arXiv subjects

Parisa Khodabakhshi

Publications and source records attributed to Parisa Khodabakhshi.

6 recordsLinked to original sources

arXiv2026

Efficient model-order reduction via graph autoencoders and operator learning for systems with sharp gradients using a novel point cloud error metric for performance assessment

This study investigates the use of graph autoencoders, a class of graph neural networks (GNNs), to perform model-order reduction (MOR) on high-dimensional systems characterized by sharp gradients for which conventional linear approximations, such as proper orthogonal decomposition (POD) bases, are less effective. Specifically, this study introduces graph neural network latent space dynamics identification (GNN-LaSDI), a MOR framework that uses an operator learning strategy to directly evaluate the temporal evolution of the graph autoencoder's latent representation. In addition to using standard error metrics, this study presents a novel point cloud error metric tailored to directly evaluate the locations of sharp gradients, such as shocks in compressible fluid simulations and solid-liquid interfaces in phase-field simulations. The performance of GNN-LaSDI is compared to that of geometric deep least-squares Petrov-Galerkin (GD-LSPG), which uses a nonlinear manifold least-squares Petrov-Galerkin projection to evaluate the temporal evolution of the graph autoencoder's latent representation, and POD latent space dynamics identification (POD-LaSDI), which combines POD-based dimensionality reduction with operator learning. A constraint-enforcement strategy is incorporated into the graph autoencoder to ensure compliance with physical laws. For the studied problems, GNN-LaSDI substantially reduces the computational cost relative to GD-LSPG by eliminating repeated evaluations of the Jacobian of the decoder, while achieving significantly greater predictive accuracy than POD-LaSDI, thereby providing a balance between predictive accuracy and computational efficiency. Additionally, the proposed point cloud error provides a more informative measure of reduced-order model accuracy in capturing the locations of sharp gradients than conventional error metrics.

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arXiv2026

Hyper-reduction-free reduced-order Newton solvers for projection-based model-order reduction of nonlinear dynamical systems

This study proposes an intrusive projection-based model-order reduction framework for nonlinear problems with a polynomial structure, solved iteratively using a Newton solver in the reduced space. It is demonstrated that for the targeted class of polynomial nonlinearities, all operators appearing in the projected approximate residual and Jacobian can be precomputed in the offline phase, eliminating the need for hyper-reduction. Additionally, the evaluation of both the projected approximate residual and its Jacobian scales only with the dimension of the reduced space, and does not depend on the dimension of the full-order model, enabling effective offline-online decomposition. The proposed hyper-reduction-free (HRF) framework is applied to both Galerkin (HRF-G) and least-squares Petrov-Galerkin (HRF-LSPG) projection schemes. The accuracy and computational efficiency of the proposed HRF schemes are evaluated in two numerical experiments and compared with a commonly used hyper-reduction scheme, namely the energy-conserving sampling and weighting method, for both the Galerkin and LSPG schemes. In the first numerical example, a parametric Burgers' equation is used to assess the predictive capabilities of the considered model reduction approaches on parameter sets not seen in the training snapshots. In the second example, a parametric heat equation with a cubic reaction term is studied, for which a lifting transformation is employed to expose the desired structure. The efficacy of the HRF methods in accurately reducing the dimensionality of the lifted formulation is investigated. For the studied problems, the results show that HRF-G and HRF-LSPG achieve two and one order of magnitude speedup, respectively, with respect to the full-order model while resulting in state prediction errors below O(10^-2).

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arXiv2025

Meshless projection model-order reduction via reference spaces for smoothed-particle hydrodynamics

