Physics-informed Neural Implicit Flow neural network for parametric PDEs.

Area of Science:

• Computational Science and Engineering
• Applied Mathematics
• Machine Learning for Scientific Computing

Background:

• Physics-informed Neural Networks (PINNs) are widely used for solving partial differential equations (PDEs).
• PINNs exhibit limitations in capturing complex spatio-temporal correlations in parametric PDEs due to network constraints.
• Existing methods struggle with robust uncertainty quantification and efficient inference for parametric PDEs.

Purpose of the Study:

• To introduce a novel Physics-Informed Neural Implicit Flow (PINIF) framework to overcome PINN limitations.
• To enable meshless, low-rank representation of parametric spatio-temporal fields using Neural Implicit Flow (NIF).
• To enhance accuracy, efficiency, and robustness in solving parametric PDEs, especially those with variable coefficients.

Main Methods:

• Development of the Physics-Informed Neural Implicit Flow (PINIF) framework.
• Integration of Neural Implicit Flow (NIF) for expressive, meshless, low-rank field representation.
• Utilization of Polynomial Chaos Expansion (PCE) for uncertainty quantification in noisy data.
• Implementation of a transfer learning approach to accelerate parametric PDE inference.

Main Results:

• PINIF demonstrates superior performance over standard PINNs in solving various PDEs, including Kolmogorov flow and those with variable coefficients.
• The framework achieves higher accuracy and significantly improved computational efficiency.
• PINIF provides a robust solution representation by effectively quantifying uncertainty using PCE.

Conclusions:

• The proposed PINIF framework offers a significant advancement in solving parametric PDEs.
• PINIF effectively addresses the limitations of traditional PINNs in spatio-temporal correlation and efficiency.
• This approach paves the way for more robust and efficient scientific machine learning applications.