Self-test loss functions for learning weak-form operators and gradient flows

Abstract: 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.

• The paper proposes self-test loss functions that automatically select parameter-dependent test functions, enhancing the identifiability of weak-form operators.
• It applies the method to high-dimensional gradient flows and diffusion models, demonstrating robust performance in the presence of noisy, discrete data.
• The results show significant improvements over traditional strong-form approaches, offering a practical tool for modeling complex systems in science and engineering.

Overview of Self-Test Loss Functions in Learning Weak-Form Operators and Gradient Flows

The paper introduces a novel approach to the construction of loss functions aimed at improving the process of learning weak-form operators in partial differential equations (PDEs) and gradient flows. The primary issue it addresses is the challenge of selecting appropriate test functions required by weak-form equations, specifically when these operators depend linearly on the function-valued parameter to be estimated. The authors propose the use of self-test loss functions that automatically employ test functions depending on the unknown parameters and the data, providing a computationally efficient and robust solution in the presence of noisy, discrete data.

Key Contributions

1. Self-Test Loss Functions: The core innovation is the self-test loss function, which automatically selects test functions that are parameter-dependent. These functions are designed to facilitate the theoretical analysis of the identifiability and well-posedness of the inverse problem. The self-test loss functions retain a quadratic form allowing for efficient regression algorithms.
2. Applications to Weak-Form Operators: The paper applies the self-test loss function framework to various weak-form operators, including high-dimensional gradient flows and diffusion models. This framework is particularly useful for problems where low-order derivatives or derivative-free operators are preferable due to noisy and discrete data inputs.
3. Robustness to Data Challenges: The proposed method demonstrates robustness to noise and data discretization due to its reliance on weak formulations, which do not require accurate high-order derivative approximations. Consequently, self-test loss functions can process real-world data more reliably than approaches based on strong-form equations.

Examples and Numerical Results

The paper presents several examples, including Wasserstein gradient flows, weak-form elliptic operators, and interacting particle systems, to illustrate the application and efficiency of self-test loss functions. Through these examples, the authors demonstrate that their method can seamlessly handle noisy inputs and produce reliable parameter estimates, showing significant improvements over traditional approaches that depend on strong-form equations:

• Wasserstein Gradient Flows: The self-test loss function conserves the system’s energy by matching the energy dissipation along the data flow, thereby maintaining system fidelity.
• Diffusion Models: For elliptic operators, the proposed method shows clear advantages in estimating diffusion rates by reducing sensitivity to noise and discretization errors.
• Interacting Particle Systems: In high-dimensional settings, self-test loss functions allow the learning of interaction potentials using ensemble data without relying on individual particle trajectory information.

Implications and Future Directions

The work has significant theoretical and practical implications. Theoretically, it offers insights into constructing quadratic loss functions, promoting further research into the identifiability and well-posedness of inverse problems in various complex systems. Practically, the approach provides a robust tool for scientific fields that require modeling from noisy and incomplete data, such as physics, biology, and earth sciences.

Future research could expand upon this framework by exploring its applicability in other types of PDEs or more complex systems with additional constraints. Additionally, integrating these methods with neural network architectures could further improve their applicability in data-driven modeling tasks. The method's robustness makes it particularly suitable for high-dimensional problems that are prevalent in modern AI applications.