arXiv subjects

Reza Najian Asl

Publications and source records attributed to Reza Najian Asl.

9 recordsLinked to original sources

arXiv2026

Data-Free Weak-Form Staggered Neural Operators for Magneto-Mechanical Coupling in Finite-Strain Elastomers

Magneto-active elastomers, as a class of smart materials, exhibit strongly coupled magnetic and mechanical behavior at finite strains. Considering variations in microstructure, material properties, and geometry can lead to computationally expensive analyses. Building on the finite operator learning (FOL) framework, this study develops a data-free, physics-informed operator-learning framework for families of coupled finite-strain magneto-mechanical boundary-value problems. The governing magnetostatic and mechanical equations are enforced through finite-element weak-form residuals, allowing the neural operators to be trained without labeled finite-element solution data. The main contribution of this study is the development of a weak-form staggered neural operator (WSNO) framework for strongly coupled magneto-mechanical saddle-point problems. Separate magnetic and mechanical neural operators are trained alternately using a staggered optimization strategy, while their physical coupling is retained through the constitutive relations and residual evaluations. The resulting framework learns mappings from parameterized material and geometric descriptions to the corresponding coupled magnetic and mechanical solution fields. The proposed framework is investigated across several settings, including heterogeneous random-inclusion microstructures, varying magnetic phase contrast, area-fraction-dependent geometries, strongly out-of-distribution material morphologies, and three-dimensional geometry-parametric problems. In addition, the learned operator is combined with a neural-initialized Newton strategy, in which the nonlinear finite-element solver is initialized using the neural prediction. The results demonstrate that the proposed operator-learning framework can accurately capture coupled magneto-mechanical responses across a broad range of parametric problem settings.

cs.CE↗

arXiv2026

Tackling multiphysics problems via finite element-guided physics-informed operator learning

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

cs.LG↗

arXiv2025

A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries

In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics-informed encoder-decoder network to establish the mapping between continuous parameter and solution spaces. The decoder constructs the parametric solution field by leveraging an implicit neural field network conditioned on a latent or feature code. Instance-specific codes are derived through a PDE encoding process based on the second-order meta-learning technique. In training and inference, a physics-informed loss function is minimized during the PDE encoding and decoding. iFOL expresses the loss function in an energy or weighted residual form and evaluates it using discrete residuals derived from standard numerical PDE methods. This approach results in the backpropagation of discrete residuals during both training and inference. iFOL features several key properties: (1) its unique loss formulation eliminates the need for the conventional encode-process-decode pipeline previously used in operator learning with conditional neural fields for PDEs; (2) it not only provides accurate parametric and continuous fields but also delivers solution-to-parameter gradients without requiring additional loss terms or sensitivity analysis; (3) it can effectively capture sharp discontinuities in the solution; and (4) it removes constraints on the geometry and mesh, making it applicable to arbitrary geometries and spatial sampling (zero-shot super-resolution capability). We critically assess these features and analyze the network's ability to generalize to unseen samples across both stationary and transient PDEs. The overall performance of the proposed method is promising, demonstrating its applicability to a range of challenging problems in computational mechanics.

cs.LG↗

arXiv2025

Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs

We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single framework. We can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities, meaning the derivatives of the solution space with respect to the design space. These capabilities enable gradient-based optimization without the typical sensitivity analysis costs, unlike adjoint methods that scale directly with the number of response functions. Our Finite Operator Learning (FOL) approach uses an uncomplicated feed-forward neural network model to directly map the discrete design space (i.e. parametric input space) to the discrete solution space (i.e. finite number of sensor points in the arbitrary shape domain) ensuring compliance with physical laws by designing them into loss functions. The discretized governing equations, as well as the design and solution spaces, can be derived from any well-established numerical techniques. In this work, we employ the Finite Element Method (FEM) to approximate fields and their spatial derivatives. Subsequently, we conduct Sobolev training to minimize a multi-objective loss function, which includes the discretized weak form of the energy functional, boundary conditions violations, and the stationarity of the residuals with respect to the design variables. Our study focuses on the steady-state heat equation within heterogeneous materials that exhibits significant phase contrast and possibly temperature-dependent conductivity. The network's tangent matrix is directly used for gradient-based optimization to improve the microstructure's heat transfer characteristics. ...

