Parameter identification for PDEs using sparse interior data and a recurrent neural network.

Area of Science:

• Computational Science
• Applied Mathematics
• Machine Learning

Background:

• Physics-informed neural networks (PINNs) integrate physical laws into neural networks for solving differential equations.
• PINNs face performance degradation with sparse interior data, limiting their application in real-world scenarios.
• Existing methods often require dense data, posing challenges for inverse problems with limited measurements.

Purpose of the Study:

• To develop a novel approach that overcomes the limitations of PINNs in scenarios with sparse interior data.
• To enhance the accuracy and robustness of parameter identification and solution retrieval for partial differential equations.
• To demonstrate the efficacy of the proposed method across various benchmark problems.

Main Methods:

• A hybrid approach combining Gated Recurrent Units (GRUs) with an implicit numerical method is proposed.
• The GRU provides an initial solution approximation, which is then refined using an implicit time-stepping scheme.
• Physical constraints are embedded within the time iteration, and the loss function incorporates errors from both the iteration scheme and sparse data points.

Main Results:

• The proposed method successfully identifies unknown parameters and obtains complete solutions even with sparse interior data.
• Numerical experiments on Burgers' equation, Allen-Cahn equation, and the non-linear Schrödinger equation validate the algorithm's effectiveness.
• The approach demonstrates improved performance compared to standard PINNs when interior data is scarce.

Conclusions:

• The integration of GRUs and implicit numerical methods offers a robust solution for data-scarce inverse problems in scientific computing.
• This hybrid approach significantly enhances the applicability of physics-informed learning in complex physical systems.
• The method provides a promising direction for advancing numerical simulations and parameter estimation in computational science.