Scientific Machine Learning: Physics-Informed Neural Networks with Craig Gin

Step-by-Step Tutorial: Applying Deep Learning Models for Global Coordinate Transformations to Linearize PDEs

1. Introduction to the Goal:

   • The goal of this work is to create a method that can be used to discover coordinate transformations that linearize Partial Differential Equations (PDEs).
   • By applying a coordinate transformation, the aim is to obtain a new function 'v' that is governed by a linear PDE.

2. Motivation:

   • Linearizing PDEs allows us to leverage established techniques for linear PDEs such as estimation, control, and uncertainty quantification.
   • Exact solutions can be obtained for linear PDEs, making them easier to work with compared to nonlinear PDEs.

3. Understanding Linearization:

   • By using an invertible coordinate transformation, it becomes possible to evolve the solution to a nonlinear PDE through a linear space.

4. Identifying Transformations:

   • Discrete time dynamical systems can be used to identify transformations that linearize PDEs.
   • Examples like the Kohlhoff transformation and the inverse scattering transform are known to linearize certain classes of PDEs.

5. Methodology:

   • The study started with the Kohlhoff transformation as an example to develop the methodology.
   • By applying the Kohlhoff transformation, the resulting function 'v' satisfies the heat equation, for which exact solutions are known.

6. Deep Learning Approach:

   • Utilizing deep learning, a neural network is trained to learn the transformation and linear dynamics architecture.
   • An autoencoder architecture is used, consisting of an encoder, linearizing transformation, and decoder for evolving the system forward in time.

7. Network Architecture:

   • The architecture includes an outer encoder for linearizing transformation, an inner encoder for dimensionality reduction, and a decoder for inversion.
   • Skip connections are incorporated for residual learning, inspired by physics principles.

8. Training Process:

   • The loss function for training the network includes unsupervised, supervised, and linearity loss functions to ensure accurate learning.
   • Training data includes various initial conditions to generalize the network's ability to handle different scenarios.

9. Results and Validation:

   • The network's performance is evaluated by comparing exact solutions with network predictions for different initial conditions.
   • Results show good agreement even with initial conditions not present in the training data, demonstrating the network's generalization capability.

10. Application to Different PDEs:

    • The methodology is extended to linearize the KS equation using a convolutional neural network architecture.
    • The results demonstrate the network's effectiveness in reproducing the dynamics of the KS equation.

11. Conclusion:

    • The developed deep autoencoder architecture offers a data-driven approach to linearize PDEs, making it interpretable and generalizable to various scenarios.
    • Key factors for successful results include diverse training data, appropriate neural network architecture, and well-defined loss functions.

12. Further Exploration:

    • For more in-depth details, refer to the paper "Deep Learning Models for Global Coordinate Transformations" in the European Journal of Applied Mathematics.

By following these steps, you can understand and apply deep learning models for global coordinate transformations to linearize PDEs effectively.