NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

Mohammad Sadegh Eshaghi, Cosmin Anitescu, Navid Valizadeh, Yizheng Wang, Xiaoying Zhuang, Timon Rabczuk

PDF

• We propose Neural Operator Warm Starts (NOWS), a hybrid strategy that harnesses learned solution operators to accelerate classical iterative solvers by producing high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. NOWS integrates seamlessly with existing discretizations (finite-difference, finite-element, isogeometric analysis, finite volume method, etc.), consistently reducing iteration counts and computational time by up to 90%, while preserving the stability and convergence guarantees of traditional numerical algorithms.

Multi-Head Neural Operator for Modelling Interfacial Dynamics

Mohammad Sadegh Eshaghi, Cosmin Anitescu, Navid Valizadeh, Yizheng Wang, Xiaoying Zhuang, Timon Rabczuk

PDF

• We introduce the Multi-Head Neural Operator (MHNO), a novel neural operator architecture specifically designed to address temporal challenges in solving time-dependent PDEs governing interfacial dynamics.

Variational Physics-informed Neural Operator (VINO) for Solving Partial Differential Equations

Mohammad Sadegh Eshaghi, Cosmin Anitescu, Manish Thombre, Yizheng Wang, Xiaoying Zhuang, Timon Rabczuk

PDF

• We propose the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs through the integration of neural operators with variational principles. This framework can be trained without any labeled data, significantly reducing computational costs.

PENCO: A Physics-Energy-Numerical-Consistent Operator for 3D Phase Field Modeling

Mostafa Bamdad, Mohammad Sadegh Eshaghi, Cosmin Anitescu, Navid Valizadeh, Timon Rabczuk

PDF

• We propose PENCO, a hybrid operator-learning framework that integrates physical laws with data-driven neural operators for solving spatiotemporal PDEs in phase-field modeling. PENCO introduces an enhanced H¹ Gauss–Lobatto collocation residual for robust enforcement of governing dynamics, a Fourier-space numerical consistency term capturing semi-implicit discretization behavior, and an energy-dissipation constraint ensuring thermodynamic consistency.

Artificial intelligence for partial differential equations in computational mechanics: A review

Yizheng Wang, Jinshuai Bai, Zhongya Lin, Qimin Wang, Cosmin Anitescu, Jia Sun, Mohammad Sadegh Eshaghi, Yuantong Gu, Xi-Qiao Feng, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu PDF

• We review AI for PDEs in computational mechanics, including solid mechanics, fluid mechanics, and biomechanics.

Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving PDEs based on Kolmogorov Arnold Networks

Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu

PDF

• We propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN’s potential for more efficient and accurate PDE solutions in AI for PDEs.

Methods for enabling real-time analysis in digital twins: A literature review

Mohammad Sadegh Eshaghi, Cosmin Anitescu, Timon Rabczuk

PDF

• This paper presents a comprehensive literature review on methods for enabling real-time analysis in digital twins—virtual models of physical systems. We systematically review and categorize methods and tools for reducing computational demands, accelerating the modeling of physical phenomena, and addressing challenges such as real-time data analysis, resource limitations, and data uncertainty to support cost reduction, risk mitigation, efficiency enhancement, and decision-making in digital twin implementations.

Applications of scientific machine learning for the analysis of functionally graded porous beams

Mohammad Sadegh Eshaghi, Mostafa Bamdad, Cosmin Anitescu, Yizheng Wang, Xiaoying Zhuang, Timon Rabczuk

PDF

• We investigate and compare different SciML approaches for analyzing functionally graded porous beams with arbitrary continuous material property variations. We implement three formulations: (a) the vector approach leading to Physics-Informed Neural Networks (PINNs), (b) the energy approach resulting in the Deep Energy Method (DEM), and (c) the data-driven approach yielding Neural Operator methods.

Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight FineTuning, and Low-Rank Adaptation

Yizheng Wang, Jinshuai Bai, Mohammad Sadegh Eshaghi, Cosmin Anitescu, Xiaoying Zhuang, Timon Rabczuk, and Yinghua Liu

PDF

• We explore the generalization capability of transfer learning in the strong and energy form of PINNs across different boundary conditions, materials, and geometries. The transfer learning methods we employ include full finetuning, lightweight finetuning, and Low-Rank Adaptation (LoRA).

