Is it possible to solve partial differential equations with AI and machine learning? If yes what is a best source for this topic?

Yes, it is possible to solve partial differential equations (PDEs) using AI and machine learning techniques. Various approaches have been developed to leverage the power of neural networks and other machine learning methods for solving PDEs. One popular method is to use deep learning architectures, such as neural networks, to approximate the solutions of PDEs.

Here are a few techniques and references for solving PDEs with AI and machine learning:

Physics-Informed Neural Networks (PINNs): Physics-Informed Neural Networks are designed to incorporate known physics and constraints into the training process. These networks are trained to satisfy both the given data and the governing PDE. PINNs have been used for a wide range of PDEs in different scientific and engineering domains. Reference: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.

Generative Adversarial Networks (GANs): GANs can be employed for generating samples that satisfy the PDE boundary conditions. By training the generator network to produce solutions consistent with the PDE, GANs can be used for solving inverse problems and generating solutions in complex domains. Reference:Yang, H., Cao, W., & Han, D. (2018). Physics-constrained generative adversarial networks for high-dimensional partial differential equations. Journal of Computational Physics, 394, 56-71.

Finite Element Networks (FEN): Finite Element Networks combine the concept of finite element methods with neural networks to solve PDEs. This approach has been applied to problems in structural mechanics, heat transfer, and fluid dynamics. Reference:Zhang, Y., Yan, W., Sturler, E., & Biros, G. (2018). Deep Learning for Solving Helmholtz Equations on Triangular Meshes. arXiv preprint arXiv:1810.04443.

Deep Galerkin Method: The Deep Galerkin Method formulates a loss function based on the residual of the PDE and uses this loss function to train a neural network to approximate the solution. Reference:Sirignano, J., & Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375, 1339-1364.

It's important to note that the choice of method depends on the specific problem, data availability, and computational requirements. Additionally, exploring academic papers and online courses on platforms like arXiv, Google Scholar, and others can provide in-depth insights into the latest developments in using AI and machine learning for solving PDEs.

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