Keywords
dislocation source
simulation
numerical methods
Rosenbrock method
physics-informed neural networks (PINN)
SciPy
How to Cite
Abstract
we present a comparative analysis of numerical methods (4th-order Rosenbrock, Radau, BDF, and LSODA from the SciPy library) and the Physics-Informed Neural Networks (PINN) approach for solving a mathematical model of high-frequency geoacoustic emission from a single dislocation source using Python. Parallel implementation of the Rosenbrock method on eight processors significantly improved its performance. We compared the methods in terms of accuracy, computational cost, and stability. The results show that the PINN approach, with an architecture specifically designed for mathematical physics problems, achieves accuracy comparable to the manually implemented Rosenbrock method. In addition, PINNs offer advantages such as simplified problem parameterization and a global approximation of the solution, ensuring smoothness and avoiding the numerical dispersion typical of grid-based methods.
References
Sadovskii V. M., Sadovskaya O. V., Detkov V. A. Analysis of Elastic Waves Generated in Frozen Grounds by Means of the Electromagnetic Pulse Source “Yenisei”. IOP Conference Series: Earth and Environmental Science. 2018;193(1):12058.
Vershinin A., Ampilov Yu., Levin V., Petrovsky K. Full Waveform Modeling in Seismic Exploration Based on a Digital Geological Model Using Spectral Element Method on GPU. 16th World Congress on Computational Mechanics and 4th Pan American Congress on Computational Mechanics. 2024. DOI: 10.23967/c.wccm.2024.056.
Lukovenkova O. O., Solodchuk A. A. Digital Signal Processing Methods for Geoacoustic Emission. 23rd International Conference on Digital Signal Processing and its Applications (DSPA), Moscow, Russian Federation. 2021:1–5. DOI: 10.1109/DSPA51283.2021.9535762.
Dobrynin E., Maksymov M., Boltenkov V. Development of a Method for Determining the Wear of Artillery Barrels by Acoustic Fields of Shots. Eastern-European Journal of Enterprise Technologies. 2020;3:6–18. DOI: 10.15587/1729-4061.2020.206114.
Prokopovich S. V., Uzdin A. M., Ivanova T. V. Setting Seismic Input Characteristics Required for Designing. Magazine of Civil Engineering. 2021;2(102):10209.
Lukovenkova O. O., Marapulets Y. V., Solodchuk A. A. Adaptive Approach to Time-Frequency Analysis of AE Signals of Rocks. Sensors. 2022;22:9798. DOI: 10.3390/s22249798.
Gapeev M. I., Senkevich Y. I., Lukovenkova O. O. Estimation of Probability Distributions of Geoacoustic Signal Characteristics. Journal of Physics: Conference Series. IOP Publishing. 2021;2096(1):012018.
Sergienko D., Parovik R. Berlage Oscillator as a Mathematical Model of High-Frequency Geoacoustic Emission with One Dislocation Source. Acoustics. 2025;7(4):65. DOI: 10.3390/acoustics7040065.
Мингазова Д. Ф., Паровик Р. И. Некоторые аспекты качественного анализа модели высокочастотной геоакустической эмиссии. Вестник КРАУНЦ. Физ.-мат. науки. 2023;42(1):191–206. DOI: 10.26117/2079-6641-2023-42-1-191-206.
Гапеев М. И., Солодчук А. А., Паровик Р. И. Связанные осцилляторы как модель высокочастотной геоакустической эмиссии. Вестник КРАУНЦ. Физ.-мат. науки. 2022;40(3):88–100. DOI: 10.26117/2079-6641-2022-40-3-88-100.
Sergienko D. F., Parovik R. Investigation of a System of Two Coupled Linear Oscillators with Nonconstant Coefficients. Journal of Mathematical Sciences. 2025. DOI: 10.1007/s10958-025-07912-z.
Сергиенко Д. Ф., Паровик Р. И. Программный комплекс QAMODEL: компьютерное моделирование высокочастотной геоакустической эмиссии. Программные продукты и системы. 2025;38(2):261–268. DOI: 10.15827/0236-235X.150.261-268.
Сергиенко Д. Ф., Паровик Р. И. Исследование эффективности численных методов для решения математической модели высокочастотной геоакустической эмиссии. Вестник КРАУНЦ. Физ.-мат. науки. 2025;50(1):169–183. DOI: 10.26117/2079-6641-2025-50-1-169-183.
Shampine L. F., Reichelt M. W. The MATLAB ODE Suite. SIAM J. Sci. Stat. Comput. 1997;18(1):1–22. DOI: 10.1137/s1064827594276424.
Hindmarsh A. C. ODEPACK, a Systematized Collection of ODE Solvers. Scientific Computing. 1983;55–64.
Petzold L. Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. SIAM J. Sci. Stat. Comput. 1983;4(1):136–148. DOI: 10.1137/0904010.
Hairer E., Wanner G. Stiff Differential Equations Solved by Radau Methods. Journal of Computational and Applied Mathematics. 1999;111(1–2):93–111. DOI: 10.1016/S03770427(99)00134-X.
Hairer E., Wanner G. Solving Ordinary Differential Equations II. Stiff and Differential – Algebraic Problems. Berlin: Springer-Verlag; 1991.
Ponalagusamy R., Ponnammal K. A Parallel Fourth Order Rosenbrock Method: Construction, Analysis and Numerical Comparison. Int. J. Appl. Comput. Math. 2015:1(1);45–68. DOI: 10.1007/s40819-0140002-x.
Tristanov A., Lukovenkova O., Marapulets Yu., Kim A. Improvement of Methods for Sparse Model Identification of Pulsed Geophysical Signals. Conf. Proc. of SPA 2019. Poznan. 2019:256–260. DOI: 10.23919/SPA.2019.8936817.