AI4PDE

Energy · collocation · transfer

Selected publications

Selected research and review articles on machine learning for PDEs, including energy methods, deep collocation, transfer learning, neural operators, and computational mechanics.

01
NATURE COMMUNICATIONS · 2026

Towards Unified AI-Driven Fracture Mechanics: The Extended 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.

XDEM unifies discrete and phase-field fracture descriptions, enriching neural fields with discontinuities and crack-tip asymptotics while using relatively sparse, uniformly distributed collocation points.

Nature Communications 17, 8492BibTeX ↓Open ↗
@article{wang2026xdem,
  title={Towards Unified AI-Driven Fracture Mechanics: The Extended Deep Energy Method (XDEM)},
  journal={Nature Communications}, volume={17}, pages={8492}, year={2026},
  doi={10.1038/s41467-026-76748-1}
}
02
AMR · 2026

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, Xiqiao Feng, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu.

A comprehensive review of PINNs, deep energy methods, operator learning, and physics-informed neural operators across solid, fluid, and biomechanical PDEs.

Applied Mechanics Reviews, 1–81BibTeX ↓Open ↗
@article{wang2026review,
  title={Artificial Intelligence for Partial Differential Equations in Computational Mechanics: A Review},
  journal={Applied Mechanics Reviews}, pages={1--81}, year={2026},
  doi={10.1115/1.4071710}
}
03
IJMSD · 2025

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

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

Full fine-tuning, lightweight fine-tuning, and LoRA are compared across changing boundary conditions, materials, and geometries in strong- and energy-form PINNs.

International Journal of Mechanical System Dynamics 5(2), 212–235BibTeX ↓Open ↗
@article{wang2025transfer,
  title={Transfer Learning in Physics-Informed Neural Networks: Full Fine-Tuning, Lightweight Fine-Tuning, and Low-Rank Adaptation},
  journal={International Journal of Mechanical System Dynamics}, volume={5}, number={2}, pages={212--235}, year={2025},
  doi={10.1002/msd2.70030}
}
04
JMPS · 2026

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.

A physics-informed neural operator provides fast initial solutions that are refined by finite elements for accuracy, robustness, and convergence.

Journal of the Mechanics and Physics of Solids 214, 106682BibTeX ↓Open ↗
@article{wang2026pfem,
  title={Pretrain Finite Element Method: A Pretraining and Warm-Start Framework for PDEs via Physics-Informed Neural Operators},
  journal={Journal of the Mechanics and Physics of Solids}, volume={214}, pages={106682}, year={2026},
  doi={10.1016/j.jmps.2026.106682}
}
05
C&S · 2026

Deep Energy Method with Large Language Model-Assisted Geometry Modeling: An Open-Source Streamlit-Based Platform for Solving Variational PDEs

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

An interactive platform connects large-model-assisted geometry generation with energy-based PDE solutions and finite-element references.

Computers & Structures 329, 108302BibTeX ↓Open ↗
@article{wang2026lmdem,
  title={Deep Energy Method with Large Language Model-Assisted Geometry Modeling},
  journal={Computers and Structures}, volume={329}, pages={108302}, year={2026},
  doi={10.1016/j.compstruc.2026.108302}
}
06
IJMS · 2026

Multi-Head Neural Operator for Modeling Interfacial Dynamics

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

Time-specific projections and explicit temporal connections enable full trajectory prediction for long-time interfacial dynamics.

International Journal of Mechanical Sciences 317, 111363BibTeX ↓Open ↗
@article{eshaghi2026mhno,
  title={Multi-Head Neural Operator for Modeling Interfacial Dynamics},
  journal={International Journal of Mechanical Sciences}, volume={317}, pages={111363}, year={2026},
  doi={10.1016/j.ijmecsci.2026.111363}
}
07
CMAME · 2026

NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers

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

Learned solution operators produce high-quality initial guesses for classical iterative solvers while preserving their numerical role.

Computer Methods in Applied Mechanics and Engineering 458, 118989BibTeX ↓Open ↗
@article{eshaghi2026nows,
  title={NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers},
  journal={Computer Methods in Applied Mechanics and Engineering}, volume={458}, pages={118989}, year={2026},
  doi={10.1016/j.cma.2026.118989}
}
08
CMAME · 2025

Kolmogorov–Arnold-Informed Neural Network: A Physics-Informed Deep Learning Framework for Solving Forward and Inverse Problems

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

KINN evaluates Kolmogorov–Arnold representations across forward and inverse PDE problems in computational mechanics.