A model-order reduction framework for the meshless smoothed-particle hydrodynamics (SPH) method is presented. The proposed framework introduces the concept of modal reference spaces to overcome the challenges of discovering low-dimensional subspaces from unstructured, dynamic, and mixing numerical topology that occurs in SPH simulations. These reference spaces enable a low-dimensional representation of the field equations while maintaining the inherent meshless qualities of SPH. Modal reference spaces are constructed by projecting snapshot data onto a reference space where low-dimensionality of field quantities can be discovered via traditional modal decomposition techniques (e.g., the proper orthogonal decomposition (POD)). Modal quantities are mapped back to the meshless SPH space via scattered data interpolation during the online predictive stage. The proposed model-order reduction framework is cast into the meshless Galerkin POD and the Adjoint Petrov-Galerkin projection model-order reduction (PMOR) formulation. The PMORs are tested on three numerical experiments: 1) the Taylor--Green vortex; 2) the lid-driven cavity; and 3) the flow past an open cavity. Results show good agreement in reconstructed and predictive velocity fields, which showcase the ability of this framework to evolve the field equations in a low-dimensional subspace on an unstructured, dynamic, and mixing numerical topology. Results also show that the pressure field is sensitive to the projection error due to the stiff weakly-compressible assumption made in the current SPH framework, but this sensitivity can be alleviated through nonlinear approximations, such as the APG approach. The proposed meshless model-order reduction framework reports up to 90,000x dimensional compression within 10% error in quantities of interest, marking a step toward drastic cost reduction in SPH simulations.

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arXiv2025

Projection-based model-order reduction via graph autoencoders suited for unstructured meshes

This paper presents the development of a graph autoencoder architecture capable of performing projection-based model-order reduction (PMOR) using a nonlinear manifold least-squares Petrov-Galerkin (LSPG) projection scheme. The architecture is particularly useful for advection-dominated flows modeled by unstructured meshes, as it provides a robust nonlinear mapping that can be leveraged in a PMOR setting. The presented graph autoencoder is constructed with a two-part process that consists of (1) generating a hierarchy of reduced graphs to emulate the compressive abilities of convolutional neural networks (CNNs) and (2) training a message passing operation at each step in the hierarchy of reduced graphs to emulate the filtering process of a CNN. The resulting framework provides improved flexibility over traditional CNN-based autoencoders because it is readily extendable to unstructured meshes. We provide an analysis of the interpretability of the graph autoencoder's latent state variables, where we find that the Jacobian of the decoder for the proposed graph autoencoder provides interpretable mode shapes akin to traditional proper orthogonal decomposition modes. To highlight the capabilities of the proposed framework, which is named geometric deep least-squares Petrov-Galerkin (GD-LSPG), we benchmark the method on a one-dimensional Burgers' model with a structured mesh and demonstrate the flexibility of GD-LSPG by deploying it on two test cases for two-dimensional Euler equations that use an unstructured mesh. The proposed framework is more flexible than using a traditional CNN-based autoencoder and provides considerable improvement in accuracy for very low-dimensional latent spaces in comparison with traditional affine projections.

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arXiv2023

Multifidelity Methods for Uncertainty Quantification of a Nonlocal Model for Phase Changes in Materials

This study is devoted to the construction of a multifidelity Monte Carlo (MFMC) method for the uncertainty quantification of a nonlocal, non-mass-conserving Cahn-Hilliard model for phase transitions with an obstacle potential. We are interested in the estimation of the expected value of an output of interest (OoI) that depends on the solution of the nonlocal Cahn-Hilliard model. As opposed to its local counterpart, the nonlocal model captures sharp interfaces without the need for significant mesh refinement. However, the computational cost of the nonlocal Cahn-Hilliard model is higher than that of its local counterpart with similar mesh refinement, inhibiting its use for outer-loop applications such as uncertainty quantification. The MFMC method augments the desired high-fidelity, high-cost OoI with a set of lower-fidelity, lower-cost OoIs to alleviate the computational burden associated with nonlocality. Most of the computational budget is allocated to sampling the cheap surrogate models to achieve speedup, whereas the high-fidelity model is sparsely sampled to maintain accuracy. For the non-mass-conserving nonlocal Cahn-Hilliard model, the use of the MFMC method results in, for a given computational budget, about one-order-of-magnitude reduction in the mean-squared error of the expected value of the OoI relative to that of the Monte Carlo method.

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arXiv2023

Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning and physics-based modeling. The parametric structure of the governing equations is embedded directly into the reduced-order model, and parameterized reduced-order operators are learned via a data-driven linear regression problem. The result is a reduced-order model that can be solved rapidly to map parameter values to approximate PDE solutions. Such parameterized reduced-order models may be used as physics-based surrogates for uncertainty quantification and inverse problems that require many forward solves of parametric PDEs. Numerical issues such as well-posedness and the need for appropriate regularization in the learning problem are considered, and an algorithm for hyperparameter selection is presented. The method is illustrated for a parametric heat equation and demonstrated for the FitzHugh-Nagumo neuron model.

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