11. Application of physics-informed neural networks in simulating heat transfer and mass diffusion
DOI:
https://doi.org/10.61591/jslhu.22.878Từ khóa:
Heat transfer; Mass diffusion; Physics-informed neural networks; Computational physics; Generalization.Tóm tắt
This paper presents a novel approach to simulating classical physical phenomena-specifically heat transfer and mass diffusion-using Physics-Informed Neural Networks (PINNs), a class of deep neural networks that incorporate physical constraints. Unlike conventional machine learning models, PINNs allow the integration of empirical data with partial differential equations (PDEs) governing the underlying physical systems. This results in models capable of making accurate predictions even in the presence of incomplete or noisy data. The study constructs and trains PINN models for two canonical problems: heat conduction in a one-dimensional (1D) rod and concentration diffusion in a closed medium. Simulation results demonstrate that the PINNs achieve significantly lower prediction errors compared to standard neural networks without physical constraints, while also exhibiting strong generalization capabilities and numerical stability. This method offers a promising new direction for simulating physical processes, particularly in scenarios where real-world data are limited-making it well-suited for applications in education, engineering, and scientific research.
Tài liệu tham khảo
Trang N T. Simulation of heat conduction using the finite element method. 2020, Journal of Science and Technology – University of Danang, no. 9(130).
Son L V. Some diffusion models in environmental engineering. 2015, Journal of Environmental Engineering Science, no. 42.
Ngo P T. Engineering Thermodynamics. 2008, Hanoi, Vietnam: Science and Technology Publishing House.
Huy N C. Simulation of heat conduction in solids using FEM. 2019, Journal of Mechanics, no. 4.
Kien N T. Application of the finite volume method to simulate heat conduction. 2011, Journal of VNU–Hanoi Science, vol. A27, no. 1.
Karniadakis G E et al. Numerical Methods for Scientific Computing. 2005, Springer.
Dehghan P. Finite difference techniques for solving diffusion-type equations. 2006, Applied Mathematics and Computation, vol. 179, no. 1.
LeCun Y, Bengio Y, Hinton G. Deep learning. 2015, Nature, vol. 521.
Goodfellow I et al. Deep Learning. 2016, MIT Press.
Raissi M, Perdikaris P, Karniadakis G E. Deep hidden physics models: Deep learning of nonlinear partial differential equations. 2018, Journal of Machine Learning Research, vol. 19.
Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. 2019, Journal of Computational Physics, vol. 378, pp. 686–707.
Sun Y et al. PINNs for incompressible Navier–Stokes equations. 2020, Computers & Fluids.
Haghighat E et al. A physics-informed deep learning framework for inverse problems in elasticity. 2021, Computer Methods in Applied Mechanics and Engineering, vol. 379.
Zhu A et al. PINNs in electromagnetic field simulations. 2021, IEEE Access, vol. 9.
Cai W et al. Physics-informed neural networks for heat conduction problems. 2022, Thermal Science, vol. 26, no. 3.
Kharazmi A, Zhang Z, Karniadakis G E. Variational PINNs for time-dependent problems. 2020, Journal of Computational Physics, vol. 419.
Tuan D H. Simulation of heat propagation using artificial neural networks. 2023, Journal of Science and Technology in Transport, no. 39.
Thu Trang N T. Predicting heat conduction using machine learning regression. 2021, Journal of Information and Communication Technology, no. 2.
Nam T V. COMSOL-based simulation of heat conduction in electronic components. 2022, Journal of Aerospace Science and Technology, no. 42.
Cong N T. Simulation of diffusion temperature in solar energy panels. 2020, Vietnam Journal of Clean Energy, no. 6.
Zhu Y, Zabaras N, Koutsourelakis P-S, Perdikaris P. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. 2019, Journal of Computational Physics, vol. 394, pp. 56–81.
Khoa N D. Diffusion Engineering and Applications. 2015, Hanoi University of Science and Technology.
Dung N V. Analysis of boundary conditions in heat conduction problems. 2019, Journal of Science and Technology of Vietnam.
Kreyszig E. Advanced Engineering Mathematics, 10th ed. 2011, Wiley.
Wang S, Teng Y, Perdikaris P. Understanding and mitigating gradient pathologies in physics-informed neural networks. 2020, arXiv:2001.04536.
Abadi M et al. TensorFlow: Large-scale machine learning on heterogeneous systems. 2016, arXiv:1603.04467.
Lu L et al. DeepXDE: A deep learning library for solving differential equations. 2021, SIAM Review, vol. 63, no. 1, pp. 208–228.
Zhu Y, Zabaras N, Koutsourelakis P-S, Perdikaris P. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. 2020, Advances in Neural Information Processing Systems (NeurIPS).
Tuan T P A. Application of machine learning in solving partial differential equations. 2022, Journal of IT & Applications.
Nhan N T. Deep learning and applications in physical simulation. 2023, Journal of Natural Sciences.
Haghighat E, Bekar A, Madenci E, Juanes R. A nonlocal physics-informed deep learning framework using the peridynamic differential operator. 2020, arXiv:2006.00446.
Khang N V. Optimizing neural networks for temperature prediction. 2022, Journal of Informatics & Control, no. 3.
Hien T N. Automatic differentiation and applications in physical simulation. 2021, Journal of Science and Technology – Military Technical Academy.
Vinh N T. Recurrent neural networks for solving heat conduction problems. 2020, Journal of Science and Technology – Can Tho University.
Kharazmi A, Zhang Z, Karniadakis G E. Variational physics-informed neural networks for time-dependent PDEs. 2020, Computer Methods in Applied Mechanics and Engineering, vol. 370.
Haghighat E, Juanes R. XPINNs: Parallel physics-informed neural networks for forward and inverse PDE problems. 2021, Journal of Computational Physics, vol. 443.
Mao Z, Dissanayake A, Karniadakis G E. fPINNs: Fractional physics-informed neural networks for fractional PDEs. 2020, Journal of Computational Physics, vol. 423.
Raissi M, Karniadakis G E. Hidden physics models: Machine learning of nonlinear partial differential equations. 2018, Journal of Computational Physics, vol. 357, pp. 125–141.
Lu L, Zou Y, Wang J, Zhang L, Deng X. Unsupervised learning with physics informed graph networks for partial differential equations. 2025, Applied Intelligence, vol. 55, Article 617.
Raissi M, Yazdani A, Karniadakis G E. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. 2020, Science, vol. 367(6481), pp. 1026–1030.
Kharazmi E, Zhang Z, Karniadakis G E. hp-VPINNs: Variational physics-informed neural networks with domain decomposition. 2021, Computer Methods in Applied Mechanics and Engineering, vol. 374, 113547.
