Unleashing the Power of Automatic Differentiation for Physics-informed Operator Learning

Are you ready to revolutionize your physics-informed operator learning? In this article, we delve into the groundbreaking technique called Zero Coordinate Shift (ZCS) for automatic differentiation. By introducing ZCS, we simplify derivative computations and unlock a new level of training efficiency. Join us as we explore the power of ZCS in physics-informed machine learning and its potential to transform the way we solve partial differential equations (PDEs) without data.

Introducing Zero Coordinate Shift (ZCS)

Discover the game-changing technique of Zero Coordinate Shift (ZCS) for automatic differentiation.

Automatic differentiation is a critical step in physics-informed machine learning, enabling us to compute high-order derivatives of network output with respect to coordinates. In this sub heading, we introduce you to an innovative and lightweight algorithm called Zero Coordinate Shift (ZCS). This technique simplifies the process of computing derivatives by introducing scalar-valued leaf variables for each spatial or temporal dimension. By doing so, ZCS significantly enhances training efficiency and memory consumption, making it a game-changer in physics-informed operator learning.

With ZCS, we shift from the traditional approach of using 'many-roots-many-leaves' to the more efficient 'one-root-many-leaves' paradigm. This optimization allows us to train physics-informed DeepONets for solving partial differential equations (PDEs) without data more effectively. ZCS is easy to implement using current deep learning libraries, and it brings down GPU memory consumption and training time by an order of magnitude. Its benefits increase as the problem scale, such as the number of functions, points, and the order of PDE, increases.

Enhancing Physics-informed Neural Networks with Operator Learning

Explore the powerful framework of Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN) for solving PDEs with sharp solutions.

Physics-informed Neural Networks (PINNs) have shown great promise in solving forward and inverse problems of partial differential equations (PDEs). However, when it comes to problems with sharp solutions, PINNs face significant challenges. In this sub heading, we introduce you to a novel framework called Operator Learning Enhanced Physics-informed Neural Networks (OL-PINN), designed specifically to tackle PDEs characterized by sharp solutions.

The OL-PINN framework combines the power of DeepONet, a solution operator learning approach, with PINN to address the challenges posed by sharp solution problems. By pre-training DeepONet on a set of smooth problems related to the PDEs, we enhance the capability of PINN to handle sharp solution problems. The integration of these two techniques results in improved accuracy, robust training, and strong generalization with minimal residual points. OL-PINN also inherits the advantage of PINN in solving inverse problems, making it suitable for solving ill-posed and complex inverse problems.