearized Dynamics with Unsteady Base Flows

Understanding the linear growth of disturbances due to external forcing is crucial for flow stability analysis, flow control, and uncertainty quantification. These applications typically require a large number of forward simulations of the forced linearized dynamics, often in a brute-force fashion. When dealing with simple steady-state or periodic base flows, there exist powerful and cost-effective solution operator techniques. Once these solution operators are constructed, they can be used to determine the response to various forcings with negligible computational cost. However, these methods do not apply to problems with arbitrarily time-dependent base flows. This paper develops and investigates reduced-order modeling with time-dependent bases (TDBs) to build low-rank solution operators for forced linearized dynamics with arbitrarily time-dependent base flows. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. Several demonstrations are included to illustrate the utility of the f-OTD low-rank approximation for performing global transient stability analysis. Additionally, we demonstrate the application of f-OTD in computing the post-transient response of linearized Navier-Stokes equations to a large number of impulses, which has applications in flow control.

physics.flu-dyn↗

arXiv2023

Learning stiff chemical kinetics using extended deep neural operators

We utilize neural operators to learn the solution propagator for the challenging chemical kinetics equation. Specifically, we apply the deep operator network (DeepONet) along with its extensions, such as the autoencoder-based DeepONet and the newly proposed Partition-of-Unity (PoU-) DeepONet to study a range of examples, including the ROBERS problem with three species, the POLLU problem with 25 species, pure kinetics of the syngas skeletal model for $CO/H_2$ burning, which contains 11 species and 21 reactions and finally, a temporally developing planar $CO/H_2$ jet flame (turbulent flame) using the same syngas mechanism. We have demonstrated the advantages of the proposed approach through these numerical examples. Specifically, to train the DeepONet for the syngas model, we solve the skeletal kinetic model for different initial conditions. In the first case, we parametrize the initial conditions based on equivalence ratios and initial temperature values. In the second case, we perform a direct numerical simulation of a two-dimensional temporally developing $CO/H_2$ jet flame. Then, we initialize the kinetic model by the thermochemical states visited by a subset of grid points at different time snapshots. Stiff problems are computationally expensive to solve with traditional stiff solvers. Thus, this work aims to develop a neural operator-based surrogate model to solve stiff chemical kinetics. The operator, once trained offline, can accurately integrate the thermochemical state for arbitrarily large time advancements, leading to significant computational gains compared to stiff integration schemes.

physics.chem-ph↗

arXiv2022

Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-$r$ reduced-order model (ROM) based on time-dependent basis, a.k.a. TDB-ROM, is roughly equal to that of solving the full-order model for $r$ random samples. As of now, this nominal performance can only be achieved for linear or quadratic SPDEs -- at the expense of a highly intrusive process. On the other hand, for problems with non-polynomial nonlinearity, the computational cost of solving the TDB evolution equations is the same as solving the full-order model. In this work, we present an adaptive sparse interpolation algorithm that enables stochastic TDB-ROMs to achieve nominal computational cost for generic nonlinear SPDEs. Our algorithm constructs a low-rank approximation for the right hand side of the SPDE using the discrete empirical interpolation method (DEIM). The presented algorithm does not require any offline computation and as a result the low-rank approximation can adapt to any transient changes of the dynamics on the fly. We also propose a rank-adaptive strategy to control the error of the sparse interpolation. Our algorithm achieves computational speedup by adaptive sampling of the state and random spaces. We illustrate the efficiency of our approach for two test cases: (1) one-dimensional stochastic Burgers' equation, and (2) two-dimensional compressible Navier-Stokes equations subject to one-hundred-dimensional random perturbations. In all cases, the presented algorithm results in orders of magnitude reduction in the computational cost.

math.NA↗

arXiv2022

Skeletal Reaction Models for Methane Combustion

A local-sensitivity-analysis technique is employed to generate new skeletal reaction models for methane combustion from the foundational fuel chemistry model (FFCM-1). The sensitivities of the thermo-chemical variables with respect to the reaction rates are computed via the forced-optimally time dependent (f-OTD) methodology. In this methodology, the large sensitivity matrix containing all local sensitivities is modeled as a product of two low-rank time-dependent matrices. The evolution equations of these matrices are derived from the governing equations of the system. The modeled sensitivities are computed for the auto-ignition of methane at atmospheric and high pressures with different sets of initial temperatures, and equivalence ratios. These sensitivities are then analyzed to rank the most important (sensitive) species. A series of skeletal models with different number of species and levels of accuracy in reproducing the FFCM-1 results are suggested. The performances of the generated models are compared against FFCM-1 in predicting the ignition delay, the laminar flame speed, and the flame extinction. The results of this comparative assessment suggest the skeletal models with 24 and more species generate the FFCM-1 results with an excellent accuracy.

physics.chem-ph↗

arXiv2022

Scalable In Situ Compression of Transient Simulation Data Using Time-Dependent Bases

Large-scale simulations of time-dependent problems generate a massive amount of data and with the explosive increase in computational resources the size of the data generated by these simulations has increased significantly. This has imposed severe limitations on the amount of data that can be stored and has elevated the issue of input/output (I/O) into one of the major bottlenecks of high-performance computing. In this work, we present an in situ compression technique to reduce the size of the data storage by orders of magnitude. This methodology is based on time-dependent subspaces and it extracts low-rank structures from multidimensional streaming data by decomposing the data into a set of time-dependent bases and a core tensor. We derive closed-form evolution equations for the core tensor as well as the time-dependent bases. The presented methodology does not require the data history and the computational cost of its extractions scales linearly with the size of data -- making it suitable for large-scale streaming datasets. To control the compression error, we present an adaptive strategy to add/remove modes to maintain the reconstruction error below a given threshold. We present four demonstration cases: (i) analytical example, (ii) incompressible unsteady reactive flow, (iii) stochastic turbulent reactive flow, and (iv) three-dimensional turbulent channel flow.

cs.CE↗

arXiv2022

Reduced Order Modeling of Turbulence-Chemistry Interactions using Dynamically Bi-Orthonormal Decomposition

The performance of the dynamically bi-orthogonal (DBO) decomposition for the reduced order modeling of turbulence-chemistry interactions is assessed. DBO is an on-the-fly low-rank approximation technique, in which the instantaneous composition matrix of the reactive flow field is decomposed into a set of orthonormal spatial modes, a set of orthonormal vectors in the composition space, and a factorization of the low-rank correlation matrix. Two factors which distinguish between DBO and the reduced order models (ROMs) based on the principal component analysis (PCA) are: (i) DBO does not require any offline data generation; and (ii) in DBO the low-rank composition subspace is time-dependent as opposed to static subspaces in PCA. Because of these features, DBO can adapt on-the-fly to intrinsic and externally excited transient changes in state of the transport variables. For demonstration, simulations are conducted of a non-premixed CO/H2 flame in a temporally evolving jet. The GRI-Mech 3.0 model with 53 species is used for chemical kinetics modeling. The results are appraised via a posteriori comparisons against data generated via full-rank direct numerical simulation (DNS) of the same flame, and the PCA reduction of the DNS data. The DBO also yields excellent predictions of various statistics of the thermo-chemical variables.

physics.flu-dyn↗