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About this paper
Hyeonjung, Jung, Jayant Gupta, Bharat Jayaprakash, Matthew Eagon, Harish Panneer Selvam, Carl Molnar, William Northrop and 1 more
A comprehensive survey and novel two-level taxonomy are presented for deep neural network (DNN) methods applied to solving and discovering differential equations. The work includes an educational tutorial on Physics-Informed Neural Networks (PINNs), demonstrating practical implementation with DeepXDE and highlighting how DNNs can provide faster, more transferable solutions for high-dimensional and inverse problems compared to traditional numerical methods.
Key insights
- Deep neural networks can act as universal function approximators, effectively solving both forward and inverse problems involving differential equations by encoding physical laws into their loss functions.
- Automatic Differentiation (AD) is a foundational enabler for DNN-DE methods, allowing efficient computation of high-order derivatives required to incorporate DE residuals and boundary conditions.
- DNN-DE approaches offer advantages in computational speed, transferability, and generalizability, especially for high-dimensional systems and inverse problems, complementing the strengths of traditional numerical methods.
Summary by alphaXiv.
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