WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS.

Area of Science:

• Applied Mathematics
• Scientific Machine Learning
• Dynamical Systems Theory

Background:

• Sparse Identification of Nonlinear Dynamics (SINDy) is a powerful tool for discovering governing equations from data.
• The weak formulation of SINDy has shown significant improvements in noise robustness.
• Extending SINDy to partial differential equations (PDEs) is crucial for complex system modeling.

Purpose of the Study:

• To extend the Weak SINDy (WSINDy) framework to identify partial differential equations (PDEs).
• To enhance the robustness and accuracy of PDE discovery in the presence of significant noise.
• To develop an efficient and scalable algorithm for PDE system identification.

Main Methods:

• Discretization of a convolutional weak form of PDEs.
• Exploitation of test function separability with the Fast Fourier Transform (FFT) for efficient model identification.
• Development of an *a priori* test function selection algorithm based on noise robustness spectra.
• Introduction of a learning algorithm for sequential-thresholding least-squares (STLS) thresholds.
• Utilization of scale invariance for identifying PDEs from poorly-scaled datasets.

Main Results:

• Machine-precision recovery of PDE model coefficients from noise-free data.
• Robust identification of PDEs even with signal-to-noise ratios approaching one.
• Demonstrated WSINDy's robustness, speed, and accuracy on challenging PDE benchmarks.
• Achieved a worst-case computational complexity of $$ for datasets with N points in D + 1 dimensions.

Conclusions:

• WSINDy provides a robust and efficient framework for discovering PDEs from noisy data.
• The Fourier-based implementation and novel algorithms enable accurate model identification across various data scales and noise levels.
• Publicly available code facilitates broader application of WSINDy for scientific discovery.