A Physics-Informed Neural Network Approach for Solving the Monge-Ampère Partial Differential Equation

This paper presents a comparative study of classical numerical and machine learning techniques for solving the nonlinear Monge–Ampère Equation (MAE). A finite difference method (FDM) implemented in MATLAB is first validated using a quadratic exact solution on a square domain. Two neural network approaches are then investigated: a supervised neural network trained on analytical solution data, and a Physics-Informed Neural Network (PINN) implemented in Python using a Streamlit interface. The PINN solves the MAE through an unsupervised learning strategy based on minimizing PDE residuals and boundary losses. The effects of tanh and sigmoid activation functions on convergence and accuracy are also examined. Numerical results show that while FDM provides a reliable benchmark, PINNs offer an effective mesh-free alternative. Compared with classical methods, PINNs trade algorithmic rigor for adaptability and dimensional scalability, offering a modern, mesh-free perspective on PDE approximation. The ReLU/Tanh networks employed here is capable of representing both convex and concave solutions, however the gradient descent procedure can get trapped in the concave branch with the hyperbolic tangent (tanh) activation function, as we discovered in our case. Overall, our study demonstrates the potential of PINNs for solving fully nonlinear partial differential equations.

Physics-Informed Neural Networks, Monge-Ampère PDE, Scientific Machine Learning, Neural PDE Solvers.

1. [1] Pogorelov, A., (1964); Monge-Ampere Equations of Elliptic Type, Published by Noordhoff, Groningen, The Netherlands. [Google Scholar] [Crossref]

2. [2] Alexandrov, A. and Kutateladze, S., (2006); Selected Works Part II: Intrinsic Geometry of Convex Surfaces. [Google Scholar] [Crossref]

3. [3] Caffarelli, L., Nirenberg, L. and Spruck, J., (1984); The Dirichlet problem for nonlinear second-order elliptic equations I: Monge-Ampère equations, Communications on Pure and Applied Mathematics Volume 37,Issue 3, pp. 339–402. [Google Scholar] [Crossref]

4. [4] Kutateladze S. S. (2006) Intrinsic Geometry of Convex Surfaces. ISBN: 978-0-415-29802-5. [Google Scholar] [Crossref]

5. [5] Ollicker, V., and Prussner, L., (1989); On the numerical solution of the equation and its discretization. I Numerical. Mathematics., 54, pp. 271-293 [Google Scholar] [Crossref]

6. [6] Brenner and Neilan (2012). Finite element approximations of the three dimensional Monge-Ampère equation. ESAIM: Mathematical Modelling and Numerical Analysis. 46(5), 979-1001. [Google Scholar] [Crossref]

7. [7] Feng and Neilan (2009). Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. Journal of Scientific Computing. 38(1), 74-98. [Google Scholar] [Crossref]

8. [8] Froese and Oberman (2011). Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher. SIAM Journal on Numerical Analysis. 49(4), 1692-1714. [Google Scholar] [Crossref]

9. [9] Brittany D. F. & Oberman A. M. (2011). Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher, SIAM Journal on Numerical Analysis. Vol. 49. [Google Scholar] [Crossref]

10. [10] Lagaris I. E., Likas A. C., and Fotiadis D.I., (1997). Artificial Neural Network for Solving Ordinary and Partial Differential Equations, arXiv:physics/9705023v1, Greece. [Google Scholar] [Crossref]

11. [11] Han J., Jentzen A., Weinan E,,(2018). Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences. 115(34). 8505–8510. [Google Scholar] [Crossref]

12. [12] 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. Vol. 378. 686–707. [Google Scholar] [Crossref]

13. [13] Hacking, R., Kusch, L., Mitra, K., Anthonissen, M., & IJzerman, W. (2024). A neural network approach for solving the Monge-Ampère equation with transport boundary condition. DOI:10.48550/arXiv.2410.19496. [Google Scholar] [Crossref]

14. [14] Chen H. and Wang T. (2024). Machine Learning Algorithm for the Monge-Ampère Equation with Transport Boundary Conditions. East Asian Journal on Applied Mathematics, 14(4), 788-819. https://doi.org/10.4208/eajam.2023-084.050923 [Google Scholar] [Crossref]

15. [15] Nielsen, Michael A. (2023). Neural Networks and Deep Learning: A Textbook. Springer. DOI: 10.1007/978-3-031-43780-5 [Google Scholar] [Crossref]

16. [16] Goodfellow I., Bengio, Y., & Courville A. (2020). Deep Learning. MIT Press. DOI: 10.7551/mitpress/11039.001.0001 [Google Scholar] [Crossref]

17. [17] Millidge B., Yuhang S., Tommaso S., Thomas L., Rafal B. (2023). Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning. International Conference on Learning Representations (ICLR). [Google Scholar] [Crossref]

18. [18] Hinton, Geoffrey E., & Vinyals, Oriol (2021). Deep Learning with Backpropagation: Progress and Challenges. IEEE Transactions on Neural Networks and Learning Systems, 32(5), 1011-1026. DOI: 10.1109/TNNLS.2020.2998540 [Google Scholar] [Crossref]

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