qed137 / piml

Physics Informed Machine Leanring

PDE Solver with Physics-Informed Neural Networks (PINNs)

Physics-Informed Neural Networks (PINNs) are a powerful scientific machine-learning technique used to solve problems involving Partial Differential Equations (PDEs). Unlike traditional numerical methods, PINNs approximate PDE solutions by training a neural network to minimize a loss function. This loss function includes terms reflecting the initial and boundary conditions along the space-time domain’s boundary and the PDE residual at selected points in the domain, known as collocation points.

Key Concept

• Residual Network: PINNs incorporate a residual network that encodes the governing physics equations, allowing them to learn the underlying physics of the problem.

• Unsupervised Training: PINNs operate as an unsupervised strategy, eliminating the need for labeled data or prior simulations.

• Mesh-Free Technique: PINNs transform the problem of directly solving PDEs into a loss function optimization problem, making them a mesh-free technique.

• Physics-Driven Learning: By integrating the mathematical model into the network and reinforcing the loss function with a residual term from the governing equation, PINNs leverage structured prior knowledge about the solution.

Benefits

• Incorporation of Physics: PINNs take into account the underlying physics of the problem, leading to physically consistent solutions.
• Flexibility: PINNs can handle complex geometries and boundary conditions without the need for extensive mesh generation.
• Data Efficiency: PINNs require minimal labeled data, making them suitable for problems with limited or expensive data availability.

Application

PINNs have found applications in various fields, including fluid dynamics, solid mechanics, heat transfer, and electromagnetics. They offer a promising approach for solving real-world engineering and scientific problems efficiently and accurately.

References

• Owhadi, H. (2022). Optimal learning machines. Constructive Approximation, 1-31.
• Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next