Computer Methods in Applied Mechanics and Engineering 433, 117518BibTeX ↓Open ↗
@article{wang2025kinn,
  title={Kolmogorov--Arnold-Informed Neural Network: A Physics-Informed Deep Learning Framework for Solving Forward and Inverse Problems},
  journal={Computer Methods in Applied Mechanics and Engineering}, volume={433}, pages={117518}, year={2025},
  doi={10.1016/j.cma.2024.117518}
}
09
CMAME · 2025

Variational Physics-Informed Neural Operator for Solving Partial Differential Equations

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

VINO embeds variational physics into neural operators and supports training without labelled solution data.

Computer Methods in Applied Mechanics and Engineering 437, 117785BibTeX ↓Open ↗
@article{eshaghi2025vino,
  title={Variational Physics-Informed Neural Operator for Solving Partial Differential Equations},
  journal={Computer Methods in Applied Mechanics and Engineering}, volume={437}, pages={117785}, year={2025},
  doi={10.1016/j.cma.2025.117785}
}
10
NEUROCOMPUTING · 2025

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.

A comparative scientific-machine-learning study for functionally graded porous beam analysis.

Neurocomputing 619, 129119BibTeX ↓Open ↗
@article{eshaghi2025beams,
  title={Applications of Scientific Machine Learning for the Analysis of Functionally Graded Porous Beams},
  journal={Neurocomputing}, volume={619}, pages={129119}, year={2025},
  doi={10.1016/j.neucom.2024.129119}
}
11
CMAME · 2020

An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications

E. Samaniego, C. Anitescu, S. Goswami, V. M. Nguyen-Thanh, H. Guo, K. Hamdia, Xiaoying Zhuang, Timon Rabczuk.

A foundational framework connecting variational energy principles, neural approximation, implementation, and applications for PDEs in computational mechanics.

Computer Methods in Applied Mechanics and Engineering 362, 112790BibTeX ↓Open ↗
@article{samaniego2020energy,
  title={An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications},
  journal={Computer Methods in Applied Mechanics and Engineering}, volume={362}, pages={112790}, year={2020},
  doi={10.1016/j.cma.2019.112790}
}
12
CMC · 2019

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

Hongwei Guo, Xiaoying Zhuang, Timon Rabczuk.

A physics-informed deep collocation formulation for fourth-order Kirchhoff plate bending equations and their boundary conditions.

Computers, Materials & Continua 59, 433–456BibTeX ↓Open ↗
@article{guo2019plate,
  title={A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate},
  journal={Computers, Materials and Continua}, volume={59}, pages={433--456}, year={2019},
  doi={10.32604/cmc.2019.06660}
}
13
EWC · 2022

Analysis of Three-Dimensional Potential Problems in Non-Homogeneous Media with Physics-Informed Deep Collocation Method Using Material Transfer Learning and Sensitivity Analysis

Hongwei Guo, Xiaoying Zhuang, Pengwan Chen, Naif Alajlan, Timon Rabczuk.

Material transfer learning improves generality across material gradations, while global sensitivity analysis identifies influential network configurations.

Engineering with Computers 38, 5423–5444BibTeX ↓Open ↗
@article{guo2022potential,
  title={Analysis of Three-Dimensional Potential Problems in Non-Homogeneous Media with Physics-Informed Deep Collocation Method Using Material Transfer Learning and Sensitivity Analysis},
  journal={Engineering with Computers}, volume={38}, pages={5423--5444}, year={2022},
  doi={10.1007/s00366-022-01633-6}
}
14
EWC · 2022

Stochastic Deep Collocation Method Based on Neural Architecture Search and Transfer Learning for Heterogeneous Porous Media

Hongwei Guo, Xiaoying Zhuang, Pengwan Chen, Naif Alajlan, Timon Rabczuk.

Sensitivity-guided architecture search and transfer learning reduce the cost of stochastic three-dimensional flow simulation in highly heterogeneous porous media.

Engineering with Computers 38, 5173–5198BibTeX ↓Open ↗
@article{guo2022stochastic,
  title={Stochastic Deep Collocation Method Based on Neural Architecture Search and Transfer Learning for Heterogeneous Porous Media},
  journal={Engineering with Computers}, volume={38}, pages={5173--5198}, year={2022},
  doi={10.1007/s00366-021-01586-2}
}