Physics-Informed Neural Networks for Solving Stochastic Differential Equations
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Page Numbers:
1312-1332
Digital Object Identifier:
10.58578/mikailalsys.v4i2.10240
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Authors:
Rishav Jha1*, Kameshwar Sahani2, Suresh Kumar Sahani3, Ravi Kumar Raj4, Dilip Kumar Sah5 
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Abstract
Stochastic differential equations (SDEs) are fundamental tools for modeling systems with inherent randomness across finance, physics, biology, and engineering; however, traditional numerical methods, including Monte Carlo simulations and Euler–Maruyama schemes, often face substantial computational challenges, particularly in high-dimensional settings. This study aims to develop and evaluate a comprehensive Physics-Informed Neural Networks (PINNs) framework as an efficient approach for solving SDEs. The proposed methodology embeds physical laws and stochastic dynamics directly into the neural network architecture through a carefully designed loss function that incorporates PDE residuals, initial conditions, and boundary constraints. The framework was evaluated through extensive numerical experiments on benchmark problems, including geometric Brownian motion, Ornstein–Uhlenbeck processes, and multi-dimensional stochastic systems. The findings indicate that PINNs achieve accuracy comparable to traditional numerical methods while providing substantial computational advantages, especially for parametric studies and real-time applications. Across all test cases, relative L2 errors consistently remained below 2%, and computational speedups reached up to 1000 times compared with Monte Carlo methods after network training. The proposed framework also demonstrated strong scalability for higher-dimensional problems, addressing the curse of dimensionality commonly associated with conventional numerical approaches. This study concludes that PINNs offer a promising paradigm for the efficient and accurate solution of stochastic differential equations. Its contribution lies in advancing scientific machine learning for stochastic modeling and opening further opportunities for uncertainty quantification and real-time computational applications.
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References
Beck, C., Becker, S., Grohs, P., Jaafari, N., & Jentzen, A. (2018). Solving stochastic differential equations and Kolmogorov equations by means of deep learning. arXiv. https://doi.org/10.48550/arXiv.1806.00421
Bihlo, A., & Popovych, R. O. (2022). Physics-informed neural networks for the shallow-water equations on the sphere. Journal of Computational Physics, 456, Article 111024. https://doi.org/10.1016/j.jcp.2022.111024
Cai, S., Mao, Z., Wang, Z., Yin, M., & Karniadakis, G. E. (2021). Physics-informed neural networks (PINNs) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12), 1727–1738. https://doi.org/10.1007/s10409-021-01148-1
E, W., Han, J., & Jentzen, A. (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics, 5(4), 349–380. https://doi.org/10.1007/s40304-017-0117-6
Glasserman, P. (2003). Monte Carlo methods in financial engineering. Springer. https://doi.org/10.1007/978-0-387-21617-1
Han, J., Jentzen, A., & E, W. (2018). Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34), 8505–8510. https://doi.org/10.1073/pnas.1718942115
Itô, K. (1944). Stochastic integral. Proceedings of the Imperial Academy, 20(8), 519–524. https://doi.org/10.3792/pia/1195572786
Jin, X., Cai, S., Li, H., & Karniadakis, G. E. (2021). NSFnets (Navier–Stokes flow nets): Physics-informed neural networks for the incompressible Navier–Stokes equations. Journal of Computational Physics, 426, Article 109951. https://doi.org/10.1016/j.jcp.2020.109951
Kloeden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Springer. https://doi.org/10.1007/978-3-662-12616-5
Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5), 987–1000. https://doi.org/10.1109/72.712178
Maruyama, G. (1955). Continuous Markov processes and stochastic equations. Rendiconti del Circolo Matematico di Palermo, 4(1), 48–90. https://doi.org/10.1007/BF02846028
Milstein, G. N. (1974). Approximate integration of stochastic differential equations. Theory of Probability & Its Applications, 19(3), 557–562. https://doi.org/10.1137/1119062