Machine learning-based estimation of soil’s true air-entry value from GSD curves

Mohammad Sadegh Eshaghi, Mohammad Rezania, Meghdad Bagheri

PDF

• We develop a machine learning predictive model for estimating the true air-entry value (AEV) of soils from grain size distribution (GSD) curves using the UNSODA database of 790 soil samples. The model incorporates bulk density and GSD parameters to predict true AEV from water content-based soil water retention curves (SWRCs), achieving high accuracy and reliability.

Multicriteria decision-making methods in selecting seismic upgrading strategy of high-rise RC wall buildings

Mohammad Sadegh Eshaghi, Mohammad Sadegh Barkhordari, Zhenyu Huang, Jianqiao Ye

PDF

• This study investigates retrofitting strategies for high-rise reinforced concrete (RC) buildings with shear walls subjected to seismic loads. Four buildings were equipped with passive energy dissipation devices and analyzed under far-field and near-field earthquake records from FEMA P-695. Using validated numerical models, structural responses were evaluated and ranked through Multicriteria Decision Making methods.

Enhanced teacher-learning based algorithm in real size structural optimization

Mohammad Sadegh Eshaghi, Alireza Salehi, Alfred Strauss

PDF

• We propose an Enhanced Teacher-Learning Based Optimization algorithm for optimizing large-scale space frame structures. The algorithm improves upon the original TLBO method by incorporating a machine learning-based crossover operation between new solutions and the best solution during the teacher phase, enabling the algorithm to escape local minima and achieve faster convergence with better solution quality. This enhancement makes ETLBO particularly suitable for practical large-scale structural optimization problems such as halls, hangars, and passenger stations, where computational efficiency and accuracy are crucial.

Factors and failure patterns analysis for undrained seismic bearing capacity of strip footing above void

Mohammad Sadegh Eshaghi, Mohsen Abbaspour, Timon Rabczuk

PDF

• This study employs finite-element limit analysis to predict the undrained seismic bearing capacity and failure mechanisms of shallow strip footings on cohesive soils placed above unsupported rectangular voids. The research computes upper and lower bounds across comprehensive ranges of geometries, horizontal earthquake accelerations, material properties, and void configurations.

Machine learning-based prediction of the seismic bearing capacity of a shallow strip footing over a void in heterogeneous soils

Mohammad Sadegh Eshaghi, Mohsen Abbaspour, Hamidreza Abbasianjahromi, Stefano Mariani

PDF

• This paper presents a machine learning framework for predicting the seismic bearing capacity of shallow strip footings positioned above voids in heterogeneous soil. Using a dataset of 38,000 finite element limit analysis simulations, various ML techniques were compared. The study accounts for variations in soil properties (undrained shear strength and internal friction angle), horizontal earthquake accelerations, and void characteristics (position, shape, and size).

Straightforward prediction for responses of the concrete shear wall buildings subject to ground motions using machine learning algorithms

Mohammad Sadegh Barkhordari, Mohammad Sadegh Eshaghi

PDF

• This study evaluates machine learning and hybrid models for predicting seismic responses of reinforced concrete shear walls under strong ground motions. Using OpenSees, four buildings (15, 20, 25, and 30 stories) with concrete shear walls were analyzed with 150 seismic records to create a comprehensive database linking record characteristics to structural responses.

Replay-Based Continual Learning for Physics-Informed Neural Operators

Mohammad Sadegh Eshaghi, Cosmin Anitescu, Navid Valizadeh, Yizheng Wang, Xiaoying Zhuang, Timon Rabczuk

PDF

• We propose a replay-based continual learning framework for ML-based PDE solvers that enables efficient adaptation to new tasks while preserving knowledge from previously learned domains.

Towards Unified AI-Driven Fracture Mechanics: TheExtended Deep Energy Method (XDEM)

Yizheng Wang, Yuzhou Lin, Somdatta Goswami, Luyang Zhao, Huadong Zhang, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu

PDF

• We propose Extended Deep Energy Method (XDEM), a unified deep learning framework that incorporates both displacement discontinuities and crack-tip asymptotics in the discrete setting, while flexibly coupling displacement and phase fields in the continuous setting.

Pretrain Finite Element Method: A Pretraining and Warm-start Framework for PDEs via Physics-Informed Neural Operators

Yizheng Wang, Zhongkai Hao, Mohammad Sadegh Eshaghi, Cosmin Anitescu, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu

PDF

• We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM).

Deep Energy Method with Large Language Model assistance: an open-source Streamlit-based platform for solving variational PDEs

Yizheng Wang, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu

PDF

• We present LM-DEM(Large-Model-assisted Deep Energy Method), an open-source, Streamlit-based platform for solving variationalpartial differential equations (PDEs) in computational mechanics.