cs.CE↗

arXiv2025

Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning

We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to approximate the nonlinear parametric solutionof the governing equations. This establishes a continuous mapping between the parameter and solution spaces, which can then be evaluated for a given parameter at any spatial resolution. Second, since the neural approximation may not be exact, it is subsequently refined using a Newton-based correction initialized by the neural output. To evaluate the effectiveness of this hybrid approach, we compare three solution strategies: (i) the standard Newton-Raphson solver used in NFEM, which is robust and accurate but computationally demanding; (ii) physics-informed neural operators, which provide rapid inference but may lose accuracy outside the training distribution and resolution; and (iii) the neural-initialized Newton (NiN) strategy, which combines the efficiency of neural operators with the robustness of NFEM. The results demonstrate that the proposed hybrid approach reduces computational cost while preserving accuracy, highlighting its potential to accelerate large-scale nonlinear simulations.

cs.LG↗

arXiv2024

A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations

To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics-informed neural networks and concept of neural operators. This approach generalizes and enhances each method, learning the parametric solution for mechanical problems without relying on data from other resources (e.g. other numerical solvers). We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics-based training yields higher accuracy compared to purely data-driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high-resolution solutions, avoiding the need to use basic interpolation techniques. First is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid point. Next, Fourier-based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. Comparisons with other well-known operator learning algorithms, further emphasize the advantages of the newly proposed method.

cs.LG↗

arXiv2022

Implicit bulk-surface filtering method for node-based shape optimization and comparison of explicit and implicit filtering techniques

This work studies shape filtering techniques, namely the convolution-based (explicit) and the PDE-based (implicit), and introduces an implicit bulk-surface filtering method to control the boundary smoothness and preserve the internal mesh quality simultaneously in the course of bulk (solid) shape optimization. To that end, volumetric mesh is filtered by the solution of pseudo-solid governing equations which are stiffened by the mesh-Jacobian and endowed with the Robin boundary condition which involves the Laplace-Beltrami operator on the mesh boundaries. Its superior performance from the non-simultaneous (sequential) treatment of boundary and internal meshes is demonstrated for the shape optimization of a complex solid structure. Well-established explicit filters, namely Gaussian and linear, and the Helmholtz/Sobolev-based (implicit) filter are critically examined in terms of consistency (rigid-body-movement production), geometric characteristics and the computational cost. It is shown that the implicit filtering is numerically more efficient and unconditionally consistent, compared to the explicit one. Supported by numerical experiments, a regularized Green's function is introduced as an equivalent explicit form of the Helmholtz/Sobolev filter. Furthermore, we give special attention to derive mesh-independent filtered sensitivities for node-based shape optimization with non-uniform meshes. It is shown that the mesh independent filtering can be achieved by scaling discrete sensitivities with the inverse of the mesh mass matrix.

math.NA↗

arXiv2019

A partitioned scheme for adjoint shape sensitivity analysis of fluid-structure interactions involving non-matching meshes

This work presents a partitioned solution procedure to compute shape gradients in fluid-structure interaction (FSI) using black-box adjoint solvers. Special attention is paid to project the gradients onto the undeformed configuration. This is due to the mixed Lagrangian-Eulerian formulation of large-displacement FSI in this work. Adjoint FSI problem is partitioned as an assembly of well-known adjoint fluid and structural problems, without requiring expensive cross-derivatives. The sub-adjoint problems are coupled with each other by augmenting the target functions with auxiliary functions, independent of the concrete choice of the underlying adjoint formulations. The auxiliary functions are linear force-based or displacement-based functionals which are readily available in well-established single-disciplinary adjoint solvers. Adjoint structural displacements, adjoint fluid displacements, and domain-based adjoint sensitivities of the fluid are the coupling fields to be exchanged between the adjoint solvers. A reduced formulation is also derived for the case of boundary-based adjoint shape sensitivity analysis for fluids. Numerical studies show that the complete formulation computes accurate shape gradients whereas inaccuracies appear in the reduced gradients, specially in regions of strong flow gradients and near singularities. Nevertheless, reduced gradient formulations are found to be a compromise between computational costs and accuracy. Mapping techniques including nearest element interpolation and the mortar method are studied in computational adjoint FSI. It is numerically shown that the mortar method does not introduce spurious oscillations in primal and sensitivity fields along non-matching interfaces, unlike the nearest element interpolation.

math.NA↗

arXiv2013

Sizing Optimization of Truss Structures using a Hybridized Genetic Algorithm

This paper presents a genetic-based hybrid algorithm that combines the exploration power of Genetic Algorithm (GA) with the exploitation capacity of a phenotypical probabilistic local search algorithm. Though not limited to a certain class of optimization problems, the proposed algorithm has been "fine-tuned" to work particularly efficiently on the optimal design of planar and space structures, a class of problems characterized by the large number of design variables and constraints, high degree of non-linearity and multitude of local minima. The proposed algorithm has been applied to the skeletal weight reduction of various planar and spatial trusses and shown to be superior in all of the cases,

math.OC↗