Deep fuzzy physics-informed neural networks for forward and inverse PDE problems

Area of Science:

• Computational physics
• Artificial intelligence
• Data science

Background:

• Physics-Informed Neural Networks (PINNs) offer a grid-independent method for solving partial differential equations (PDEs) by integrating physical laws and data.
• Traditional PINNs assume data reliability, which is often not the case with data from commercial simulations, leading to inaccuracies.

Purpose of the Study:

• To develop a robust method that accounts for data uncertainty in Physics-Informed Neural Networks (PINNs).
• To improve the accuracy of solving forward and inverse PDE problems when dealing with ambiguous or inaccurate data.

Main Methods:

• Introduced Deep Fuzzy Physics-Informed Neural Networks (FPINNs) to capture data uncertainty.
• Implemented fuzzy representation using fuzzy membership function and fuzzy rule layers.
• Integrated fuzzy representation with deep neural network representations.
• Utilized a loss function combining physical equation residuals and data errors.

Main Results:

• FPINNs effectively explore and represent data uncertainty.
• The proposed method demonstrates superior performance in solving both forward and inverse PDE problems compared to existing methods.
• Experiments conducted on four widely used datasets validate the effectiveness of FPINNs.

Conclusions:

• FPINNs provide a more reliable approach to solving PDEs when data uncertainty is present.
• The integration of fuzzy logic enhances the capability of PINNs to handle real-world, imperfect data.
• This work offers a promising direction for applying AI in scientific simulations with noisy data.