Parameter estimation in the pollutant dispersion problem with Physics-Informed Neural Networks

Authors

DOI:

https://doi.org/10.5902/2179460X74615

Keywords:

Inverse problem, Parameter estimation, Physics-Informed neural network

Abstract

In this work, the inverse problem of parameter estimation in the advection-dispersion-reaction equation, modelling the pollutant dispersion in a river, is studied with a Neural Network approach. In the direct problem, the dispersion, velocity and reaction parameters are known and then the initial and boundary value problem is solved by classical numerical methods, where it is used as input dataset for the inverse problem and formulation. In the inverse problem, we know the dispersion and the velocity parameters and also the information about the pollutant concentration from the synthetic experimental data, and then the aim is to estimate the reaction parameter in the advection-dispersion-reaction equation. This inverse problem is solved by an usual Artificial Neural Network (ANN) and by a Physics-Informed Neural Network (PINN), which is a special type of neural networks that includes in its formulation the physical laws that describe the phenomena involved. Numerical experiments with both the ANN and PINN are presented, demonstrating the feasibility of the approach considered.

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Author Biographies

Roberto Mamud Guedes da Silva, Federal University of Rio de Janeiro

Roberto Mamud is professor at Polytechnic Institute, Universidade Federal do Rio de Janeiro since 2013.
He is Bachelor and Master in Mathematics, both from Federal University of Rio de Janeiro and Doctor of Science in Nuclear Engineering at Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa em Engenharia. Professor Roberto Mamud main current research areas are: Inverse Problems; Heat and Mass Transfer; Computational Intelligence.

Helio dos Santos Migon, Federal University of Rio de Janeiro

Emeritus professor Helio S. Migon works at the Universidade Federal do Rio de Janeiro. In addition to his M.Sc. in Statistics from Universidade de São Paulo, he also received a doctorate in Statistics from Warwick University in the UK. Applied probability and statistics,  are the main research interests for Professor Helio S. Migon right presently. These topics mainly include Bayesian inference, dynamic models and forecasting, finite population sampling, applied econometrics in finance, and actuarial science.

Antônio José da Silva Neto, Rio de Janeiro State University

Antônio J. Silva Neto is a Full Professor in the Department of Mechanical Engineering and Energy, Polytechnic Institute, Universidade do Estado do Rio de Janeiro. He has Ph.D. in Mechanical Engineering (North Carolina State University), with a minor in Computational Mathematics; M.Sc. in Nuclear Engineering (Universidade Federal do Rio de Janeiro); and Mechanical/Nuclear Engineer (UFRJ). His main current research areas are: Inverse Problems; Heat and Mass Transfer; Radiative Transfer; Computational Intelligence; and Environmental Modelling.

References

Abadi, M., Agarwal, A., Barham, P., et. al (2015). Tensorflow: Large-scale machine learning on heterogeneous systems. Software available from tensorflow.org.

Baydin, A. G., Pearlmutter, B. A., Radul, A. A., Siskind, J. M. (2018). Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18, 1–43.

Chollet, F. (2018). Deep Learning with Python, 1st edn. Manning Publications, Shelter Island, NY.

Colton, D., Kress, R. (2013). Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, New York.

Evans, L. C. (1998). Partial Differential Equations, 1st edn. Americal Mathematical Society.

Goodfellow, I., Bengio, Y., Courville, A. (2017). Deep Learning, 1st edn. MIT Press, Cambridge, MA.

Hadamard, J. (1902). Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, pp. 49–52.

Isakov, V. (1990). Inverse Source Problem, 1st edn. American Mathematical Society, Providence, Rhode Island.

Kingma, D. P., Ba, J. L. (2015). Adam: A method for stochastic optimization. In: Proceedings of the International Conference on Learning Representations - ICLR 2015, San Diego, California.

LeCun, Y., Bengio, Y., Hinton, G. (2015). Deep learning. Nature, 521, 436–444.

Li, Z., Liu, F., Yang, W., Peng, S., Zhou, J. (2022). A survey of convolutional neural networks: Analysis, applications, and prospects. IEEE Transactions on Neural Networks and Learning Systems, 33(12), 6999–7019.

Maas, A. L., Hannun, A. Y., Ng, A. Y. (2013). Rectifier nonlinearities improve neural network acoustic models. In: Proceedings of the 30 th International Conference on Machine Learning, Atlanta, Georgia, USA.

Moura Neto, F. D., Silva Neto, A. J. (2013). An Introduction to Inverse Problems with Applications, 1st edn. Springer, New York.

Nielsen, M. A. (2015). Neural Networks and Deep Learning, 1st edn. Determination Press.

Raissi, M., Perdikaris, P., Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse probems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707.

Rojas, C. J., Bittencourt, M. L., Boldrini, J. L. (2021). Parameter identification for a damage model using a physics informed neural network. arXiv:210708781v1, pp. 1–31.

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Published

2023-12-01

How to Cite

Silva, R. M. G. da, Migon, H. dos S., & Silva Neto, A. J. da. (2023). Parameter estimation in the pollutant dispersion problem with Physics-Informed Neural Networks. Ciência E Natura, 45(esp. 3), e74615. https://doi.org/10.5902/2179460X74615

Issue

Vol. 45 No. esp. 3 (2023): ENMC/ECTM/MCSul/SEMENGO

Section

Special Edition

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