Ming, X., Liu, C., Li, Q., & Liu, C. (2022). Physics-informed neural networks for solving the Boltzmann equation. arXiv. https://doi.org/10.48550/arXiv.2205.10481
Øksendal, B. (2003). Stochastic differential equations: An introduction with applications (6th ed.). Springer. https://doi.org/10.1007/978-3-642-14394-6
Purandare, R., Ratnaparkhi, A., & Bang, A. (2023). Resource optimization using the Taguchi technique for channel allocation. International Journal of Intelligent Systems and Applications in Engineering, 11(3s), 93–99. https://www.ijisae.org/index.php/IJISAE/article/view/2535
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045
Sah, B. K., & Sahani, S. K. (2023). Development and characterization of a new linear exponential distribution for reliability analysis of complex systems. International Journal of Intelligent Systems and Applications in Engineering, 11(11s), 854–865. https://doi.org/10.17762/ijisae.v11i11s.7713
Sahani, S. K. (2019). Runge–Kutta-based dynamic simulation of a multi-degree-of-freedom vibrating system. International Journal of Intelligent Systems and Applications in Engineering, 7(4), 275–284. https://doi.org/10.17762/ijisae.v7i4.7733
Sahani, S. K. (2020). Nonlinear dynamic analysis of multistory structures using Runge–Kutta integration. International Journal of Intelligent Systems and Applications in Engineering, 8(4), 375–385. https://doi.org/10.17762/ijisae.v8i4.7732
Sahani, S. K. (2021). Differential equation on astrophysics: A fundamental approach to understanding cosmic structures and their dynamic evolution. Letters in High Energy Physics, 2021, 1–13. https://doi.org/10.52783/lhep.2021.1468
Sahani, S. K. (2022). A mathematical framework for incorporating neural networks into root-finding algorithms. Journal of Electrical Systems, 18(1), 99–109. https://journal.esrgroups.org/jes/article/view/8947
Sahani, S. K. (2023). Application of numerical methods in structural health monitoring using IoT sensors. Journal of Electrical Systems, 19(1), 194–207. https://doi.org/10.52783/jes.8941
Sahani, S. K. (2023). Neural network surrogates for weather prediction using numerical solutions of the shallow water equations. International Journal of Intelligent Systems and Applications in Engineering, 11(3s), 356–368. https://doi.org/10.17762/ijisae.v11i3s.7686
Sahani, S. K. (2024). AI-enhanced finite element method (FEM) for structural analysis. Journal of Electrical Systems, 20(1), 661–676. https://journal.esrgroups.org/jes/article/view/8946
Sahani, S. K., & Sah, D. K. (2022). A comprehensive study on predicting numerical integration errors using machine learning approaches. Letters in High Energy Physics, 2022, 96–103. https://doi.org/10.52783/lhep.2022.1465
Sahani, S. K., Oruganti, S. K., Kumar, K. S., Sahani, K., Pandey, B. K., & Pandey, D. (2025). Case study on mechanical and operational behavior in steel production: Performance and process behavior in steel manufacturing plant. Reports in Mechanical Engineering, 6(1), 180–197. https://rme-journal.org/index.php/asd/article/view/496
Sahani, S. K., Raj, R. K., Sathishkumar, K., Keshava Murthy, G. N., Manojkumar, S. B., Naveen, K. B., Sah, B. K., Jayanthiladevi, A., Mandal, P., & Sahani, K. (2026). Mechanical process control and statistical process control for reducing butter-oil defects in industrial production. Reports in Mechanical Engineering, 7(1), 169–184. https://rme-journal.org/index.php/asd/article/view/575
Sirignano, J., & Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375, 1339–1364. https://doi.org/10.1016/j.jcp.2018.08.029
Tian, X., Conibear, L., & Steward, J. (2023). A neural-network based MPAS—shallow water model and its 4D-Var data assimilation system. Atmosphere, 14(1), Article 157. https://doi.org/10.3390/atmos14010157
Wang, S., Yu, X., & Perdikaris, P. (2022). When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449, Article 110768. https://doi.org/10.1016/j.jcp.2021.110768
Yang, L., Zhang, D., & Karniadakis, G. E. (2020). Physics-informed generative adversarial networks for stochastic differential equations. SIAM Journal on Scientific Computing, 42(1), A292–A317. https://doi.org/10.1137/18M1225409
Zhang, D., Guo, L., & Karniadakis, G. E. (2020). Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks. SIAM Journal on Scientific Computing, 42(2), A639–A665. https://doi.org/10.1137/19M1260141