Fisher Kolmogorov Equation Theory Simulation Using Deep Learning

Authors

  • Conny Tria Shafira Telkom University, Bandung
  • Putu Harry Gunawan Telkom University, Bandung
  • Aditya Firman Ihsan Telkom University, Bandung

DOI:

https://doi.org/10.30865/mib.v7i1.5432

Keywords:

Simulation, Neural Network (NN), Physics-Informed Neural Network (PINN), Fisher-Kolmogorov, Partial Differential Equations (PDEs), Mean Squared Error (MSE)

Abstract

Neural Networks (NNs), a powerful tool for identifying non-linear systems, derive their computational power through a parallel distributed structure. The Physics-Informed Neural Network (PINN) technique can solve the Partial Differential Equation (PDP) in the Fisher Kolmogorov equation. By testing several hyperparameter changes, the formula is correct, and the visualization results can be consistent. Shows that an accurate value can be obtained from the results of the Mean Squared Error (MSE) on the formula loss value (loss f) and data loss (loss u). In experiment 1 the MSE obtained was 0.00001657 (Loss f) and 0.00000038 (loss u), as well as the MSE values obtained in experiment 4, is 0.00005865 (Loss f) and 0.00000216 (Loss u). It can be said to be accurate if the MSE value is close to 0. A formula is proven correct if it displays consistent data in random input data, but with the condition that it uses the same parameters. The author conducted research to simulate the Fisher-Kolmogorov equation with deep learning using the PINN technique. So the purpose of the research conducted was to simulate the Fisher-Kolmogorov equation with the deep learning method using the PINN technique. From the research, it can be concluded that Fisher-Kolmogorov's equation proves to be true if the simulation is carried out in deep learning and produces a visualization that is consistent with the functions used for visualization.

References

M. R. Ebert and M. Reissig, Basics for Partial Differential Equations. 2018.

A. Hasan, M. Pereira, and R. Ravier, “LEARNING PARTIAL DIFFERENTIAL EQUATIONS FROM DATA USING NEURAL NETWORKS Vahid Tarokh Department of Biomedical Engineering, Duke University, Durham NC 27708 Department of Electrical and Computer Engineering, Duke University, Durham NC 27708,†pp. 3962–3966, 2020.

H. P. Breuer, W. Huber, and F. Petruccione, “Fluctuation effects on wave propagation in a reaction-diffusion process,†Phys. D Nonlinear Phenom., vol. 73, no. 3, pp. 259–273, 1994, DOI: 10.1016/0167-2789(94)90161-9.

Gallouët, T., Herbin, R., Latché, J.C. and Nguyen, T.T., “Playing with Burgers’s equationâ€. In Finite Volumes for Complex Applications VI Problems & Perspectives (pp. 523-531). Springer, Berlin, Heidelberg, 2011.

R. Herbin, J. C. Latché, and T. T. Nguyen, “Consistent segregated staggered schemes with explicit steps for the isentropic and full Euler equations,†ESAIM Math. Model. Numer. Anal., vol. 52, no. 3, pp. 893–944, 2018, DOI: 10.1051/m2an/2017055.

T. Nguyen, B. Pham, T. T. Nguyen, and B. T. Nguyen, “A deep learning approach for solving Poisson’s equations,†Proc. - 2020 12th Int. Conf. Knowl. Syst. Eng. KSE 2020, pp. 213–218, 2020, DOI: 10.1109/KSE50997.2020.9287419.

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations,†no. Part I, pp. 1–22, 2017, [Online]. Available: http://arxiv.org/abs/1711.10566.

A. F. Ihsan, “On the Neural Network Solution of One-Dimensional Wave Problem,†J. RESTI (Rekayasa Sist. dan Teknol. Informasi), vol. 5, no. 6, pp. 1106–1112, 2021, DOI: 10.29207/rest.v5i6.3565.

A. F. Ihsan, “Performance Analysis of the Neural Network Solution of Advection-Diffusion-Reaction Problem.â€

A. M. Tartakovsky, C. O. Marrero, P. Perdikaris, G. D. Tartakovsky, and D. Barajas-Solano, “Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks,†2018, [Online]. Available: http://arxiv.org/abs/1808.03398.

M. Shakeel, “Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion,†Appl. Math., vol. 04, no. 08, pp. 148–160, 2013, DOI: 10.4236/am.2013.48a021.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,†Neural Networks, vol. 2, no. 5, pp. 359–366, 1989, DOI: 10.1016/0893-6080(89)90020-8.

Hakim, D.M., "Optical Music Recognition Pada Citra Notasi Musik Menggunakan Convolutional Neural Network" (Doctoral dissertation, Universitas Komputer Indonesia), 2019.

S. Amini Niaki, E. Haghighat, T. Campbell, A. Poursartip, and R. Vaziri, “Physics-informed neural network for modeling the thermochemical curing process of composite-tool systems during manufacture,†Comput. Methods Appl. Mech. Eng., vol. 384, p. 113959, 2021, DOI: 10.1016/j.cma.2021.113959.

L. Yang and A. Shami, “On hyperparameter optimization of machine learning algorithms: Theory and practice,†Neurocomputing, vol. 415, pp. 295–316, 2020, DOI: 10.1016/j.neucom.2020.07.061.

D. Irfan, R. Rosnelly, M. Wahyuni, J. T. Samudra, and A. Rangga, “Perbandingan Optimasi Sgd, Adadelta, Dan Adam Dalam Klasifikasi Hydrangea Menggunakan Cnn,†J. Sci. Soc. Res., vol. 5, no. 2, p. 244, 2022, doi: 10.54314/jssr.v5i2.789.

T. Yu and H. Zhu, “Hyper-Parameter Optimization: A Review of Algorithms and Applications,†pp. 1–56, 2020, [Online]. Available: http://arxiv.org/abs/2003.05689.

M. El-Hachem, S. W. Mccue, W. Jin, Y. Du, M. J. Simpson, and M. J. Simpson, “Revisiting the Fisher – Kolmogorov – Petrovsky – Piskunov equation to interpret the spreading – extinction dichotomy Subject Areas : Author for correspondence :†2019.

J. H. Lagergren, J. T. Nardini, G. Michael Lavigne, E. M. Rutter, and K. B. Flores, “Learning partial differential equations for biological transport models from noisy Spatio-temporal data,†Proc. R. Soc. A Math. Phys. Eng. Sci., vol. 476, no. 2234, 2020, DOI: 10.1098/rspa.2019.0800.

L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, “DeepXDE: A deep learning library for solving differential equations,†SIAM Rev., vol. 63, no. 1, pp. 208–228, 2021, DOI: 10.1137/19M1274067.

Downloads

Published

2023-01-28

Issue

Vol. 7 No. 1 (2023): Januari 2023

Section

Articles

License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License

Authors who publish with this journal agree to the following terms:

  1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under Creative Commons Attribution 4.0 International License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
  2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
  3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (Refer to The Effect of Open Access).