Self-test loss functions for learning weak-form operators and gradient flows
This work addresses a specific problem in computational mathematics for researchers modeling PDEs and gradient flows, offering an incremental improvement by optimizing test function selection in loss construction.
The paper tackles the challenge of constructing loss functions for data-driven modeling of weak-form operators in PDEs and gradient flows by introducing self-test loss functions that use test functions dependent on unknown parameters, resulting in a quadratic loss that enables theoretical analysis and efficient algorithms, with numerical experiments showing robustness against noisy and discrete data.
The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge by introducing self-test loss functions, which employ test functions that depend on the unknown parameters, specifically for cases where the operator depends linearly on the unknowns. The proposed self-test loss function conserves energy for gradient flows and coincides with the expected log-likelihood ratio for stochastic differential equations. Importantly, it is quadratic, facilitating theoretical analysis of identifiability and well-posedness of the inverse problem, while also leading to efficient parametric or nonparametric regression algorithms. It is computationally simple, requiring only low-order derivatives or even being entirely derivative-free, and numerical experiments demonstrate its robustness against noisy and discrete data.